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9 defined, 445 of a prism, 446, 459 of a pyramid, 445, 447–448 of a sphere, 546–547 and volume, relationship of, 599–602 surfaces, area and, 445 surveying land, 36, 94, 453, 469, 649 Swenson, Sue, 422 Syllogism, Law of, 611, 612–613 Symbolic Logic, Part I (Dodgson), 612 symbols angle, 38 approximately equal to, 40 arc, 68 conditional statement, 552 congruence, 31, 40, 59 equals, use of, 31 glide reflection, 398 image point label, 358 line, 28 line segment, 31 of logic, 551–553, 611, 612, 613 measure, 31, 40 negation (logic), 552 parallel, 48 perpendicular, 48 pi, 331 plane, 28 point, 28 ray, 32 same measure, 31 similarity, 565 slant height, 447 therefore, 552 triangle, 54 symmetric property of congruence, 671 symmetric property of equality, 670 symmetry, 3–4, 361–362 bilateral, 3 glide-reflectional, 376 in an isosceles trapezoid, 269 in an isosceles triangle, 242 in a kite, 266 line of, 3, 361 of polygons, 362 reflectional, 3, 4, 361 rotational, 3–4, 361 Symmetry Drawing E25 (Escher), 393 Symmetry Drawing E99 (Escher), 395 Symmetry Drawing E103 (Escher), 197 Symmetry Drawing E105 (Escher), 389 Symmetry Drawing E108 (Escher), 399 systems of linear equations, 285–286, 401–402 Szent-Györgyi, Albert, 120 Szyk, Arthur, 675 T Taj Mahal, 6 Take Another Look angle measures, 253 area, 459 circumference, 355 congruence shortcuts, 253 cyclic quadrilaterals, 355 dilation, 617 polygon conjectures, 303–304 Pythagorean Theorem, 501–502 quadrilateral conjectures, 253 similarity, 617–618 tangents, 355 triangle conjectures, 253, 303 trigonometry, 665–666 volume, 557–558 Talmud, 417 Tan, Amy, 67 tangent(s) defined, 69 external, 315 internal, 315 point of, 69 proofs involving, 699–700 properties of, 313–315 and Pythagorean Theorem, 492 radius and, 313–314 segments, 314 as term, use of, 69 tangent circles, 315 Tangent Conjecture, 313, 314 tangent ratio (tan), 620, 621–622, 624 tangent segments, 314 Tangent Segments Conjecture, 314 Tangent Theorem, 699–700 tangram puzzle, 484 Taoism, 316 tatami, 386 technology applications, 52, 112, 131, 235, INDEX 761 263, 271, 283, 314, 317, 323, 339, 351, 423, 435, 498, 583, 607, 628, 630 exercises, 123, 145, 150, 160, 170, 180, 181, 186, 201, 259, 284, 316, 318, 323, 345, 382–383, 397, 427, 429, 436, 550, 585, 610, 645, 648, 650, 691, 710, 711 telecommunications, 112, 435, 498 temari balls, 720 temperature conversion, 210 term, nth, 106–108 Tessellating Quadrilaterals Conjecture, 385 Tessellating Triangles Conjecture, 384 tessellation (tiling) creation of, 21, 388, 389–391, 393–396 defined, 20 dual of, 382 glide reflection, 398–399 monohedral, 379–380, 384–385 nonperiodic, 388 with nonregular polygons, 384–385 regular, 380 rotation, 393–395 semiregular, 380–381 translation, 389–390 vertex arrangement, 380, 381 test problems, writing, 254 tetrahedron, 505 Thales of Miletus, 233, 583, 668 theodolite, 628 Theorem of Pythagoras. See Pythagorean Theorem theorem(s), 668 defined, 463 logical family tree of, 682–684 proving. See proof(s) See also specific theorems listed by name therefore, symbol for, 552 Thiebaud, Wayne, 514 thinking backward, 294 Third Angle Conjecture, 200–201 Third Angle Theorem, 682–684 30°-60°-90° triangle, 476–477 30°-60°-90° Triangle Conjecture, 476–477 Thomas, Calista, 647 Thompson, Rewi, 709 Three Midsegments Conjecture, 273 3-uniform tiling, 381 Three Worlds (Escher), 27 Tibet, 576 tiling. See tessellation transformation(s), 358 nonrigid, 358, 566–567, 578–580 762 INDEX rigid. See isometry transitive property of congruence, 671 transitive property of equality, 670 transitive property of similarity, 706 translation, 358–359 and composition of isometries, 373–374, 376 defined, 358 direction of, 358 distance of, 358 tessellations by, 389–390 as type of isometry, 358 vector, 358 transversal line, 126 trapezium, 268 trapezoid(s) arch design and, 271 area of, 417–418 base angles of, 267 bases of, 267 defined, 62 diagonals of, 269, 699 height of, 417 isosceles, 268–269 linkages of, 283 midsegments of, 273, 274–275 proofs involving, 699 properties of, 268–269 Trapezoid Area Conjecture, 418 Trapezoid Consecutive Angles Conjecture, 268 Trapezoid Midsegment Conjecture, 275 tree diagrams, 78 triangle(s) acute, 60, 636, 641–642 adjacent interior angles of, 215–216 altitudes of, 154, 177, 401, 586 angle bisectors of, 176–179, 586, 587–588 area of, 411, 417, 454, 634–635 centroid of, 183–185, 189–190, 402 circumcenter. See circumcenter circumscribed, 71, 179 congruence of, 168–169, 219–222, 225–227, 230–231 constructing, 143, 168–169, 205 definitions of, 60–61 determining parts of, 168 drawing, 134 elliptic geometry and, 720 equiangular. See equilateral triangle(s) equilateral. See equilateral triangle(s) exterior angles of, 215–216 height of, 154, 634–635 incenter, 177–179 included angle, 219 included side, 219 inequalities, 213–216 inscribed, 71, 179 interior angles of, 215–216 isosceles. See isosceles triangle(s) medians of, 149, 183–185, 402, 586–587 midsegments of, 149, 273–274, 275 naming of, 54 obtuse, 60, 641–642 orthocenter. See orthocenter parallel lines and proportions of obtuse, 641–642 perpendicular bisectors of, 149, 176–178 points of concurrency of. See point(s) of concurrency proofs involving, 681–684, 686–688, 706–709 relationships of, 78 remote interior angles of, 215–216 right. See right triangle(s) scalene, 384 similarity of, 200–201, 572–574, 586–588 sum of angles of, 199–200 symbol for, 54 tessellations with, 379–381, 384, 394–395 vertex angle, 62, 242–243 Triangle Area Conjecture, 417 Triangle Exterior Angle Conjecture, 216 Triangle Inequality Conjecture, 214 Triangle Midsegment Conjecture, 274, 275 Triangle Sum Conjecture, 198–201 Triangle Sum Theorem, 681–682 triangular numbers, 115 triangular prism, 506 triangular pyramid, 506 triangulation, 229 trigonometry, 620 adjacent side, 620 cosine (cos), 621–622 graphs of functions, 654 inverse cosine (cos1), 624 inverse sine (sin1), 624 inverse tangent (tan1), 624 Law of Cosines, 641–643, 647 Law of Sines, 634–637, 647 opposite side, 620 and periodic phenomena, 654 problem solving with, 627, 647 ratios, 620–624 sine (sin), 621–622 tables and calculators for, 622–624, 654 tangent (tan), 620, 621–622, 624 unit circle and, 651–654 vectors and, 647 truncated pyramid, 685 Tsiga series (Vasarely), 3 Turkey, 379, 668 Twain, Mark, 59, 104 two-column proof, 655, 687–688 two-point perspective, 174–175 2-uniform tiling, 381 Tyson, Cicely, 157 U undecagon, 54 unit circle, 651–654 units area and, 413 nautical mile, 351 not stated, 31 volume and, 514 Using Your Algebra Skills Coordinate Proof, 712–717 Finding the Circumcenter, 329–330 Finding the Orthocenter and Centroid, 401–403 Midpoint, 36–37 Proportion and Reasoning, 560–561 Radical Expressions, 473–474 Slope, 133–134 Slopes of Parallel and Perpendicular Lines, 165–166 Solving Systems of Linear Equations, 285–286 Writing Linear Equations, 210–211 Uzbekistan, 60 V VA Theorem (Vertical Angles Theorem), 679–680 valid argument, 100, 102, 551 valid reasoning. See logic vanishing point(s), 172, 173, 174 Vasarely,Victor, 3, 13 vector(s) defined, 280 diagrams with, 280–281 resultant, 281 translation, 358 trigonometry with, 647 vector sum, 281 velocity and speed calculations, 134, 293, 302, 337, 338, 340, 344, 345, 351, 392, 483, 497, 660, 661 velocity vectors, 280–281 Venn diagram, 78 Venters, Diane, 56 Verblifa tin (Escher), 503 Verne, Jules, 337 vertex (vertices) of a cone, 508 consecutive, 54 defined, 38 naming angles by, 38 of a polygon, 54 of a polyhedron, 505 of a pyramid, 506 tessellation arrangement, 380, 381 vertex angle(s) bisector of, 242–243 of an isosceles triangle, 62, 242–243 of a kite, 266 Vertex Angle Bisector Conjecture, 242 vertex arrangement, 380, 381 vertical angles, 50, 121, 679–680 Vertical Angles Conjecture, 121–122, 129 Vertical Angles Theorem (VA Theorem), 679–680 vintas, 144 Vichy-Chamrod, Marie de, 142 Vietnam Veterans Memorial Wall, 130 volume of a cone, 522–524 of a cylinder, 515–517 defined, 514 displacement and density and, 535–536 of a hemisphere, 542–543 maximizing, 538 of a prism, 515–517 problems in, 531 proportion and, 593–594 of a pyramid, 522–524 of a sphere, 542–543 and surface area, relationship of, 599–602 units used to measure, 514 Vries, Jan Vredeman de, 172 W Walker, Mary Willis, 546 Wall Drawing #652 (LeWitt), 61 Warhol, Andy, 507 water and buoyancy, 537 and volume, 520, 537 Water Series (Greve), 306 Waterfall (Escher), 461 Weyl, Hermann, 358 Wick, Walter, 66 wigwams, 548 Wilcox, Ella Wheeler, 647 Wilde, Oscar, 462 Wiles, Andrew, 74 Williams, William T., 79 woodworking, 34 work, 484 World Book Encyclopedia, 26 Wright, Frank Lloyd, 9 Wright, Steven, 514 writing test problems, 254 Y y-intercept, 210 yin-and-yang symbol, 316 Z zero product property of equality, 670 Zhoubi Suanjing, 502 zillij, 22 zoology and animal care, 15, 435, 536, 576, 694 I n d e x INDEX 763 Photo Credits Abbreviations: top (T), center (C), bottom (B), left (L), right (R). Cover Background image: Doug Wilson/Corbis; Construction image: Sonda Dawes/The Image Works; All other images: Ken Karp Photography. Front Matter v (T): Ken Karp Photography; v (C): Cheryl Fenton; v (B): Cheryl Fenton; vi (T): Ken Karp Photography; vii (T): Ken Karp Photography; vii (C): Courtesy, St. John’s Episcopal Church; vii (B): Hillary Turner; viii (T): Ken Karp Photography; viii (B): Corbis/Stockmarket; ix (T): Cheryl Fenton; ix (B): Cheryl Fenton; x (T): Ken Karp Photography; xi (T): Ken Karp Photography; xii (T): Ken Karp Photography; xii (B): Perry Collection/Photo by Cheryl Fenton; xiii: Cheryl Fenton. Chapter 0 1: Print Gallery, M. C. Escher, 1956/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 2 (B): ©1993 Metropolitan Museum of Art, Bequest of Edward C. Moore, 1891 (91.1.)2064 2 (C): Cheryl Fenton; 2 (TL): NASA; 3 (BL): Christie’s Images; 3 (T): Tsiga I,II,III (1991),Victor Vasarely, Courtesy of the artist.; 4 (CR): Hillary Turner; 4 (TL): Cheryl Fenton; 4 (TR): Cheryl Fenton; 5
(B): ©Andy Goldsworthy, Courtesy of the artist and Galerie Lelong; 5 (C): Cheryl Fenton; 5 (CL): Cheryl Fenton; 5 (TC): Cheryl Fenton; 5 (TL): Cheryl Fenton; 6: Corbis; 7 (BL): Dave Bartruff/Stock Boston; 7 (BR): Robert Frerck/Woodfin Camp & Associates; 7 (CL): Rex Butcher/Bruce Coleman Inc.; 7 (CR): Randy Juster; 9: Schumacher & Co./Frank Lloyd Wright Foundation; 10 (R): Sean Sprague/Stock Boston; 10 (TC): Christie’s Images/Corbis; 12: W. Metzen/Bruce Coleman Inc.; 13 (L): Hesitate, Bridget Riley/Tate Gallery, London/Art Resource, NY; 13 (R): Harlequin by Victor Vasarely, Courtesy of the artist.; 15 (C): National Tourist Office of Spain; 15 (T): Tim Davis/Photo Researchers; 16: Cheryl Fenton; 17: Snakes, M. C. Escher, 1969/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 18: Will & Deni McIntyre/ Photo Researchers Inc.; 19: SEKI/PY XVIII (1978), Kunito Nagaoka/ Courtesy of the artist.; 19 (L): Cheryl Fenton; 19 (R): Cheryl Fenton; 20 (B): Corbis; 20 (T): Nathan Benn/Corbis; 22 (B): Ken Karp Photography; 22 (C): Peter Sanders Photography; 22 (TL): Peter Sanders Photography; 22 (TR): Peter Sanders Photography; 23: Photo Researchers Inc.; 24: Hot Blocks (1966–67) ©Edna Andrade, Philadelphia Museum of Art, Purchased by Philadelphia Foundation Fund; 25 (L): Comstock; 25 (R): Scala/Art Resource; 29 (T): George Lepp/Photo Researchers Inc. Chapter 1 27: Three Worlds, M. C. Escher, 1955/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 28 (B): Spencer Grant/Photo Researchers Inc.; 28 (C): Cheryl Fenton; 28 (T): Hillary Turner; 29: By permission of Johnny Hart and Creators Syndicate, Inc.; 30: Bachman/Photo Researchers Inc.; 32: S. Craig/Bruce Coleman Inc.; 33 (R): Grafton Smith/Corbis Stock Market; 33 (L): Michael Daly/Corbis Stock Market; 34: Addison Geary/Stock Boston; 35: Bob Stovall/Bruce Coleman Inc.; 36: Archivo Iconografico, S. A./Corbis; 38 (TL): Bruce Coleman Inc.; 38 (TR): David Leah/Getty Images; 39 (B): Hillary Turner; 39 (BR): Cheryl Fenton; 39 (BR): Comstock; 39 (T): Ken Karp Photography; 41: Pool & Billiard Magazine; 47: Illustration by John Tenniel; 48: Osentoski & Zoda/Envision; 49: Christie’s Images; 50: Corbis; 51: Hillary Turner; 54: Cheryl Fenton; 55: Ira Lipsky/International Stock Photography; 56: Quilt by Diane Venters/More Mathematical Quilts; 56 (C): Cheryl Fenton; 56 (TCL): Hillary Turner; 56 (TCR): Hillary Turner; 56 (TL): Hillary Turner; 56 (TR): Hillary Turner; 59: Spencer Swanger/Tom Stack & Associates; 60: Gerard Degeorge/Corbis; 61: Sol LeWitt—Wall Drawing #652—On three walls, continuous forms with color ink washes superimposed, Color in wash. Collection: Indianapolis Museum of Art, Indianapolis, IN. September, 1990. Courtesy of the Artist.; 62 (C): Michael Moxter/Photo Researchers Inc.; 62 (L): Larry Brownstein/Rainbow; 62 (R): Stefano Amantini/Bruce Coleman Inc.; 63 (B): Cheryl Fenton; 63 (T): Cheryl Fenton; 65: Friedensreich ©Erich Lessing/Art Resource, NY; 66: From WALTER WICK’S OPTICAL TRICKS. Published by Cartwheel books, a division of Scholastic Inc. ©1998 by Walter Wick. Reprinted by permission.; 67 (BL): Joel Tribhout/ Agence Vandystadt/Getty Images; 67 (BR): Terry Eggers/Corbis Stock Market; 67 (T): By permission of Johnny Hart and Creators Syndicate Inc.; 68 (L): Cheryl Fenton; 68 (R): Getty Images; 70 (CL): Corbis; 70 (CR): Alfred Pasieka/Photo Researchers Inc.; 70 (T): Corbis; 73: Bookplate for Albert Ernst Bosman, M. C. Escher, 1946/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 74 (BC): Stock Montage, Inc.; 74 (BL): Stock Montage, Inc.; 74 (BR): AP/Wide World; 79: William Thomas Williams, “DO YOU THINK A IS B,” Acrylic on Canvas, 1975–77, Fisk University Galleries, Nashville, Tennessee; 80 (C): Cheryl Fenton; 80 (L): Cheryl Fenton; 80 (R): Cheryl Fenton; 81: Cheryl Fenton; 82: Cheryl Fenton; 83: Courtesy of Kazumata Yamashita, Architect; 84 (B): Ken Karp Photography; 84 (C): Mike Yamashita/Woodfin Camp & Associates; 85: T. Kitchin/Tom Stack & Associates; 86 (C): Cheryl Fenton; 86 (T): Ken Karp Photography; 88: Paul Steel/Corbis-Stock Market; 90: Cheryl Fenton. Chapter 2 93: Hand with Reflecting Sphere (Self-Portrait in Spherical Mirror), M. C. Escher/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 94 (B): Barry Rosenthal/FPG; 94 (T): Ken Karp Photography; 95: Andrew McClenaghan/Photo Researchers Inc.; 100: Bob Daemmrich/Stock Boston; 101: California Institute of Technology and Carnegie Institution of Washington; 104: NASA; 106: Drabble reprinted by permission of United Feature Syndicate, Inc.; 112 (L): National Science Foundation Network; 112 (R): Hank Morgan/Photo Researchers Inc.; 113: Hillary Turner; 115: Cheryl Fenton; 118 (B): Ken Karp Photography; 118 (T): Culver Pictures; 121: Ken Karp Photography; 126 (B): Ken Karp Photography; 126 (T): Alex MacLean/Landslides; 128: Hillary Turner; 130: James Blank/Bruce Coleman Inc.; 134 (L): Photo Researchers; 134 (R): Mark Gibson/Index Stock; 135 (B): Ted Scott/Fotofile, Ltd.; 135 (C): Ken Karp Photography; 137: Art Matrix. Chapter 3 141: Drawing Hand, M. C. Escher, 1948/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 142 (L): Hillary Turner; 142 (R): Hillary Turner; 142 (T): Bettmann/Corbis; 143 (L): Hillary Turner; 143 (R): Hillary Turner; 144: Travel Ink/Corbis; 154: Overseas Highway by Crawford Ralston, 1939 by Crawford Ralston/Art Resource, NY; 156: Rick Strange/Picture Cube; 159: Corbis; 172 (BL): Ken Karp Photography; 172 (BR): Dover Publications; 172 (T): Greg Vaughn/Tom Stack & Associates; 174: Art Resource; 175: Timothy Eagan/Woodfin Camp & Associates; 176: Ken Karp Photography; 177: Rounds and Triangles by Rudolf Bauer /Christie’s Images; 180: Corbis; 185 (B): Corbis; 185 (T): Ken Karp Photography; 186: Keith Gunnar/Bruce Coleman Inc.; 189: Ken Karp Photography; 191: Victoria & Albert Museum, London/Art Resource, NY. Chapter 4 197: Symmetry Drawing E103, M. C. Escher,1959/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 198 (B): Courtesy, St. John’s Presbyterian Church; 198 (C): Jim Corwin/Stock Boston; 198 (TL): Cheryl Fenton; 198 (TR): Cheryl Fenton; 199: Ken Karp Photography; 202: The Far Side® by Gary Larson ©1987 FarWorks, Inc. All rights reserved. Used with permission.; 203: Hillary Turner; PHOTO CREDITS 765 204 (L): David L. Brown/Picture Cube; 204 (R): Joe Sohm/The Image Works; 207: Art Stein/Photo Researchers Inc.; 213: Robert Frerck/Woodfin Camp & Associates; 217: Judy March/Photo Researchers Inc.; 220: Adamsmith Productions/Corbis; 222: Carolyn Schaefer/Gamma Liaison; 227: Cheryl Fenton; 234 (B): Ken Karp Photography; 234 (T): AP/Wide World; 240: Ken Karp Photography; 241: Eric Schweikardt/The Image Bank; 247: Archivo Iconografico, SA/Corbis; 249 (T): Ken Karp Photography; 251: Ken Karp Photography; 252 (L): Cheryl Fenton; 252 (R): Cheryl Fenton; 253: Cheryl Fenton. Chapter 5 255: Still Life and Street, M. C. Escher, 1967–68/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 256: Ken Karp Photography; 256 (T): Cheryl Fenton; 260: Emerging Order by Hannah Hoch/Christie’s Images/Corbis; 263: Richard Megna/Fundamental Photographs; 265: Courtesy of the artist, Teresa Archuleta-Sagel; 266: Steve Dunwell/The Image Bank; 268 (C): Lindsay Hebberd/Woodfin Camp & Associates; 268 (T): Photo Researchers Inc.; 271: Cheryl Fenton; 278: Malcolm S. Kirk/Peter Arnold Inc.; 284: ©Paula Nadelstern/Photo by Bobby Hansson; 286: Corbis; 289: Ken Karp Photography; 290: Cheryl Fenton; 291: Bill Varie/The Image Bank; 292: Bill Varie/The Image Bank; 292: Alex MacLean/Landslides; 298: Cheryl Fenton; 299: Red and Blue Puzzle (1994) Mabry Benson/Photo by Carlberg Jones; 302: Ken Karp Photography; 303: Boy With Birds by David C. Driskell/Collection of Mr. and Mrs. David C. Driskell. Chapter 6 305: Curl-Up, M. C. Escher,1951/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 306 (B): Stephen Saks/Photo Researchers, Inc.; 306 (C): Corbis; 308: Tom Sanders/Corbis Stock Market; 311: NASA; 312: Tony Freeman/PhotoEdit; 313 (L): Stephen A. Smith; 313 (R): Mark Burnett/Stock Boston; 314: NASA; 315: Gray Mortimore/Getty Images; 316 (L): Corbis; 316 (L): Jim Zuckerman/Corbis; 319: Xinhua/Gamma Liaison; 328 (L): Cheryl Fenton; 328 (R): Cheryl Fenton; 331 (B): Cheryl Fenton; 331 (C): Ken Karp Photography; 332: Hillary Turner; 334 (B): Don Mason/Corbis Stock Market; 334 (C): David R. Frazier/Photo Researchers Inc.; 334 (T): Ken Karp Photography; 337: Photofest; 339: Calvin and Hobbes ©Watterson. Reprinted with permission of UNIVERSAL PRESS SYNDICATE. All rights reserved.; 340: Guido Cozzi/Bruce Coleman Inc.; 341: Grant Heilman Photography; 344 (B): Nick Gunderson/Corbis; 344 (T): Charles Feil/Stock Boston; 345 (B): Bob Daemmrich/ Stock Boston; 345 (T): Cathedrale de Reims; 346: Gamma Liaison; 349 (B): Ken Karp Photography; 349 (T): Cheryl Fenton; 351: Ken Karp Photography; 352: The Far Side® by Gary Larson ©1990 FarWorks, Inc. All rights reserved. Used with permission.; 353: David Malin/Anglo-Australian Telescope Board; 355: Private collection, Berkeley, California/Ceramist, Diana Hall. Chapter 7 357: Magic Mirror, M. C. Escher,1946/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 358: Giraudon/Art Resource; 358 (B): Ken Karp Photography; 360: Corbis; 361: Grant Heilman Photography; 363 (BC): Denver Museum of Natural History; 363 (BL): Michael Lustbader/Photo Researchers; 363 (BR): Richard Cummins/Corbis; 363 (TL): Phil Cole/Getty Images; 363 (TR): Corbis; 364: Oxfam; 369: Corbis; 372: By Holland ©1976 Punch Cartoon Library; 373: Gina Minielli/Corbis; 378: Ken Karp Photography; 379 (L): W. Treat Davison/Photo Researchers; 379 (R): Ancient Art and Architecture Collection; 381: SpringerVerlag; 385: Courtesy Doris Schattschneider; 386: Lucy Birmingham/ Photo Researchers Inc.; 388 (B): Carleton College/photo by Hilary N. Bullock; 388 (T): Ashton, Ragget, McDougall Architects; 389 (C): Symmetry Drawing E105, M. C. Escher 1960/©2002 Cordon Art B.V.–Baarn–Hollan
d. All rights reserved.; 389 (T): Brickwork, Alhambra, M. C. Escher/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 393: Symmetry Drawing E25, M. C. Escher, 1939/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 394: Reptiles, M. C. Escher, 1943/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 395 (T): Symmetry Drawing E99, M. C. Escher, 1954/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 396 (L): Interwoven patterns–V– structure 17 by Rinus Roelofs/Courtesy of the artist & ©2002 Artist Rights Society (ARS), New York/Beeldrecht, Amsterdam; 396 (R): Impossible structures–III–structure 24 by Rinus Roelofs/ Courtesy of the artist & ©2002 Artist Rights Society (ARS), New York/Beeldrecht, Amsterdam; 398 (B): Horseman sketch, M. C. Escher/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 398 (T): Horseman, M. C. Escher, 1946/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 399: Symmetry Drawing E108, M. C. Escher, 1967/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 404: King by Minne Evans/Corbis & Luise Ross Gallery; 405 (C): Paul Schermeister/Corbis; 405 (TR): Comstock; 406: Ken Karp Photography; 407 (B): Starbuck Goldner; 407 (C): Day and Night, M. C. Escher, 1938/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved. Chapter 8 409: Square Limit, M. C. Escher, 1964/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 410: Spencer Grant/Stock Boston; 414 (B): Karl Weatherly/Corbis; 414 (T): Courtesy Naoko Hirakura Associates; 421: Ken Karp Photography; 422: Joseph Nettis/Photo Researchers Inc.; 423 (B): NASA; 423 (C): Ken Karp Photography; 424: Phil Schermeister/Corbis; 428: Cheryl Fenton; 434: Cheryl Fenton; 435 (B): Reuters NewMedia Inc./Corbis; 435 (C): Georg Gerster/Photo Researchers Inc.; 437: Shahn Kermani/ Gamma Liaison; 440: Eye Ubiquitous/Corbis; 442: Ken Karp Photography; 444: Cheryl Fenton; 445 (BL): Stefano Amantini/ Bruce Coleman Inc.; 445 (BR): Alain Benainous/Gamma Liaison; 445 (CB): Joan Iaconetti/Bruce Coleman Inc.; 445 (CT): Bill Bachmann/The Image Works; 446: Greg Pease/Corbis; 447 (L): Sonda Dawes/The Image Works; 447 (R): John Mead/ Photo Researchers Inc.; 449: Library of Congress; 451: Bryn Campbell/Getty Images; 452: Douglas Peebles; 458: Ken Karp Photography. Chapter 9 461: Waterfall, M. C. Escher, 1961/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 462: FUNKY WINKERBEAN by Batiuk. Reprinted with special permission of North American Syndicate; 463: Corbis; 463 (L): Corbis; 464: Will & Deni McIntyre/ Photo Researchers Inc.; 467 (C): Corbis; 467 (T): Ken Karp Photography; 468: Ken Karp Photography; 469: Rare Book and Manuscript Library/Columbia University; 470: John Colett/Stock Boston; 472: Christie’s Images/©2002 Sol LeWitt/Artists Rights Society (ARS), New York; 475: Turner Entertainment ©1939. All rights reserved.; 482: FUNKY WINKERBEAN by Batiuk. Reprinted with special permission of North American Syndicate; 483: Bob Daemmrich/The Image Works; 484 (B): Private collection, Berkeley, California/Photo by Cheryl Fenton; 484 (T): Ken Karp Photography; 490: David Muench/Corbis; 494: Bruno Joachim/ Gamma Liaison; 495: Ken Karp Photography; 498: Ken Karp Photography. Chapter 10 503: Verblifa tin, M. C. Escher, 1963/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 504 (BL): J. Bernholc, etal./North Carolina State/Photo Researchers Inc.; 504 (CB): Ken Eward/Photo Researchers Inc.; 504 (CT): Runk-Schoenberger/Grant Heilman Photography; 504 (CT): Ken Eward/Photo Researchers; 504 (L): Archivo Iconografico, S. A. /Corbis; 504 (T): Cheryl Fenton; 505: Esaias Baitel/Gamma Liaison; 506: Cheryl Fenton; 507: Burstein Collection/Corbis/© Andy Warhol/Artists Rights Society (ARS), New York; 510: ©1996 C. Herscovici, Brussels, Artists Rights Society (ARS) NY/Photo courtesy Minneapolis Institute of 766 PHOTO CREDITS Art; 512: Ken Karp Photography; 514: Ken Karp Photography; 514: Christie’s Images/Corbis; 516: Hillary Turner; 519 (B): Jeff Tinsly/Names Project; 519 (C): Larry Lee Photography/ Corbis; 520 (B): Thomas Kitchin/Tom Stack & Associates; 520 (C): Eye Ubiquitous/Corbis; 521 (B): Cheryl Fenton; 521 (C): Cheryl Fenton; 522: Ken Karp Photography; 525: Masao Hayashi/Photo Researchers Inc.; 527: Charles Lenars/Corbis; 528 (B): Ken Karp Photography; 528 (T): Fitzwilliam Museum, Cambridge; 532 (T): David Sutherland/Stone/Getty Images; 533: Parson’s School of Design; 537: David Morris/Gamma Liaison; 539 (B): C. J. Allen/Stock Boston; 539 (T): Ken Karp Photography; 540: David Hockney, Sunday Morning Mayflower Hotel, N.Y., Nov. 28, 1982/Photographic collage, ED: 20/50 × 77 ©David Hockney; 541: Ken Karp Photography; 542 (B): Ken Karp Photography; 544 (B): The Far Side® by Gary Larson ©1986 FarWorks, Inc. All rights reserved. Used with permission.; 544 (T): Culver Pictures/ Picture Quest; 546 (C): John Cooke/Comstock; 546 (T): NASA; 548: Larry Lefevre/Grant Heilman Photography; 549: Arte & Immagini/Corbis; 551 (L): Ken Karp Photography; 551 (T): Illustration by Sidney Paget from The Strand Magazine, 1892; 555: Piranha Club by B. Grace. Reprinted with special permission of King Features Syndicate; 558: Douglas Peebles/Corbis. Chapter 11 559: Path of Life I, M. C. Escher, 1958/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 564 (R): J. Greenberg/The Image Works; 564 (R): Chromosohn/Photo Researchers Inc.; 567 (B): National Geographic Society; 567 (C): Giaudon/Art Resource, NY; 567 (T): Erich Lessing/Art Resource, NY; 569: John Elk III/Stock Boston; 570: Daniel Sheehan/Black Star Publishing/PictureQuest; 571 (C): Ken Karp Photography; 571 (T): Michael S.Yamashita/Corbis; 575: Patricia Lanza/Bruce Coleman Inc.; 576 (B): Thomas Dove/Douglas Peebles; 576 (L): Rossi & Rossi; 576 (R): Rossi & Rossi; 577: Dave Bartruff/Corbis; 583: Ken Karp Photography; 586: Ken Karp Photography; 592: Alon Reininger/Contact Press Images/PictureQuest; 594: Ken Karp Photography; 595: David Fraser/Photo Researchers Inc.; 597 (B): Dr. Paul A. Zahl/Photo Researchers Inc.; 597 (T): La Petite Chatelaine, Version a La Natte Courbe by Camille Claudel/Christie’s Images/© 2002 Artists Rights Society (ARS), New York/ADAGP, Paris; 599: Ken Karp Photography; 600 (B): ©1996 Demart Pro Arte Geneva/© Salvador Dali, Gala-Salvador Dali Foundation/ARS NY; 600 (T): Corbis; 601 (C): Kurt Krieger/Corbis; 601 (T):Bettmann/ Corbis; 607: Charles Feil/Stock Boston; 611 (B): Ken Karp Photography; 612: Alice in Wonderland; 615 (B): Robert Holmes/ Corbis; 615 (T): Dan McCoy/Rainbow; 616 (L): Chris Lisle/Corbis; 616 (R): Alex Webb/Magnum; 617: Underwood & Underwood/ Corbis; 618: NASA. Chapter 12 619: Belvedere, M. C. Escher, 1958/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 620: Bettmann/Corbis; 621 (B): Perry Collection; 621 (T): Gary Braasch/Woodfin Camp & Associates; 623: Cheryl Fenton; 627: Ken Karp Photography; 628 (B): Ecoscene/ Corbis; 628 (C): H. Reinhard/Photo Researchers; 629: Breezing Up by Winslow Homer/Photo by Francis G. Mayer/Corbis; 630: Dennis Marsico/Corbis; 631 (L): Wendell Metzen/Bruce Coleman Inc.; 631 (R): Corbis; 632 (C): Ken Karp Photography; 633: Greg Rynders; 638 (C): Courtesy of Association for the Preservation of Virginia Antiquities; 639 (B): Stevan Stefanovic/Photo Researchers Inc.; 639 (T): Steve Owlett/Bruce Coleman Inc.; 644 (B): Grant Heilman Photography; 646: Photograph by Hiroshi Umeoka; 647: Sonda Dawes/The Image Works; 649: James A. Sugar/Black Star Publishing/Picture Quest; 651: Angelo Hornak/Corbis; 652: Walter Hodges/Corbis; 654: Courtesy California Academy of Sciences; 655: Ken Karp Photography; 660: © Marty Sohl; 661: B. Christensen/Stock Boston. Chapter 13 667: Another World (Other World), M. C. Escher/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 668: Springer-Verlag; 668: Araldo de Luca/CORBIS; 669 (C): Ken Karp Photography; 669 (R): Ken Karp Photography; 670: Charles & Josette Lenars/ Corbis; 671: Art Resource; 675: Library of Congress; 679: ©1977 by Sidney Harris, American Scientist Magazine; 682: Ken Karp Photography; 684: Ken Karp Photography; 703: M. Dillon/Corbis; 709: Paul A. Souders/Corbis; 718 (L): Springer-Verlag; 718 (R): Ken Karp Photography; 720: Collection of Suzanne Summer/Cheryl Fenton Photography PHOTO CREDITS 767
es? Round to the nearest thousandth gram. Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. DAY 5 Which conjecture about polygons is NOT true? The area of a parallelogram is the product of its base and height. A rhombus has four right angles. A square has four congruent sides. A trapezoid has exactly one pair of parallel sides. Which Pythagorean triple would be most helpful in finding the value of a? 3-4-5 5-12-14 8-15-17 7-24-25 DAY 4 Natalia plans to install glass doors across the front of her square fireplace opening and then seal the perimeter of the opening with a special caulk that can sustain high temperatures. What is the perimeter of the opening? Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. Countdown to TAKS TX21 TX21 �� Countdown to TAKS WEEK 19 DAY 1 Which two line segments are congruent? ̶̶ AB and ̶̶ CE and ̶̶̶ GH and ̶̶ CD and ̶̶ DF ̶̶̶ GH ̶̶ AB ̶̶ DE DAY 2 DAY 3 Based on the table, which algebraic expression best represents the number of triangles formed by drawing all of the diagonals from one vertex in a polygon with n sides? At a certain time of the day, a 24-foot tree casts an 18-foot shadow. How long is the shadow cast by a 4-foot mailbox at the same time of day? 3 1 4 2 5 3 8 6 No. of sides No. of triangles formed n 2n - 1 n - 2 n + 2 _ 2 �� Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. DAY 4 DAY 5 A school increases the width of its rectangular playground from 25 meters to 40 meters and the length from 45 meters to 60 meters. By how much does the perimeter of the playground increase? Trey is using triangular tiles to floor his bathroom. What is x? 30 meters 60 meters 200 meters 225 meters TX22 TX22 Countdown to TAKS Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. �� Countdown to TAKS WEEK 20 DAY 1 The figure shows the measure of each interior angle for several regular polygons. Which algebraic expression best represents the measure of an interior angle of a regular polygon with n sides? (n - 2) 180 __ n 360n _ n + 2 (n - 2) 180 180n _ 2 DAY 2 DAY 3 Which coordinates represent a vertex of the hexagon? The two triangles in the figure are similar. What is the length of ̶̶̶ MN ? (0, 2) (4, -2) (3, 2) (-2, 2) Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. DAY 4 DAY 5 Two regular pentagons have perimeters of 30 and 75 respectively. What scale factor relates the smaller figure to the larger one? Alissa is painting a diagonal line across a square tile. What is the length of the line in centimeters? Round to the nearest thousandth of a centimeter. 1 : 2.5 1 : 6 1 : 15 1 : 21 Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. Countdown to TAKS TX23 TX23 �� Countdown to TAKS WEEK 21 DAY 1 The table lists the measure of an exterior angle for the given regular polygon. Which expression best represents the measure of an exterior angle of a regular polygon with n sides? Figure Quadrilateral Pentagon Decagon Exterior angle 90° 72° 36° 360 _ n - 2 360 + n _ 2 + n 360n 360 _ n DAY 2 DAY 3 When y = 65, x = 8. If y varies directly with x, what is y when x equals 15? Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. A word game uses a bag of 80 tiles. Forty of the tiles have a consonant on them, and the remaining 40 have a vowel: A, E, I, O, or U. There is an equal number of each vowel tile. What is the probability as a percent that Shelly selects an A tile and then a U tile from the bag without replacement? Round to the nearest hundredth of a percent. Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. DAY 4 DAY 5 Which equation best describes the line containing the hypotenuse of this triangle? ̶̶ AB . What are the coordinates of the The center of circle C is the midpoint of midpoint TX24 TX24 Countdown to TAKS (0, 4) (1, 4) (2, 4) (3, 3) �� Countdown to TAKS WEEK 22 DAY 1 If this pattern is continued, how many shaded triangles will there be in the fourth element of the pattern? 9 13 27 40 DAY 2 DAY 3 What is the slope of the line? A delivery truck travels 13.5 mi east and then 18 mi north. How far in miles is the truck from its starting point Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. DAY 4 DAY 5 What are the side lengths of the triangle? 3, 4, and 5 2, 3, and 5 3, 3, and 3 3, 3, and 3 √  2 An 18-foot ladder reaches the top of a building when placed at an angle of 45° with the horizontal. What is the approximate height of the building in feet? Round to the nearest tenth of a foot. Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. Countdown to TAKS TX25 TX25 �� Countdown to TAKS WEEK 23 DAY 1 DAY 2 △RST is a 30°-60°-90° triangle. What is the y-coordinate of R if a = -5 and c = -2? What is x if y is 12.8 and z is 16 in the right triangle below? 3.2 4.0 9.6 12. DAY 3 How does the slope of the hypotenuse of △ABC compare with that of △DBC? They have the same value and sign. They have opposite signs. One is a multiple of the other. They are reciprocals. DAY 4 DAY 5 How many sides does a regular polygon have if each interior angle measures 120°? Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. A piñata in the shape of a basketball is filled with treats for a game during Hanj’s birthday party. If the diameter of the piñata is 7 inches, what is the volume of the piñata in cubic inches? Round to the nearest tenth. Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. TX26 TX26 Countdown to TAKS �� Countdown to TAKS WEEK 24 DAY 1 Quadrilaterals ABCD and WXYZ are similar. What is XY? 3.5 21 24.5 35 DAY 2 DAY 3 The volume of a square pyramid is 108 cubic millimeters. What is the height of the pyramid in millimeters if one side on the base is 4.5 millimeters? Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. What is the value of x in the regular pentagon below? 54° 90° 108° 180° DAY 4 DAY 5 What is the second term in a proportion in which the first, third, and fourth terms are 3, 9, and 12, respectively? Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. The endpoints of a segment are Q (-2, 6) and R (5, -4) . What is the length of the segment to the nearest tenth? 3.6 units 4.1 units 8.5 units 12.2 units Countdown to TAKS TX27 TX27 �� Texas Friendship Texas Essential Knowledge and Skills for Geometry a Basic understandings. 1 Foundation concepts for high school mathematics. As presented in Grades K–8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences. 2 Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry; geometric figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them. 3 Geometric figures and their properties. Geometry consists of the study of geometric figures of zero, one, two, and three dimensions and the relationships among them. Students study properties and relationships having to do with size, shape, location, direction, and orientation of these figures. The State Capitol in Austin Texas wildflowers Houston skyline TX28 The state bird is the Mockingbird. Texas Essential Knowledge and Skills The Bluebonnet is the state flower. 4 The relationship between geometry, other mathematics, and other disciplines. Geometry can be used to model and represent many mathematical and real-world situations. Students perceive the connection between geometry and the real and mathematical worlds and use geometric ideas, relationships, and properties to solve problems. 5 Tools for geometric thinking. Techniques Statue of a Texas longhorn for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to solve meaningful problems by representing and transforming figures and analyzing relationships. 6 Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problemsolving, language and communication, connections within and outside mathematics, and reasoning (justification and proof ). Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem solving contexts. TX29 Texas Friendship B recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes; and C compare and contrast the structures and implications of Euclidean and non-Euclidean geometries. G.2 Geometric structure. The student analyzes geometr
ic relationships in order to make and verify conjectures. The student is expected to: A use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships; and b Knowledge and skills. G.1 Geometric structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to: A develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems; Big Bend National Park TX30 Texas Essential Knowledge and Skills B make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.3 Geometric structure. The student applies logical reasoning to justify and prove mathematical statements. C use logical reasoning to prove statements are true and find counter examples to disprove statements that are false; The student is expected to: A determine the validity of a conditional statement, its converse, inverse, and contrapositive; D use inductive reasoning to formulate a conjecture; and E use deductive reasoning to prove a statement. B construct and justify statements about geometric figures and their properties; TX31 Texas Friendship West Texas TX32 G.4 Geometric structure. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: A select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. G.5 Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: A use numeric and geometric patterns to develop algebraic expressions representing geometric properties; B use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles; Texas Essential Knowledge and Skills C use properties of B use nets to represent The student is expected to: transformations and their compositions to make connections between mathematics and the real world, such as tessellations; and D identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples. G.6 Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to: A describe and draw the intersection of a given plane with various three-dimensional geometric figures; and construct three-dimensional geometric figures; and C use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems. G.7 Dimensionality and the geometry of location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. A use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures; B use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons; and C derive and use formulas involving length, slope, and midpoint. SeaWorld TX33 G.10 Congruence and the geometry of size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to: A use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane; and B justify and apply triangle congruence relationships. Texas Friendship G.9 Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures. The student is expected to: A formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models; B formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models; C formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models; and D analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models. G.8 Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: A find areas of regular polygons, circles, and composite figures; B find areas of sectors and arc lengths of circles using proportional reasoning; C derive, extend, and use the Pythagorean Theorem; and D find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations. TX34 Texas Essential Knowledge and Skills Texas State Fair G.11 Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to: A use and extend similarity properties and transformations to explore and justify conjectures about geometric figures; B use ratios to solve problems involving similar figures; C develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods; and D describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems. TX35 Foundations for Geometry KEYWORD: MG7 TOC ARE YOU READY TEKS G.7.A G.2.A G.3.B G.3.B G.1.A Euclidean and Construction Tools 1-1 Understanding Points, Lines, and Planes . . . . . . . . . . . . . . . . . . . 6 Explore Properties Associated with Points . . . . . . . 12 1-2 Measuring and Constructing Segments . . . . . . . . . . . . . . . . . . . 13 1-3 Measuring and Constructing Angles . . . . . . . . . . . . . . . . . . . . . . 20 1-4 Pairs of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Coordinate and Transformation Tools G.8.A 1-5 Using Formulas in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 G.1.A G.5.C G.2.A On Track for TAKS: Algebra Graphing in the Coordinate Plane. . . . . . . . . . . . . . . . . . . . . . 42 1-6 Midpoint and Distance in the Coordinate Plane . . . . . . . . . . . 43 1-7 Transformations in the Coordinate Plane. . . . . . . . . . . . . . . . . . 50 Explore Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 56 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Study Guide: Preview Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Tools for Success Reading Math 5 Writing Math 10, 18, 26, 33, 40, 48, 54 Vocabulary 3, 4, 9, 17, 24, 31, 38, 47, 53, 60 Know-It Notes 6, 7, 8, 13, 14, 16, 20, 21, 22, 24, 28, 29, 31, 36, 37, 43, 44, 45, 46, 50, 52 Test Prep Exercises 11, 19, 26, 33, 40–41, 49, 55 Multi-Step TAKS Prep 10, 18, 26, 33, 34, 39, 48, 54, 58 Graphic Organizers 8, 16, 24, 31, 37, College Entrance Exam Practice 65 46, 52 Homework Help Online 9, 17, 24, 31, 38, 47, 53 TAKS Tackler 66 TAKS Prep 68 Geometric Reasoning ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 TEKS Inductive and Deductive Reasoning G.3.D 2-1 Using Inductive Reasoning to Make Conjectures . . . . . . . . . . . 74 On Track for TAKS: Number Theory G.3.A G.3.E G.4.A G.3.A Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2-2 Conditional Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2-3 Using Deductive Reasoning to Verify Conjectures. . . . . . . . . . 88 Solve Logic Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2-4 Biconditional Statements and Definitions . . . . . . . . . . . . . . . . . 96 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Mathematical Proof G.3.E G.1.A G.1.A G.1.A 2-5 Algebraic Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2-6 Geometric Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Design Plans for Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2-7 Flowchart and Paragraph Proofs . . . . . . . . . . . . . . . . . . . . . . . . 118 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 EXT Introduction to Symbolic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 128 G.4.A Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 72 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Problem Solving on Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Tools for Success KEYWORD: MG7 TOC Reading Math 73 Writing Math 78, 81, 86, 92, 96, 100, 109, 111, 115, 125 Vocabulary 71, 72, 77, 84, 91, 99, 107, 113, 122, 130 Know-It Notes 75, 76, 81, 83, 84, 89, 90, 98, 104, 106, 107, 110, 111, 112, 113, 118, 120, 122, 128 Test Prep Exercises 79, 86, 93, 101, 109, 116, 125 Multi-Step TAKS Prep 78, 85, 92, 100, 102, 109, 115, 124, 126 Graphic Organizers 76, 84, 90, 98, 107, College Entrance Exam Practice 135 113, 122 Homework Help Online 77, 84, 91, 99, 107, 113, 122 TAKS Tackler 136 TAKS Prep 138 KEYWORD: MG7 TOC Parallel and Perpendicular Lines ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 TEKS Lines with Transversals 3-1 Lines and Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 On Track for TAKS: Algebra Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 G.9.A G.3.C G.3.C G.2.A G.1.A G.2.A Explore Parallel Lines and Transversals . . . . . . . . . 154 3-2 Angles Formed by Parallel Lines and Transversals. . . . . . . . . 155 3-3 Proving Lines Parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Construct Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 3-4 Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Construct Perpendicular Lines. . . . . . . . . . . . . . . . . . . . . . . . 179 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Coordinate Geometry G.7.B G.7.B G.7.B 3-5 Slopes of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Explore Parallel and Perpendicular Lines . . . . . . . . . 188 3-6 Lines in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 On Track for TAKS: Data Analysis Scatter Plots and Lines of Best Fit . . . . . . . . . . . . . . . . . . . . . 198 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Tools for Success Writing Math 150, 160, 168, 177, Study Strategy 145 186, 196 Vocabulary 143, 144, 148, 175, 185, 194, 202 Know-It Notes 146, 147, 148, 155, 156, 157, 162, 163, 173, 174, 182, 184, 185, 190, 192, 193 Graphic Organizers 148, 157, 165, 174, 185, 193 Test Prep Exercises 150–151, 160–161, 168–169, 177–178, 187, 196–197 Multi-Step TAKS Prep 150, 160, 168, 176, 180, 186, 196, 200 College Entrance Exam Practice 207 TAKS Tackler 208 Homework Help Online 148, 158, 166, TAKS Prep 210 175, 185, 194 KEYWORD: MG7 TOC Triangle Congruence ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 TEKS G.1.A G.9.B G.2.B G.10.B G.9.B G.10.B G.9.B G.10.B G.1.A G.2.B G.2.B Triangles and Congruence 4-1 Classifying Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Develop the Triangle Sum Theorem . . . . . . . . . . . . . . . . . . 222 4-2 Angle Relationships in Triangles . . . . . . . . . . . . . . . . . . . . . . . . 223 4-3 Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Proving Triangle Congruence Explore SSS and SAS Triangle Congruence . . . . . . . . . . . 240 4-4 Triangle Congruence: SSS and SAS . . . . . . . . . . . . . . . . . . . . . . 242 Predict Other Triangle Congruence Relationships . . . . 250 4-5 Triangle Congruence: ASA, AAS, and HL . . . . . . . . . . . . . . . . . 252 4-6 Triangle Congruence: CPCTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 On Track for TAKS: Algebra Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 4-7 Introduction to Coordinate Proof. . . . . . . . . . . . . . . . . . . . . . . . 267 4-8 Isosceles and Equilateral Triangles. . . . . . . . . . . . . . . . . . . . . . . 273 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 EXT Proving Constructions Valid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 G.2.A Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Problem Solving on Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Tools for Success Reading Math 215, 273 Writing Math 220, 229, 236, 248, 258, 264, 271, 278 Vocabulary 213, 214, 219, 227, 234, 245, 256, 262, 270, 276, 284 Know-It Notes 216, 217, 218, 223, 224, 225, 226, 231, 233, 242, 243, 245, 252, 254, 255, 262, 267, 269, 273, 274, 275, 276 Graphic Organizers 218, 226, 233, 245, 255, 262, 269, 276 Test Prep Exercises 221, 230, 236, 248, 258–259, 264–265, 272, 279 Multi-Step TAKS Prep 220, 229, 236, 238, 247, 258, 264, 271, 278, 280 College Entrance Exam Practice 289 TAKS Tackler 290 Homework Help Online 219, 227, 234, TAKS Prep 292 245, 256, 262, 270, 276 KEYWORD: MG7 TOC Properties and Attributes of Triangles ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 TEKS G.3.B G.3.B G.3.B G.2.A G.7.B Segments in Triangles 5-1 Perpendicular and Angle Bisectors . . . . . . . . . . . . . . . . . . . . . . 300 5-2 Bisectors of Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 5-3 Medians and Altitudes of Triangles . . . . . . . . . . . . . . . . . . . . . . 314 Medians and Altitudes of Triangles Special Points in Triangles . . . . . . . . . . . . . . . . . . . . . . 321 5-4 The Triangle Midsegment Theorem . . . . . . . . . . . . . . . . . . . . . . 322 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Relationships in Triangles On Track for TAKS: Algebra Solving Compound Inequalities . . . . . . . . . . . . . . . . . . . . . . . 330 G.9.B G.3.B G.3.B Explore Triangle Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 331 5-5 Indirect Proof and Inequalities in One Triangle . . . . . . . . . . . 332 5-6 Inequalities in Two Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 On Track for TAKS: Algebra Simplest Radical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 G.8.C G.8.C G.5.D G.2.A Hands-on Proof of the Pythagorean Theorem . . . . . . . . 347 5-7 The Pythagorean Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 5-8 Applying Special Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . 356 Graph Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 Tools for Success Reading Math 299, 300 Writing Math 306, 313, 318, 325, 338, 344, 354, 361 Vocabulary 297, 298, 304, 311, 317, 324, 336, 352, 366 Know-It Notes 300, 301, 303, 307, 309, 310, 314, 317, 323, 324, 333, 334, 335, 340, 342, 350, 351, 352, 356, 358, 359 Test Prep Exercises 306, 313, 319, 326, 339, 345, 355, 362 Multi-Step TAKS Prep 305, 312, 319, 326, 328, 338, 344, 354, 361, 364 Graphic Organizers 303, 310, 317, 324, College Entrance Exam Practice 371 335, 342, 352, 359 Homework Help Online 304, 311, 317, 324, 336, 343, 352, 360 TAKS Tackler 372 TAKS Prep 374 Polygons and Quadrilaterals ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 TEKS G.2.A G.5.B Polygons and Parallelograms Construct Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . 380 6-1 Properties and Attributes of Polygons . . . . . . . . . . . . . . . . . . . 382 On Track for TAKS: Algebra Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 G.9.B G.3.B G.3.B Explore Properties of Parallelograms . . . . . . . . . . . . . . . . 390 6-2 Properties of Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 6-3 Conditions for Parallelograms. . . . . . . . . . . . . . . . . . . . . . . . . . . 398
MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 KEYWORD: MG7 TOC G.3.B G.2.A G.3.B G.2.A G.3.B Other Special Quadrilaterals 6-4 Properties of Special Parallelograms . . . . . . . . . . . . . . . . . . . . . 408 Predict Conditions for Special Parallelograms . . . 416 6-5 Conditions for Special Parallelograms . . . . . . . . . . . . . . . . . . . 418 Conditions for Special Parallelograms Explore Isosceles Trapezoids . . . . . . . . . . . . . . . . . . . . 426 6-6 Properties of Kites and Trapezoids . . . . . . . . . . . . . . . . . . . . . . 427 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Problem Solving on Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 Tools for Success Writing Math 379, 388, 397, 404, 414, 424, 434 Vocabulary 377, 378, 386, 395, 412, 432, 438 Know-It Notes 383, 384, 385, 391, 392, 394, 398, 399, 401, 408, 409, 411, 418, 419, 421, 427, 429, 431 Graphic Organizers 385, 394, 401, 411, Test Prep Exercises 388, 397, 405, 414–415, 425, 434–435 Multi-Step TAKS Prep 387, 396, 404, 406, 414, 424, 434, 436 College Entrance Exam Practice 443 421, 431 TAKS Tackler 444 Homework Help Online 386, 395, 402, TAKS Prep 446 412, 422, 432 Similarity KEYWORD: MG7 TOC ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 TEKS Similarity Relationships G.11.B 7-1 Ratio and Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 G.5.B G.5.B G.11.A G.11.B Explore the Golden Ratio . . . . . . . . . . . . . . . . . . . . . . . 460 7-2 Ratios in Similar Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Predict Triangle Similarity Relationships . . . . . . . . 468 7-3 Triangle Similarity: AA, SSS, and SAS . . . . . . . . . . . . . . . . . . . . 470 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 Applying Similarity Investigate Angle Bisectors of a Triangle . . . . . . . . 480 7-4 Applying Properties of Similar Triangles . . . . . . . . . . . . . . . . . 481 7-5 Using Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . . 488 7-6 Dilations and Similarity in the Coordinate Plane . . . . . . . . . . 495 G.5.B G.11.B G.11.D G.11.A On Track for TAKS: Algebra Direct Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 Tools for Success Reading Math 453, 455, 456 Writing Math 459, 463, 466, 476, 486, 493, 499 Know-It Notes 455, 457, 462, 464, 470, 471, 473, 481, 482, 483, 484, 490, 497 Graphic Organizers 457, 464, 473, 484, Vocabulary 451, 452, 457, 465, 491, 498, 490, 497 504 Homework Help Online 457, 465, 474, 484, 491, 498 Test Prep Exercises 459, 467, 477, 487, 493, 500 Multi-Step TAKS Prep 458, 466, 476, 478, 486, 492, 499, 502 College Entrance Exam Practice 509 TAKS Tackler 510 TAKS Prep 512 KEYWORD: MG7 TOC Right Triangles and Trigonometry ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 TEKS Trigonometric Ratios G.11.C 8-1 Similarity in Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 G.2.A G.11.C Explore Trigonometric Ratios . . . . . . . . . . . . . . . . . . . 524 8-2 Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 On Track for TAKS: Algebra G.11.C Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 8-3 Solving Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Applying Trigonometric Ratios G.11.C G.11.C G.11.C G.7.A 8-4 Angles of Elevation and Depression . . . . . . . . . . . . . . . . . . . . . 544 Indirect Measurement Using Trigonometry . . . . . . . . . . 550 8-5 Law of Sines and Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . 551 8-6 Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 EXT Trigonometry and the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . 570 G.11.A Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 Problem Solving on Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 Tools for Success Reading Math 517, 534, 570 Writing Math 523, 525, 531, 540, 548, 557, 566, 571 Vocabulary 515, 516, 521, 529, 547, 563, 572 Know-It Notes 518, 519, 520, 525, 528, 537, 546, 552, 553, 554, 561, 563 Graphic Organizers 520, 528, 537, 546, 554, 563 Test Prep Exercises 523, 532, 540, 549, 558, 567 Multi-Step TAKS Prep 522, 530, 539, 542, 548, 557, 565, 568 College Entrance Exam Practice 577 Homework Help Online 521, 529, 537, 547, 555, 563 TAKS Tackler 578 TAKS Prep 580 KEYWORD: MG7 TOC Extending Perimeter, Circumference, and Area ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 TEKS G.5.A G.5.A G.8.A G.8.A G.8.A Developing Geometric Formulas On Track for TAKS: Algebra Literal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 9-1 Developing Formulas for Triangles and Quadrilaterals . . . . 589 Develop π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 9-2 Developing Formulas for Circles and Regular Polygons . . . . 600 9-3 Composite Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 Develop Pick’s Theorem for Area of Lattice Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 Applying Geometric Formulas G.7.A G.11.D 9-4 Perimeter and Area in the Coordinate Plane. . . . . . . . . . . . . . 616 9-5 Effects of Changing Dimensions Proportionally . . . . . . . . . . 622 On Track for TAKS: Probability G.8.A G.8.A Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 9-6 Geometric Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 Use Geometric Probability to Estimate π. . . . . . . . . . . . . 637 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 Tools for Success Writing Math 596, 605, 611, 620, Study Strategy 587 626, 635 Vocabulary 585, 586, 603, 609, 633, 640 Know-It Notes 589, 590, 591, 593, 600, 601, 602, 608, 619, 623, 624, 630, 633 Graphic Organizers 593, 602, 608, 619, 624, 633 Test Prep Exercises 596–597, 605, 611–612, 621, 627, 636 Multi-Step TAKS Prep 595, 604, 610, 614, 620, 626, 635, 638 College Entrance Exam Practice 645 TAKS Tackler 646 Homework Help Online 593, 603, 609, TAKS Prep 648 619, 625, 633 KEYWORD: MG7 TOC Spatial Reasoning ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 TEKS G.6.A G.9.D G.6.B G.7.C Three-Dimensional Figures 10-1 Solid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 10-2 Representations of Three-Dimensional Figures . . . . . . . . . . . 661 Use Nets to Create Polyhedrons . . . . . . . . . . . . . . . . . . . . . . 669 10-3 Formulas in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 670 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 Surface Area and Volume G.8.D G.9.D G.8.D G.8.D G.8.D 10-4 Surfa
ce Area of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . 680 Model Right and Oblique Cylinders . . . . . . . . . . . . . . . . . . 688 10-5 Surface Area of Pyramids and Cones . . . . . . . . . . . . . . . . . . . . 689 10-6 Volume of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . 697 10-7 Volume of Pyramids and Cones . . . . . . . . . . . . . . . . . . . . . . . . . 705 On Track for TAKS: Algebra G.8.D G.11.D Functional Relationships in Formulas . . . . . . . . . . . . . . . . . . 713 10-8 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 Compare Surface Areas and Volumes . . . . . . . . . . . 722 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 EXT Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 G.1.C Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 Problem Solving on Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 Tools for Success Writing Math 653, 659, 667, 676, 686, 695, 703, 711, 720 Vocabulary 651, 657, 665, 674, 684, 693, 701, 709, 718, 730 Know-It Notes 654, 656, 664, 670, 671, 672, 673, 680, 681, 683, 689, 690, 692, 697, 699, 700, 705, 707, 708, 714, 716, 717, 726, 727 Graphic Organizers 656, 664, 673, 683, 692, 700, 708, 717 Test Prep Exercises 659, 667, 677, 687, 695, 703–704, 712, 721 Multi-Step TAKS Prep 658, 666, 675, 678, 686, 695, 703, 711, 720, 724 College Entrance Exam Practice 735 TAKS Tackler 736 Homework Help Online 657, 665, 674, TAKS Prep 738 684, 693, 701, 709, 718 KEYWORD: MG7 TOC Circles ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 TEKS Lines and Arcs in Circles G.9.C 11-1 Lines That Intersect Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 On Track for TAKS: Data Analysis Circle Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 11-2 Arcs and Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 11-3 Sector Area and Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 Angles and Segments in Circles Inscribed Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 11-4 Explore Angle Relationships in Circles . . . . . . . . . . 780 11-5 Angle Relationships in Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . 782 Explore Segment Relationships in Circles . . . . . . . 790 11-6 Segment Relationships in Circles . . . . . . . . . . . . . . . . . . . . . . . . 792 11-7 Circles in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 799 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 G.1.A G.8.B G.5.B G.2.A G.5.B G.9.C G.5.A G.2.B READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807 EXT Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 G.1.A Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 Tools for Success Reading Math 745, 748 Writing Math 754, 756, 762, 769, 778, 788, 797, 804 Vocabulary 743, 744, 751, 760, 767, 776, 810 Know-It Notes 746, 747, 748, 749, 750, 756, 757, 759, 764, 765, 766, 772, 773, 774, 775, 782, 783, 784, 785, 786, 792, 793, 794, 795, 799, 801 Graphic Organizers 750, 759, 766, 775, 786, 795, 801 Test Prep Exercises 754, 763, 769, 778, 789, 798, 804 Multi-Step TAKS Prep 753, 762, 768, 770, 777, 788, 797, 803, 806 College Entrance Exam Practice 815 TAKS Tackler 816 Homework Help Online 751, 760, 767, TAKS Prep 818 776, 786, 795, 802 KEYWORD: MG7 TOC Extending Transformational Geometry ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 TEKS G.10.A G.10.A G.10.A G.10.A G.10.A Congruence Transformations 12-1 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 12-2 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 On Track for TAKS: Algebra Transformations of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 838 12-3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 Explore Transformations with Matrices . . . . . . . . . . . . 846 12-4 Compositions of Transformations . . . . . . . . . . . . . . . . . . . . . . . 848 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 Patterns G.10.A G.5.C G.5.C G.11.A 12-5 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856 12-6 Tessellations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 Use Transformations to Extend Tessellations . . . . . . . . . 870 12-7 Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872 MULTI-STEP TAKS PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 EXT Using Patterns to Generate Fractals . . . . . . . . . . . . . . . . . . . . . 882 G.5.C Study Guide: Preview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888 Problem Solving on Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894 Tools for Success Writing Math 829, 836, 844, 852, 861, Study Strategy 823 868, 878, 883 Vocabulary 821, 822, 827, 851, 859, 866, 875, 884 Know-It Notes 825, 826, 832, 833, 840, 841, 848, 849, 850, 856, 857, 858, 866, 873, 874 Graphic Organizers 826, 833, 841, 850, 858, 866, 874 Test Prep Exercises 829–830, 836–837, 845, 853, 862, 869, 878 Multi-Step TAKS Prep 829, 835, 843, 853, 854, 861, 868, 876, 880 College Entrance Exam Practice 889 TAKS Tackler 890 Homework Help Online 827, 834, 842, TAKS Prep 892 851, 859, 866, 875 WHO USES MATHEMATICS? The Career Path features are a set of interviews with young adults who are either preparing for or just beginning in different career fields. These people share what math courses they studied in high school, how math is used in their field, and what options the future holds. Also, many exercises throughout the book highlight skills used in various career fields. KEYWORD: MG7 Career ELECTRICIAN p. 320 Electricians install and maintain the systems that provide many of the modernday comforts we rely on, such as climate control, lighting, and technology. Look on page 320 to find out how Alex Peralta got started on this career path. TECHNICAL WRITER p. 612 Have you ever wondered who writes manuals for operating televisions or stereos? A technical writer not only writes manuals for operating electronics, but also documents maintenance procedures for airplanes. Look at the Career Path on page 612 to find out how to become a technical writer. FURNITURE MAKER p. 805 A furniture maker must take precise measurements and be aware of spatial relationships in order to build a quality finished product. The Career Path on page 805 describes the kind of experience needed to be successful as a furniture maker. Career Applications Advertising 499 Agriculture 765 Animation 53, 835, 842 Anthropology 802 Archaeology 262, 787, 793 Architecture 47, 467, 667 Art 483, 657, 873 Aviation 277, 546, 564 Business 108, 194, 625 Carpentry 168, 418, 836 City Planning 305, 827 Communication 634, 802 Computer Graphics 495 Design 311, 317, 318 Electronics 692 Engineering 260, 554 Finance 108, 522 Forestry 548 Graphic Design 498, 752 Health 343 Industry 344 Interior Decorating 609, 867 Landscaping 607, 686, 702 Manufacturing 38, 754 Marine Biology 698, 720 Mechanics 434 Meteorology 801 Music 24 Navigation 228, 567, 729 Nutrition 107 Oceanography 174 Optometry 877 Photography 385, 459, 475 Political Science 79, 93 Real Estate 486 Surveying 25, 263, 556 xviii xviii Who Uses Mathematics? WHY LEARN MATHEMATICS? Links to interesting topics, including some in Texas, may accompany realworld applications in the text. These links help you see how math is used in the real world. For a complete list of all applications in Holt Geometry, see page S162 in the Index. Real-World Animation 835 Archaeology 787 Architecture 159, 220, 695 Astronomy 752 Bicycles 337 Bird-Watching 401 Biology 100, 604 Chemistry 828 Conservation 271 Design 313 Ecology 248 Electronics 692 Engineering 115, 233 Entertainment 149, 683, 803, 833 Fitness 539 Navigation 278 Oceanography 174 Pets 361 Racing 392 Recreation 92, 674 Food 195 Geography 626 Geology 86, 804 History 4
8, 413, 531, 566, 595 Kites 428 Landscaping 607 Marine Biology 698, 720 Math History 41, 78, 257, 318, 493, 611, 703, 768 Measurement 404 Mechanics 434 Meteorology 476, 675, 797 Monument 466 Mosaics 876 Shuffleboard 305 Space Shuttle 548 Sports 19, 530, 635 Surveying 353, 556 Transportation 183 Travel 458 Why Learn Mathematics xix xix Each frame of a computer-animated feature represents 1__24of a second of film.Source: www.pixar.comAnimationThe Top Thrill Dragster is 420 feet tall and includes a 400-foot vertical drop. It twists 270° as it drops. It is one of 16 roller coasters at Cedar Point amusement park.RecreationThis mosaic of the seal of the Republic of Texas is one of six tile mosaics that were installed on the front façade of the Sam Houston Regional Library and Research Center in Liberty, Texas, in fall 2001.MosaicsEach frame of a computer-animated feature represents 1__24of a second of film.Source: www.pixar.comAnimationThe Top Thrill Dragster is 420 feet tall and includes a 400-foot vertical drop. It twists 270° as it drops. It is one of 16 roller coasters at Cedar Point amusement park.RecreationThis mosaic of the seal of the Republic of Texas is one of six tile mosaics that were installed on the front façade of the Sam Houston Regional Library and Research Center in Liberty, Texas, in fall 2001.MosaicsEach frame of a computer-animated feature represents 1__24of a second of film.Source: www.pixar.comAnimationThe Top Thrill Dragster is 420 feet tall and includes a 400-foot vertical drop. It twists 270° as it drops. It is one of 16 roller coasters at Cedar Point amusement park.RecreationThis mosaic of the seal of the Republic of Texas is one of six tile mosaics that were installed on the front façade of the Sam Houston Regional Library and Research Center in Liberty, Texas, in fall 2001.Mosaics476Chapter 7 Similarity 26.Critical Thinking�ABC is not similar to �DEF, and �DEF is not similar to�XYZ. Could �ABC be similar to �XYZ? Why or why not? Make a sketch to support your answer. 27.Recreation To play shuffleboard, two teams take ��turns sliding disks on a court. The dimensions of the scoring area for a standard shuffleboard court are shown. What are JK and MN? 28. Prove the Transitive Property of Similarity. Given:�ABC∼�DEF,�DEF∼�XYZProve:�ABC∼�XYZ 29. Draw and label �PQR and �STU such that PQ___ST= QR___TUbut�PQR is NOT similar to �STU. 30.Given:�KNJ is isosceles with ��∠N as the vertex angle.∠H�∠LProve:�GHJ∼�MLK31.MeteorologySatellite photography makes it possible ��to measure the diameter of a hurricane. The figure shows that a camera’s aperture YX is 35 mm and its focal lengthWZ is 50 mm. The satellite W holding the camera is 150 mi above the hurricane, centered at C. a. Why is �XYZ∼�ABZ? What assumption must you make about the position of the camera in order to make this conclusion? b. What other triangles in the figure must be similar? Why? c. Find the diameter AB of the hurricane. 32./////ERROR ANALYSIS///// Which solution for the ��value of y is incorrect? Explain the error.�� 33.Write About It Two isosceles triangles have congruent vertex angles. Explain why the two triangles must be similar.476 25. This problem will prepare you for the ��Multi-Step TAKS Prep on page 478. The set for an animated film includes three small triangles that represent pyramids. a. Which pyramids are similar? Why? b. What is the similarity ratio of the similar pyramids?This satellite image shows Hurricane Lili as it moves across the Gulf of Mexico. In October 2002, an estimated 500,000 people evacuated in advance of Lili’s hitting Texas.Meteorology HOW TO STUDY GEOMETRY This book has many features designed to help you learn and study effectively. Becoming familiar with these features will prepare you for greater success on your exams. Learn The vocabulary is listed at the beginning of every lesson. Look for the Know-It-Note icons to identify important information. Practice Use a graphic organizer to summarize each lesson. Refer to the examples from the lesson to solve the Guided Practice exercises. Review Study and review vocabulary from the entire chapter. xx xx How To Study Geometry 4-4 Triangle Congruence: SSS and SAS TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.2.A, G.3.B, G.3.E Objectives Apply SSS and SAS to construct triangles and to solve problems. Prove triangles congruent by using SSS and SAS. Vocabulary triangle rigidity included angle Who uses this? Engineers used the property of triangle rigidity to design the internal support for the Statue of Liberty and to build bridges, towers, and other structures. (See Example 2.) In Lesson 4-3, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate. Postulate 4-4-1 Side-Side-Side (SSS) Congruence Side-Side-Side (SSS) Congruence POSTULATE POSTULATE POSTULATE POSTULATE HYPOTHESIS HYPOTHESIS HYPOTHESIS HYPOTHESIS CONCLUSION CONCLUSION CONCLUSION CONCLUSION If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. � �� FDE �ABC � �FDE �ABC �FDE � � � E X A M P L E 1 Using SSS to Prove Triangle Congruence Adjacent triangles Adjacent triangles share a side, so share a side, so you can apply the Reflexive Property to get a pair of congruent parts. Use SSS to explain why �PQR � �PSR. −− −− SR. By QR � It is given that −− PR � the Reflexive Property of Congruence, Therefore �PQR � �PSR by SSS. −− PS and that −− PQ � −− PR. 1. Use SSS to explain why �ABC � �CDA. � � � � � An included angle is an angle formed by two adjacent sides of a polygon. ∠B is the included angle between sides −− AB and −− BC. � � � � 242 242 Chapter 4 Triangle Congruence THINK AND DISCUSS 1. Describe three ways you could prove that �ABC � �DEF.DEF.DEF prove that �ABC � �DEF. 2. Explain why the SSS and SAS 2. Explain why the SSS and SAS � � Postulates are shortcuts for Postulates are shortcuts for proving triangles congruent. proving triangles congruent. 3. GET ORGANIZED Copy and 3. GET ORGANIZED Copy and complete the graphic organizer. complete the graphic organizer. Use it to compare the SSS and Use it to compare the SSS and SAS postulates. SAS postulates. � � � � � �� 4-4 4-4 Exercises Exercises Exercises Exercises Exercises Exercises Exercises KEYWORD: MG7 4-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In �RST which angle is the included angle of sides −− ST and −− TR Use SSS to explain why the triangles in each pair are congruent. p. 242 2. �ABD � �CDB 3. �MNP � �MQP � � � � � � � � . Design This Texas flag consists of a blue, p. 243 perpendicular stripe with a white star in the center. The star consists of five triangles. GJ = LG = 20 in., and GK = GH = 13 in. Use SAS to explain why �JGK � �LGH Show that the triangles are congruent for the given value of the variable. p. 244 5. �GHJ � �IHJ, x = 4 6. �RST � �TUR, x = 18 � � �� Study the examples to apply new concepts and skills. Examples include stepped out solutions. Test your understanding of examples by trying the Check It Out problems. Check your work in the Selected Answers. If you get stuck, use the internet for Homework Help Online. 4-4 Triangle Congruence: SSS and SAS 245 245 For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary acute triangle acute triangle . . . . . . . . . . . . . . 216 CPCTC . . . . . . . . . . . . . . . . . . . . . 260 isosceles triangle . . . . . . . . . . . 217 auxiliary line auxiliary line . . . . . . . . . . . . . . . 223 equiangular triangle . . . . . . . . 216 legs of an isosceles triangle . . 273 base base . . . . . . . . . . . . . . . . . . . . . . . 273 equilateral triangle . . . . . . . . . 217 obtuse triangle . . . . . . . . . . . . . 216 base angle base angle . . . . . . . . . . . . . . . . . . 273 exterior . . . . . . . . . . . . . . . . . . . . 225 remote interior angle . . . . . . . 225 congruent polygons congruent polygons . . . . . . . . . 231 exterior angle . . . . . . . . . . . . . . 225 right triangle . . . . . . . . . . . . . . . 216 coordinate proof coordinate proof . . . . . . . . . . . . 267 included angle . . . . . . . . . . . . . . 242 scalene triangle . . . . . . . . . . . . . 217 corollary corollary . . . . . . . . . . . . . . . . . . . 224 included side . . . . . . . . . . . . . . . 252 triangle rigidity . . . . . . . . . . . . . 242 corresponding angles corresponding angles . . . . . . . 231 interior . . . . . . . . . . . . . . . . . . . . 225 vertex angle . . . . . . . . . . . . . . . . 273 corresponding sides corresponding sides . . . . . . . . . 231 interior angle . . . . . . . . . . . . . . . 225 Complete the sentences below with vocabulary words from the list above. Complete the sentences below with vocabulary words from the list above. 1. A(n) 1. A(n) ? is a triangle with at least two congruent sides. −−−− 2. A name given to matching angles of congruent triangles is ? . −−−− 3. A(n) ? is the common side of two consecutive angles in a polygon. −−−− 4-1 Classifying Triangles (pp. 216–221) TEKS G.1.A TEKS G.1.A E X A M P L E EXERCISES � Classify t
he triangle by its angle measures and side lengths. isosceles right triangle Classify each triangle by its angle measures and side lengths. 4. 5. �� 4-2 Angle Relationships in Triangles (pp. 223–230) TEKS G.1.A, G.2.B TEKS G.1.A, G.2.B E X A M P L E � Find m∠S. 12x = 3x + 42 + 6x �� 12x = 9x + 42 �� 3x = 42 x = 14 m∠S = 6(14) = 84° EXERCISES Find m∠N. 6. � �� 284 284 Chapter 4 Triangle Congruence 7. In�LMN, m∠L = 8x °, m∠M = (2x + 1)°, and m∠N = (6x - 1)°. Use the list on p. S82 to review the postulates and theorems found in the chapter. Test yourself with practice problems from every lesson in the chapter. TOOLS OF GEOMETRY In geometry, it is important to use tools correctly in order to measure accurately and produce accurate figures. One important tool is your pencil. Always use a sharp pencil with a good eraser. Ruler The ruler shown has a mark every 1 __ 8 inch, so the accuracy is to the nearest 1 __ 8 inch. Protractor To use a protractor to measure an angle, you may need to extend the sides of the angle. For acute angles, use the smaller measurement. For obtuse angles, use the larger measurement. � � � � � � � � � � � �� Line up one ray with 0. � � � � �� Line up one end with 0, not the edge. Choose the measurement that is the closest. Place the center of your protractor on the vertex. Compass A compass is used to draw arcs and circles. If you have trouble keeping the point in place, try keeping the compass still and turning the paper. Straightedge A straightedge is used to draw a line through two points. If you use a ruler as a straightedge, do not use the marks on the ruler. Keep your wrist flexible. Turn the compass with your index finger and thumb. Tilt the compass slightly. First place your pencil on one points. Place the straightedge against your pencil and the other point. Draw the line. � � � � � � � � � � � � � � � � � � � �� Geometry Software Geometry software can be used to create figures and explore their properties. Use the toolbar to select, draw; and label figures. Drag points to explore properties of a figure. Use the menus to construct, transform, and measure figures. The parts of each figure are linked. To avoid deleting a whole figure, hide parts instead of deleting them. Tools of Geometry xxixxi �� Scavenger H Use this scavenger hunt to discover a few of the many tools in the Texas Edition of Holt Geometry that you can use to become an independent learner. On a separate sheet of paper, write the answers to each question below. Within each answer, one letter will be in a yellow box. After you have answered every question, identify the letters that would be in yellow boxes and rearrange them to reveal the answer to the question at the bottom of the page. What theorem are you asked about in the Know-It Note on page 352? What keyword should you enter for Homework Help for Lesson 3-3? What is the first Vocabulary term in the Study Guide: Preview for Chapter 1? In Lesson 8-2, what is Example 4 teaching you to find? 1. ■ ■ ■ ■ ■ 2. ■ ■ ■ ■ ■ ■ 3. ■ ■ ■ ■ ■ ■ ■ 4. ■ ■ ■ ■ ■ ■ 8. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ In the Study Guide: Review for Lesson 11-1, what do the lines intersect? Whose job is described in the Career Path on page 612? What advice does Chapter 1’s TAKS Tackler give about how to answer a multiple choice test item you don’t know how to solve? What mathematician is featured in the Math History link on page 318? What did the little acorn say when it grew up? ■ ■ ■ ■ ■ ■ ■ ■ xxii xxii Scavenger Hunt You can practice using the four-step Problem Solving Plan to solve problems in the Problem Solving on Location feature located at the end of selected chapters. Each page focuses on interesting people, and facts from the Lone Star State. You can follow the flight path of the hot air balloons in the Great Texas Balloon Race that starts in Longview. �� Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List The Great Texas Balloon Race The Great Texas Balloon Race The Great Texas Balloon Race The annual Great Texas Balloon Race is one of the most The annual Great Texas Balloon Race is one of the most The annual Great Texas Balloon Race is one of the most exciting hot air balloon events in Texas. “Balloon Glow,” exciting hot air balloon events in Texas. “Balloon Glow,” exciting hot air balloon events in Texas. “Balloon Glow,” exciting hot air balloon events in Texas. “Balloon Glow,” in which balloons are tethered and illuminated in an in which balloons are tethered and illuminated in an in which balloons are tethered and illuminated in an evening display, was begun in Longview, the race’s evening display, was begun in Longview, the race’s evening display, was begun in Longview, the race’s starting point, in 1980. Traditionally held in July, the starting point, in 1980. Traditionally held in July, the starting point, in 1980. Traditionally held in July, the race attracts balloonists who compete to fly the race attracts balloonists who compete to fly the race attracts balloonists who compete to fly the obstacle course the most accurately. obstacle course the most accurately. Choose one or more strategies to solve each problem. Choose one or more strategies to solve each problem. 1. The event starts in Longview, and ends near Estes, 1. The event starts in Longview, and ends near Estes, Texas. The balloons do not fly from the start to the Texas. The balloons do not fly from the start to the finish in a straight line. They follow a zigzag course finish in a straight line. They follow a zigzag course finish in a straight line. They follow a zigzag course finish in a straight line. They follow a zigzag course to take advantage of the wind. Suppose one of the to take advantage of the wind. Suppose one of the to take advantage of the wind. Suppose one of the balloons leaves Longview at a bearing of N 50° E balloons leaves Longview at a bearing of N 50° E and follows the course shown. At what bearing does the balloon approach Estes? � � �� 2. The speed of the balloon depends on the current wind speed. One event in The Great Texas Balloon Race requires the balloonist to fly to a pole that is 2 mi from the starting point. The balloonist must drop a small ring around the pole, which is 20 ft tall. A second target is 1 mi from the first, a third target is another 3 mi from the second, and a final target is 5 mi farther. If the wind speed is 3.5 mi/h, how long will it take the balloonist to finish the course? Round to the nearest hundredth 3. During the race, one of the balloons leaves Longview L, flies to X, and then flies to Y. The team discovers a problem with the balloon, so it must return directly to Longview. Does the table contain enough information to determine the return course to L? Explain. of an hour. � � � � L to X X to Y Y to L Bearing N 42° E S 59° E Distance (mi) 3.1 2.4 Problem Solving on Location 295 295 T E X A S TAKS Grades 9–11 Obj. 10 Port Isabel Point Isabel Lighthouse Point Isabel Lighthouse The Point Isabel Lighthouse was built in 1853 on a prominent The Point Isabel Lighthouse was built in 1853 on a prominent bluff on the mainland. Today, the fully restored lighthouse is bluff on the mainland. Today, the fully restored lighthouse is the only one in Texas that is open for climbing and viewing. the only one in Texas that is open for climbing and viewing. Choose one or more strategies to solve each problem. Choose one or more strategies to solve each problem. 1. Suppose the beam from the lighthouse is visible for up to 1. Suppose the beam from the lighthouse is visible for up to 15 miles at sea. To the nearest square mile, what is the area of water covered by the beam as it rotates by an angle of 60°? of water covered by the beam as it rotates by an angle of 60°? 2. Given that Earth’s radius is approximately 4000 miles 2. Given that Earth’s radius is approximately 4000 miles and that the top of the tower of a lighthouse is 65 ft above and that the top of the tower of a lighthouse is 65 ft above sea level, find the distance from the top of the tower to sea level, find the distance from the top of the tower to the horizon. Round your answer to the nearest mile. the horizon. Round your answer to the nearest mile. (Hint: (Hint: 65 feet = 0.01 miles) For 3, use the table. For 3, use the table. 3. Most lighthouses use Fresnel lenses, named after their 3. Most lighthouses use inventor, Augustine Fresnel. The table shows the sizes, or inventor, Augustine Fresnel. The table shows the sizes, or orders orders, of the circular lenses. The diagram shows some measurements of the Fresnal lens used in the Point Isabel measurements of the Fresnal lens used in the Point Isabel Lighthouse. What is the order of the lens? Lighthouse. What is the order of the lens? Fresnel Lenses Order �� Lens Diameter 6 ft 1 in. 4 ft 7 in. 3 ft 3 in. 1 ft 8 in. 1 ft 3 in. 1 ft 0 in. First Second Third Fourth Fifth Sixth 894 894 Chapter 12 Extending Transformational Geometry 11 4 1 Learn about the sizes and order of lighthouse lenses, such as the one at the Point Isabel Lighthouse. Take a tour of the show caves that are located in the Texas Hill Country between San Antonio and Austin Te Te Ta Ve To ow n a s t h e Te ow ow ow c a v e i n Te Ja 5_ Ja ow Te x a s s h ow ’’t I f y o u d o n’ Te ? �� Problem Solving on Location xxixxxix Foundations for Geometry 1A Euclidean and Construction Tools 1-1 Understanding Points, Lines, and Planes Lab Explore Properties Associated with Points 1-2 Measuring and Co
nstructing Segments 1-3 Measuring and Constructing Angles 1-4 Pairs of Angles 1B Coordinate and Transformation Tools 1-5 Using Formulas in Geometry 1-6 Midpoint and Distance in the Coordinate Plane 1-7 Transformations in the Coordinate Plane Lab Explore Transformations KEYWORD: MG7 ChProj The lights encasing the geodesic dome at the top of Reunion Tower are a familiar feature of the Dallas skyline. 2 2 Chapter 1 Vocabulary Match each term on the left with a definition on the right. 1. coordinate A. a mathematical phrase that contains operations, numbers, 2. metric system of measurement 3. expression 4. order of operations and/or variables B. the measurement system often used in the United States C. one of the numbers of an ordered pair that locates a point on a coordinate graph D. a list of rules for evaluating expressions E. a decimal system of weights and measures that is used universally in science and commonly throughout the world Measure with Customary and Metric Units For each object tell which is the better measurement. 5. length of an unsharpened pencil 2 in. or 9 3__ 7 1__ 4 in. 7. length of a soccer field 100 yd or 40 yd 9. height of a student’s desk 30 in. or 4 ft Combine Like Terms Simplify each expression. 11. -y + 3y - 6y + 12y 13. -5 - 9 - 7x + 6x 1 m or 2 1__ 6. the diameter of a quarter 2 cm 8. height of a classroom 5 ft or 10 ft 10. length of a dollar bill 15.6 cm or 35.5 cm 12. 63 + 2x - 7 - 4x 14. 24 - 3 + y + 7 Evaluate Expressions Evaluate each expression for the given value of the variable. 15. x + 3x + 7x for x = -5 16. 5p + 10 for p = 78 17. 2a - 8a for a = 12 18. 3n - 3 for n = 16 Ordered Pairs Write the ordered pair for each point. 19. A 20. B 21. C 23. E 22. D 24. F Foundations for Geometry 3 3 �� Key Vocabulary/Vocabulario angle area ángulo área coordinate plane plano cartesiano line perimeter plane point línea perímetro plano punto transformation transformación undefined term término indefinido Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. A definition is a statement that gives the meaning of a word or phrase. What do you think the phrase undefined term means? 2. Coordinates are numbers used to describe a location. A plane is a flat surface. How can you use these meanings to understand the term coordinate plane ? 3. A point is often represented by a dot. What real-world items could represent points? 4. Trans- is a prefix that means “across,” as in movement. A form is a shape. How can you use these meanings to understand the term transformation ? Geometry TEKS G.1.A Geometric structure* develop an awareness of the structure of a mathematical system, connecting definitions, postulates ... 1-2 Tech. Lab Les. 1-1 ★ Les. 1-2 Les. 1-3 Les. 1-4 Les. 1-5 Les. 1-6 Les. 1-7 ★ ★ ★ ★ ★ 1-7 Tech. Lab G.1.B Geometric structure* recognize the historical development of ★ ★ geometric systems and know mathematics is developed for a variety of purposes G.2.A Geometric structure* use constructions to explore attributes ★ ★ ★ of geometric figures ... G.2.B Geometric structure* make conjectures about angles, lines ... ★ ★ ★ ★ and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational ... G.3.B Geometric structure* construct and justify statements about ★ ★ ★ geometric figures and their properties G.5.C Geometric patterns* use properties of transformations ... to make connections between mathematics and the real world ... G.7.A Dimensionality and the geometry of location* use one- and ★ two-dimensional coordinate systems to represent points, lines, rays, line segments ... G.7.C Dimensionality and the geometry of location* develop and use formulas involving length, slope, and midpoint ★ G.8.A Congruence and the geometry of size* find areas of regular ★ polygons, circles ... G.8.C Congruence and the geometry of size* derive, extend, and use the Pythagorean Theorem ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 4 4 Chapter 1 Reading Strategy: Use Your Book for Success Understanding how your textbook is organized will help you locate and use helpful information. As you read through an example problem, pay attention to the notes in the margin. These notes highlight key information about the concept and will help you to avoid common mistakes. The Glossary is found in the back of your textbook. Use it when you need a definition of an unfamiliar word or phrase. The Index is located at the end of your textbook. If you need to locate the page where a particular concept is explained, use the Index to find the corresponding page number. The Skills Bank is located in the back of your textbook. Look in the Skills Bank for help with math topics that were taught in previous courses, such as the order of operations. Try This Use your textbook for the following problems. 1. Use the index to find the page where right angle is defined. 2. What formula does the Know-It Note on the first page of Lesson 1-6 refer to? 3. Use the glossary to find the definition of congruent segments. 4. In what part of the textbook can you find help for solving equations? Foundations for Geometry 5 5 1-1 Understanding Points, Lines, and Planes TEKS G.7.A Dimensionality and the geometry of location: use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments .... Also G.1.A Objectives Identify, name, and draw points, lines, segments, rays, and planes. Apply basic facts about points, lines, and planes. Who uses this? Architects use representations of points, lines, and planes to create designs of buildings. Light, color, and geometric shapes are reflected in the design of the Central Library in San Antonio, Texas. Vocabulary undefined term point line plane collinear coplanar segment endpoint ray opposite rays postulate The most basic figures in geometry are undefined terms , which cannot be defined by using other figures. The undefined terms point, line, and plane are the building blocks of geometry. Undefined Terms TERM NAME DIAGRAM A point names a location and has no size. It is represented by a dot. A capital letter point P A line is a straight path that has no thickness and extends forever. A lowercase letter or two points on the line line ℓ,   XY or   YX A plane is a flat surface that has no thickness and extends forever. A script capital letter or three points not on a line plane R or plane ABC Points that lie on the same line are collinear . K, L, and M are collinear. K, L, and N are noncollinear. Points that lie in the same plane are coplanar . Otherwise they are noncoplanar. E X A M P L E 1 Naming Points, Lines, and Planes A plane may be named by any three noncollinear points on that plane. Plane ABC may also be named BCA, CAB, CBA, ACB, or BAC. Refer to the architectural design of the Central Library building. A Name four coplanar points. K, L, M, and N all lie in plane R. B Name three lines.   AB ,   BC , and   CA . 1. Use the diagram to name two planes Chapter 1 Foundations for Geometry �� Segments and Rays DEFINITION NAME DIAGRAM A segment , or line segment, is the part of a line consisting of two points and all points between them. The two endpoints ̶̶ BA ̶̶ AB or An endpoint is a point at one end of a segment or the starting point of a ray. A capital letter C and D A ray is a part of a line that starts at an endpoint and extends forever in one direction. Its endpoint and any other point on the ray  RS Opposite rays are two rays that have a common endpoint and form a line. The common endpoint and any other point on each ray  EF and  EG E X A M P L E 2 Drawing Segments and Rays Draw and label each of the following. A a segment with endpoints U and V B opposite rays with a common endpoint Q 2. Draw and label a ray with endpoint M that contains N. A postulate , or axiom, is a statement that is accepted as true without proof. Postulates about points, lines, and planes help describe geometric properties. Postulates Points, Lines, and Planes 1-1-1 Through any two points there is exactly one line. 1-1-2 Through any three noncollinear points there is exactly one plane containing them. 1-1-3 If two points lie in a plane, then the line containing those points lies in the plane. E X A M P L E 3 Identifying Points and Lines in a Plane Name a line that passes through two points. There is exactly one line n passing through G and H. 3. Name a plane that contains three noncollinear points. 1- 1 Understanding Points, Lines, and Planes 7 7 �� Recall that a system of equations is a set of two or more equations containing two or more of the same variables. The coordinates of the solution of the system satisfy all equations in the system. These coordinates also locate the point where all the graphs of the equations in the system intersect. An intersection is the set of all points that two or more figures have in common. The next two postulates describe intersections involving lines and planes. Postulates Intersection of Lines and Planes 1-1-4 If two lines intersect, then they intersect in exactly one point. 1-1-5 If two planes intersect, then they intersect in exactly one line. Use a dashed line to show the hidden parts of any figure that you are drawing. A dashed line will indicate the part of the figure that is not seen. E X A M P L E 4 Representing Intersections Sketch a figure that shows each of the following. A A line intersects a plane, but does not lie in the plane. B Two planes intersect in one line. 4. Sketch a figure that shows two lines intersect in one point in a plane, but only one of the lines lies in the plane. THINK AND DISCUSS 1. Explain why any two points are collinear. 2. Which postulate explains the fact that two straight roads cannot cross each other more than once? 3. Explain why points
and lines may be coplanar even when the plane containing them is not drawn. 4. Name all the possible lines, segments, and rays for the points A and B. Then give the maximum number of planes that can be determined by these points. 5. GET ORGANIZED Copy and complete the graphic organizer below. In each box, name, describe, and illustrate one of the undefined terms. 8 8 Chapter 1 Foundations for Geometry �� 1-1 Exercises Exercises KEYWORD: MG7 1-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Give an example from your classroom of three collinear points. 2. Make use of the fact that endpoint is a compound of end and point and name the endpoint of  ST . Use the figure to name each of the following. p. 6 3. five points 4. two lines 5. two planes 6. point on   BD Draw and label each of the following. p. 7 7. a segment with endpoints M and N 8. a ray with endpoint F that passes through Use the figure to name each of the following. p. 7 9. a line that contains A and C 10. a plane that contains A, D, and Sketch a figure that shows each of the following. p. 8 11. three coplanar lines that intersect in a common point 12. two lines that do not intersect PRACTICE AND PROBLEM SOLVING Independent Practice Use the figure to name each of the following. For See Exercises Example 13. three collinear points 13–15 16–17 18–19 20–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S4 Application Practice p. S28 14. four coplanar points 15. a plane containing E Draw and label each of the following. 16. a line containing X and Y 17. a pair of opposite rays that both contain R Use the figure to name each of the following. 18. two points and a line that lie in plane T 19. two planes that contain ℓ Sketch a figure that shows each of the following. 20. a line that intersects two nonintersecting planes 21. three coplanar lines that intersect in three different points 1- 1 Understanding Points, Lines, and Planes 9 9 �� 22. This problem will prepare you for the Multi-Step TAKS Prep on page 34. Name an object at the archaeological site shown that is represented by each of the following. a. a point b. a segment c. a plane Draw each of the following. 23. plane H containing two lines that intersect at M 24.  ST intersecting plane M at R Use the figure to name each of the following. 25. the intersection of TV and US 26. the intersection of 27. the intersection of  US and plane R ̶̶ TU and ̶̶ UV Write the postulate that justifies each statement. 28. The line connecting two dots on a sheet of paper lies on the same sheet of paper as the dots. 29. If two ants are walking in straight lines but in different directions, their paths cannot cross more than once. 30. Critical Thinking Is it possible to draw three points that are noncoplanar? Explain. Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 31. If two planes intersect, they intersect in a straight line. 32. If two lines intersect, they intersect at two different points. 33.  AB is another name for   BA . 34. If two rays share a common endpoint, then they form a line. 35. Art Pointillism is a technique in which tiny dots of complementary colors are combined to form a picture. Which postulate ensures that a line connecting two of these points also lies in the plane containing the points? 36. Probability Three of the labeled points are chosen at random. What is the probability that they are collinear? 37. Campers often use a cooking stove with three legs. Which postulate explains why they might prefer this design to a stove that has four legs? 38. Write About It Explain why three coplanar lines may have zero, one, two, or three points of intersection. Support your answer with a sketch. 10 10 Chapter 1 Foundations for Geometry �� 39. Which of the following is a set of noncollinear points? P, R, T Q, R, S P, Q, R S, T, U 40. What is the greatest number of intersection points four coplanar lines can have? 6 4 2 0 41. Two flat walls meet in the corner of a classroom. Which postulate best describes this situation? Through any three noncollinear points there is exactly one plane. If two points lie in a plane, then the line containing them lies in the plane. If two lines intersect, then they intersect in exactly one point. If two planes intersect, then they intersect in exactly one line. 42. Gridded Response What is the greatest number of planes determined by four noncollinear points? CHALLENGE AND EXTEND Use the table for Exercises 43–45. Figure Number of Points Maximum Number of Segments 2 1 3 3 4 43. What is the maximum number of segments determined by 4 points? 44. Multi-Step Extend the table. What is the maximum number of segments determined by 10 points? 45. Write a formula for the maximum number of segments determined by n points. 46. Critical Thinking Explain how rescue teams could use two of the postulates from this lesson to locate a distress signal. SPIRAL REVIEW 47. The combined age of a mother and her twin daughters is 58 years. The mother was 25 years old when the twins were born. Write and solve an equation to find the age of each of the three people. (Previous course) Determine whether each set of ordered pairs is a function. (Previous course)   ⎬ ⎨ (0, 1) , (1, -1) , (5, -1) , (-1, 2) 48.   Find the mean, median, and mode for each set of data. (Previous course)   ⎬ ⎨ (3, 8) , (10, 6) , (9, 8) , (10, -6) 49.   50. 0, 6, 1, 3, 5, 2, 7, 10 51. 0.47, 0.44, 0.4, 0.46, 0.44 1- 1 Understanding Points, Lines, and Planes 11 11 �� 1-2 Use with Lesson 1-2 Activity Explore Properties Associated with Points The two endpoints of a segment determine its length. Other points on the segment are between the endpoints. Only one of these points is the midpoint of the segment. In this lab, you will use geometry software to measure lengths of segments and explore properties of points on segments. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.2.B, G.3.B KEYWORD: MG7 Lab1 1 Construct a segment and label its endpoints A and C. 2 Create point B on ̶̶ AC . 3 Measure the distances from A to B and from B to C. Use the Calculate tool to calculate the sum of AB and BC. 4 Measure the length of ̶̶ AC . What do you notice about this length compared with the measurements found in Step 3? 5 Drag point B along ̶̶ AC . Drag one of the endpoints ̶̶ AC . What relationships do you think are true of about the three measurements? 6 Construct the midpoint of ̶̶ AC and label it M. 7 Measure ̶̶̶ AM and ̶̶̶ MC . What relationships do you ̶̶̶ AM , and think are true about the lengths of Use the Calculate tool to confirm your findings. ̶̶ AC , ̶̶̶ MC ? 8 How many midpoints of ̶̶ AC exist? Try This 1. Repeat the activity with a new segment. Drag each of the points in your figure (the endpoints, the point on the segment, and the midpoint). Write down any relationships you observe about the measurements. ̶̶ AC . Measure ̶̶ AD , What do you think has to be true about D for the relationship to always be true? ̶̶ AC . Does AD + DC = AC? 2. Create a point D not on ̶̶ DC , and 12 12 Chapter 1 Foundations for Geometry 1-2 Measuring and Constructing Segments TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.7.C Objectives Use length and midpoint of a segment. Construct midpoints and congruent segments. Why learn this? You can measure a segment to calculate the distance between two locations. Maps of a race are used to show the distance between stations on the course. (See Example 4.) Vocabulary coordinate distance length congruent segments construction between midpoint bisect segment bisector A ruler can be used to measure the distance between two points. A point corresponds to one and only one number on the ruler. This number is called a coordinate . The following postulate summarizes this concept. Postulate 1-2-1 Ruler Postulate The points on a line can be put into a one-to-one correspondence with the real numbers. The distance between any two points is the absolute value of the difference of the coordinates. If the coordinates of points A and B are a and b, then the distance between A and B is ⎜a - b⎟ or ⎜b - a⎟ . The distance between A and B is also called the length of ̶̶ AB , or AB. E X A M P L E 1 Finding the Length of a Segment � � �� B EF EF = ⎜-4 - (-1) ⎟ = ⎜-4 + 1⎟ = ⎜-3⎟ = 3 Find each length. A DC DC = ⎜4.5 - 2⎟ = ⎜2.5⎟ = 2.5 Find each length. 1a. XY 1b. XZ _ PQ PQ represents a number, while represents a geometric figure. Be sure to use equality for numbers (PQ = RS) and congruence for _ PQ ≅ figures ( _ RS ). Congruent segments are segments that have the same length. In the diagram, PQ = RS, so you can write is congruent to segment RS.” Tick marks are used in a figure to show congruent segments. _ RS . This is read as “segment PQ _ PQ ≅ 1- 2 Measuring and Constructing Segments 13 13 �� You can make a sketch or measure and draw a segment. These may not be exact. A construction is a way of creating a figure that is more precise. One way to make a geometric construction is to use a compass and straightedge. Construction Congruent Segment Construct a segment congruent to _ AB .    Draw ℓ. Choose a point on ℓ and label it C. Open the compass to distance AB. Place the point of the compass at C and make an arc through ℓ. Find the point where the arc and ℓ intersect and label it D. _ CD ≅ _ AB E X A M P L E 2 Copying a Segment Sketch, draw, and construct a segment congruent to _ MN . Step 1 Estimate and sketch. _ MN and sketch Estimate the length of _ PQ approximately the same length. Step 2 Measure and draw. Use a ruler to measure _ MN . MN appears _ XY t
o to be 3.1 cm. Use a ruler and draw have length 3.1 cm. Step 3 Construct and compare. Use a compass and straightedge to construct _ ST congruent to _ MN . _ PQ and _ XY are A ruler shows that approximately the same length as _ ST is precisely the same length. but _ MN , 2. Sketch, draw, and construct a segment congruent to _ JK . In order for you to say that a point B is between two points A and C, all three of the points must lie on the same line, and AB + BC = AC. Postulate 1-2-2 Segment Addition Postulate If B is between A and C, then AB + BC = AC. 14 14 Chapter 1 Foundations for Geometry �� E X A M P L E 3 Using the Segment Addition Postulate A B is between A and C, AC = 14, and BC = 11.4. Find AB. AC = AB + BC 14 = AB + 11.4 - 11.4 ̶̶̶̶̶ - 11.4 ̶̶̶̶̶̶̶ 2.6 = AB Seg. Add. Post. Substitute 14 for AC and 11.4 for BC. Subtract 11.4 from both sides. Simplify. B S is between R and T. Find RT. - 2x ̶̶̶̶̶̶̶ RT = RS + ST 4x = (2x + 7) + 28 4x = 2x + 35 - 2x ̶̶̶̶ 2x = 35 = 35 _ 2x _ 2 2 x = 35 _ 2 RT = 4x , or 17.5 Seg. Add. Post. Substitute the given values. Simplify. Subtract 2x from both sides. Simplify. Divide both sides by 2. Simplify. = 4 (17.5) = 70 Substitute 17.5 for x. 3a. Y is between X and Z, XZ = 3, and XY = 1 1 __ 3 3b. E is between D and F. Find DF. . Find YZ. _ AB is the point that bisects , or divides, the segment into The midpoint M of two congruent segments. If M is the midpoint of AM = MB. So if AB = 6, then AM = 3 and MB = 3. _ AB , then E X A M P L E 4 Recreation Application The map shows the route for a race. You are 365 m from drink station R and 2 km from drink station S. The first-aid station is located at the midpoint of the two drink stations. How far are you from the first-aid station? Let your current location be X and the location of the first-aid station be Y. XR + RS = XS 365 + RS = 2000 - 365 ̶̶̶̶̶ RS = 1635 RY = 817.5 - 365 ̶̶̶̶̶̶̶̶ Seg. Add. Post. Substitute 365 for XR and 2000 for XS. Subtract 365 from both sides. Simplify. Y is the mdpt. of _ RS , so RY = 1 __ RS. 2 XY = XR + RY = 365 + 817.5 = 1182.5 m Substitute 365 for XR and 817.5 for RY. You are 1182.5 m from the first-aid station. 4. What is the distance to a drink station located at the midpoint between your current location and the first-aid station? 1- 2 Measuring and Constructing Segments 15 15 XR365 mYS2 kmKaren Minot(415)883-6560Final art file 11/18/04Marathon RouteHolt Rinehart WinstonGeometry SE 2007 Texasge07sec01l02002a�� A segment bisector is any ray, segment, or line that intersects a segment at its midpoint. It divides the segment into two equal parts at its midpoint. Construction Segment Bisector    Draw _ XY on a sheet of paper. Fold the paper so that Y is on top of X. Unfold the paper. The line represented by the crease bisects _ XY . Label the midpoint M. XM = MY E X A M P L E 5 Using Midpoints to Find Lengths B is the midpoint of ̶̶ AC , AB = 5x, and BC = 3x + 4. Find AB, BC, and AC. Step 1 Solve for x. AB = BC 5x = 3x + 4 - 3x ̶̶̶̶ 2x = 4 = 4 _ 2x _ 2 2 x = 2 - 3x ̶̶̶̶̶̶ B is the mdpt. of _ AC . Substitute 5x for AB and 3x + 4 for BC. Subtract 3x from both sides. Simplify. Divide both sides by 2. Simplify. Step 2 Find AB, BC, and AC. AB = 5x = 5 (2) = 10 BC = 3x + 4 AC = AB + BC = 3 (2) + 4 = 10 = 10 + 10 = 20 5. S is the midpoint of RT, RS = -2x, and ST = -3x - 2. Find RS, ST, and RT. THINK AND DISCUSS 1. Suppose R is the midpoint of _ ST . Explain how SR and ST are related. 2. GET ORGANIZED Copy and complete the graphic organizer. Make a sketch and write an equation to describe each relationship. 16 16 Chapter 1 Foundations for Geometry �� 1-2 Exercises Exercises KEYWORD: MG7 1-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. _ XY into two equal parts. Name a pair of congruent _ XY at M and divides 1. Line ℓ bisects segments. 2. __?__ is the amount of space between two points on a line. It is always expressed as a nonnegative number. (distance or midpoint Find each length. p. 13 3. AB 4. BC . 14 5. Sketch, draw, and construct a segment congruent to _ RS . . B is between A and C, AC = 15.8, and AB = 9.9. Find BC. p. 15 7. Find MP. 15 8. Travel If a picnic area is located at the midpoint between Lubbock and Amarillo, find the distance to the picnic area from the sign. 16 9. Multi-Step K is the midpoint of and JK = 7. Find x, KL, and JL. _ JL , JL = 4x - 2, 10. E bisects _ DF , DE = 2y, and EF = 8y - 3. Find DE, EF, and DF. Independent Practice For See Exercises Example 11–12 13 14–15 16 17–18 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S4 Application Practice p. S28 PRACTICE AND PROBLEM SOLVING Find each length. 11. DB 12. CD 13. Sketch, draw, and construct a segment twice the length of _ AB . 14. D is between C and E, CE = 17.1, and DE = 8. Find CD. 15. Find MN. 16. Sports During a football game, a quarterback standing at the 9-yard line passes the ball to a receiver at the 24-yard line. The receiver then runs with the ball halfway to the 50-yard line. How many total yards (passing plus running) did the team gain on the play? 17. Multi-Step E is the midpoint of _ DF , DE = 2x + 4, and EF = 3x - 1. Find DE, EF, and DF. _ PR , PQ = 3y, and PR = 42. Find y and QR. 18. Q bisects 1- 2 Measuring and Constructing Segments 17 17 �� 19. This problem will prepare you for the Multi-Step TAKS Prep on page 34. Archaeologists at Valley Forge were eager to find what remained of the winter camp that soldiers led by George Washington called home for several months. The diagram represents one of the restored log cabins. a. How is C related to b. If AC = 7 ft, EF = 2 (AC) + 2, and AB = 2 (EF) - 16, _ AE ? what are AB and EF? Use the diagram for Exercises 20–23. 20. GD = 4 2 _ 3 _ _ CD ≅ DF , E bisects . Find GH. 21. _ DF , and CD = 14.2. Find EF. 22. GH = 4x - 1, and DH = 8. Find x. _ GH bisects _ CF , CF = 2y - 2, and CD = 3y - 11. Find CD. 23. Tell whether each statement is sometimes, always, or never true. Support each of your answers with a sketch. 24. Two segments that have the same length must be congruent. 25. If M is between A and B, then M bisects _ AB . 26. If Y is between X and Z, then X, Y, and Z are collinear. 27. /////ERROR ANALYSIS///// Below are two statements about the midpoint of _ AB . Which is incorrect? Explain the error. 28. Carpentry A carpenter has a wooden dowel that is 72 cm long. She wants to cut it into two pieces so that one piece is 5 times as long as the other. What are the lengths of the two pieces? 29. The coordinate of M is 2.5, and MN = 4. What are the possible coordinates for N? 30. Draw three collinear points where E is between D and F. Then write an equation using these points and the Segment Addition Postulate. Suppose S is between R and T. Use the Segment Addition Postulate to solve for each variable. 31. RS = 7y - 4 ST = y + 5 RT = 28 32. RS = 3x + 1 ST = 1 _ 2 RT = 18 x + 3 33. RS = 2z + 6 ST = 4z - 3 RT = 5z + 12 34. Write About It In the diagram, B is not between A and C. Explain. 35. Construction Use a compass and straightedge to construct a segment whose length is AB + CD. 18 18 Chapter 1 Foundations for Geometry �� 36. Q is between P and R. S is between Q and R, and R is between Q and T. PT = 34, QR = 8, and PQ = SQ = SR. What is the length of 18 10 9 _ RT ? 37. C is the midpoint of What is the length of _ AD ? _ AD . B is the midpoint of _ AC . BC = 12. 12 24 36 38. Which expression correctly states that XY ≅ VW _ XY ≅ _ VW _ XY is congruent to _ VW _ XY = _ VW ? 22 48 XY = VW 39. A, B, C, D, and E are collinear points. AE = 34, BD = 16, and AB = BC = CD. What is the length of _ CE ? 10 16 18 24 CHALLENGE AND EXTEND 40. 40. HJ is twice JK. J is between H and K. If HJ = 4x and HK = 78, find JK. 41. 41. A, D, N, and X are collinear points. D is between N and A. NA + AX = NX. Draw a diagram that represents this information. Sports Use the following information for Exercises 42 and 43. The table shows regulation distances between hurdles in women’s and men’s races. In both the women’s and men’s events, the race consists of a straight track with 10 equally spaced hurdles. Distance of Race 100 m 110 m Distance from Start to First Hurdle 13.00 m 13.72 m Distance Between Hurdles 8.50 m 9.14 m Event Women’s Men’s Distance from Last Hurdle to Finish 42. Find the distance from the last hurdle to the finish line for the women’s race. 43. Find the distance from the last hurdle to the finish line for the men’s race. 44. Critical Thinking Given that J, K, and L are collinear and that K is between J and L, is it possible that JK = JL? If so, draw an example. If not, explain. Sports Joanna Hayes, of the United States, clears a hurdle on her way to winning the gold medal in the women’s 100 m hurdles during the 2004 Olympic Games. SPIRAL REVIEW Evaluate each expression. (Previous course) 45. ⎜20 - 8⎟ 46. ⎜-9 + 23⎟ 47. - ⎜4 - 27⎟ Simplify each expression. (Previous course) 48. 8a - 3 (4 + a) - 10 49. x + 2 (5 - 2x) - (4 + 5x) Use the figure to name each of the following. (Lesson 1-1) 50. two lines that contain B 51. two segments containing D 52. three collinear points 53. a ray with endpoint C 1- 2 Measuring and Constructing Segments 19 19 �� 1-3 Measuring and Constructing Angles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.1.A, G.1.B, G.2.A, G.2.B Objectives Name and classify angles. Measure and construct angles and angle bisectors. Who uses this? Surveyors use angles to help them measure and map the ear
th’s surface. (See Exercise 27.) Vocabulary angle vertex interior of an angle exterior of an angle measure degree acute angle right angle obtuse angle straight angle congruent angles angle bisector A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a surveyor can measure the angle formed by his or her location and two distant points. An angle is a figure formed by two rays, or sides, with a common endpoint called the vertex (plural: vertices). You can name an angle several ways: by its vertex, by a point on each ray and the vertex, or by a number. The set of all points between the sides of the angle is the interior of an angle . The exterior of an angle is the set of all points outside the angle. Angle Name ∠R, ∠SRT, ∠TRS, or ∠1 You cannot name an angle just by its vertex if the point is the vertex of more than one angle. In this case, you must use all three points to name the angle, and the middle point is always the vertex. E X A M P L E 1 Naming Angles A surveyor recorded the angles formed by a transit (point T) and three distant points, Q, R, and S. Name three of the angles. ∠QTR, ∠QTS, and ∠RTS 1. Write the different ways you can name the angles in the diagram. The measure of an angle is usually given in degrees. Since there are 360° in a circle, one degree is 1 ___ 360 of a circle. When you use a protractor to measure angles, you are applying the following postulate. Postulate 1-3-1 ___ ___ ‹ › ‹ › Given AB , all rays that can be drawn from O can be put into AB and a point O on a one-to-one correspondence with the real numbers from 0 to 180. Protractor Postulate 20 20 Chapter 1 Foundations for Geometry �� Using a Protractor Most protractors have two sets of numbers around the edge. When I measure an angle and need to know which number to use, I first ask myself whether the angle is acute, right, or obtuse. For example, ∠RST looks like it is obtuse, so I know its measure must be 110°, not 70°. � � � José Muñoz Lincoln High School You can use the Protractor Postulate to help you classify angles by their measure. The measure of an angle is the absolute value of the difference of the real numbers that the rays correspond   OC corresponds with on a protractor. If with c and m∠DOC = ⎜d - c⎟ or ⎜c - d⎟ .   OD corresponds with d, Types of Angles Acute Angle Right Angle Obtuse Angle Straight Angle Measures greater than 0° and less than 90° Measures 90° Measures greater than 90° and less than 180° Formed by two opposite rays and meaures 180° E X A M P L E 2 Measuring and Classifying Angles Find the measure of each angle. Then classify each as acute, right, or obtuse. A ∠AOD m∠AOD = 165° ∠AOD is obtuse. B ∠COD m∠COD = ⎜165 - 75⎟ = 90° ∠COD is a right angle. Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse. 2a. ∠BOA 2b. ∠DOB 2c. ∠EOC 1- 3 Measuring and Constructing Angles 21 21 �� Congruent angles are angles that have the same measure. In the diagram, m∠ABC = m∠DEF, so you can write ∠ABC ≅ ∠DEF. This is read as “angle ABC is congruent to angle DEF.” Arc marks are used to show that the two angles are congruent. Construction Congruent Angle Construct an angle congruent to ∠A.      Use a straightedge to draw a ray with endpoint D. Place the compass point at A and draw an arc that intersects both sides of ∠A. Label the intersection points B and C. Using the same compass setting, place the compass point at D and draw an arc that intersects the ray. Label the intersection E. Place the compass point at B and open it to the distance BC. Place the point of the compass at E and draw an arc. Label its intersection with the first arc F. Use a straightedge to   DF . draw ∠D ≅ ∠A The Angle Addition Postulate is very similar to the Segment Addition Postulate that you learned in the previous lesson. Postulate 1-3-2 Angle Addition Postulate If S is in the interior of ∠PQR, then m∠PQS + m∠SQR = m∠PQR. (∠ Add. Post.) E X A M P L E 3 Using the Angle Addition Postulate m∠ABD = 37° and m∠ABC = 84°. Find m∠DBC. m∠ABC = m∠ABD + m∠DBC ∠ Add. Post. 84° = 37° + m∠DBC - 37 ̶̶̶̶ - 37 ̶̶̶̶̶̶̶̶̶̶̶ 47° = m∠DBC Substitute the given values. Subtract 37 from both sides. Simplify. 3. m∠XWZ= 121° and m∠XWY = 59°. Find m∠YWZ. 22 22 Chapter 1 Foundations for Geometry �� An angle bisector is a ray that divides an angle into two   JK bisects ∠LJM; thus ∠LJK ≅ ∠KJM. congruent angles. Construction Angle Bisector Construct the bisector of ∠A.    Use a straightedge to draw ___ › AD bisects ∠A. ___ › AD . Place the point of the compass at A and draw an arc. Label its points of intersection with ∠A as B and C. Without changing the compass setting, draw intersecting arcs from B and C. Label the intersection of the arcs as D. E X A M P L E 4 Finding the Measure of an Angle   BD bisects ∠ABC, m∠ABD = (6x + 3) °, and m∠DBC = (8x - 7) °. Find m∠ABD. Step 1 Find x. m∠ABD = m∠DBC (6x + 3) ° = (8x - 7) ° + 7 + 7 ̶̶̶̶̶̶ ̶̶̶̶̶̶̶ 6x + 10 = 8x - 6x ̶̶̶̶̶̶̶ 10 = 2x = 2x _ 10 _ 2 2 5 = x - 6x ̶̶̶̶̶̶ Step 2 Find m∠ABD. m∠ABD = 6x + 3 Def. of ∠ bisector Substitute the given values. Add 7 to both sides. Simplify. Subtract 6x from both sides. Simplify. Divide both sides by 2. Simplify. = 6 (5) + 3 = 33° Substitute 5 for x. Simplify. Find the measure of each angle. 4a. 4b. ___ › QS bisects ∠PQR, m∠PQS = (5y - 1) °, and m∠PQR = (8y + 12) °. Find m∠PQS. __ › JK bisects ∠LJM, m∠LJK = (-10x + 3) °, and m∠KJM = (-x + 21) °. Find m∠LJM. 1- 3 Measuring and Constructing Angles 23 23 �� THINK AND DISCUSS 1. Explain why any two right angles are congruent. ___ › BD bisects ∠ABC. How are m∠ABC, m∠ABD, and m∠DBC related? 2. 3. GET ORGANIZED Copy and complete the graphic organizer. In the cells sketch, measure, and name an example of each angle type. 1-3 Exercises Exercises KEYWORD: MG7 1-3 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. ∠A is an acute angle. ∠O is an obtuse angle. ∠R is a right angle. Put ∠A, ∠O, and ∠R in order from least to greatest by measure. 2. Which point is the vertex of ∠BCD? Which rays form � the sides of ∠BCD. 20 3. Music Musicians use a metronome to keep time as they play. The metronome’s needle swings back and forth in a fixed amount of time. Name all of the angles in the diagram. 21 Use the protractor to find the measure of each angle. Then classify each as acute, right, or obtuse. � � � � � 4. ∠VXW 5. ∠TXW 6. ∠RXU . 22 L is in the interior of ∠JKM. Find each of the following. 7. m∠JKM if m∠JKL = 42° and m∠LKM = 28° 8. m∠LKM if m∠JKL = 56.4° and m∠JKM = 82.5. 23 Multi-Step   BD bisects ∠ABC. Find each of the following. 9. m∠ABD if m∠ABD = (6x + 4) ° and m∠DBC = (8x - 4) ° 10. m∠ABC if m∠ABD = (5y - 3) ° and m∠DBC = (3y + 15) ° 24 24 Chapter 1 Foundations for Geometry �� Independent Practice For See Exercises Example 11 12–14 15–16 17–18 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S4 Application Practice p. S28 PRACTICE AND PROBLEM SOLVING 11. Physics Pendulum clocks have been used since 1656 to keep time. The pendulum swings back and forth once or twice per second. Name all of the angles in the diagram. � � � � � � Use the protractor to find the measure of each angle. Then classify each as acute, right, or obtuse. 12. ∠CGE 13. ∠BGD 14. ∠AGB T is in the interior of ∠RSU. Find each of the following. 15. m∠RSU if m∠RST = 38° and m∠TSU = 28.6° 16. m∠RST if m∠TSU = 46.7° and m∠RSU = 83.5°   SP bisects ∠RST. Find each of the following. Multi-Step 17. m∠RST if m∠RSP= (3x - 2) ° and m∠PST = (9x - 26) ° 18. m∠RSP if m∠RST = 5 __ 2 y ° and m∠PST = (y + 5) ° Estimation Use the following information for Exercises 19–22. Assume the corner of a sheet of paper is a right angle. Use the corner to estimate the measure and classify each angle in the diagram. 19. ∠BOA 21. ∠EOD 20. ∠COA 22. ∠EOB Use a protractor to draw an angle with each of the following measures. 23. 33° 24. 142° 25. 90° 26. 168° 27. Surveying A surveyor at point S discovers that the angle between peaks A and B is 3 times as large as the angle between peaks B and C. The surveyor knows that ∠ASC is a right angle. Find m∠ASB and m∠BSC. 28. Math History As far back as the 5th century B.C., mathematicians have been fascinated by the problem of trisecting an angle. It is possible to construct an angle with 1 __ 4 the measure of a given angle. Explain how to do this. Find the value of x. 29. m∠AOC = 7x - 2, m∠DOC = 2x + 8, m∠EOD = 27 30. m∠AOB = 4x - 2, m∠BOC = 5x + 10, m∠COD = 3x - 8 31. m∠AOB = 6x + 5, m∠BOC = 4x - 2, m∠AOC = 8x + 21 32. Multi-Step Q is in the interior of right ∠PRS. If m∠PRQ is 4 times as large as m∠QRS, what is m∠PRQ? 1- 3 Measuring and Constructing Angles 25 25 �� 33. This problem will prepare you for the Multi-Step TAKS Prep on page 34. An archaeologist standing at O looks for clues on where to dig for artifacts. a. What value of x will make the angle between the pottery and the arrowhead measure 57°? b. What value of x makes ∠LOJ ≅ ∠JOK? c. What values of x make ∠LOK an acute angle? L O 3xº J (2x + 12)º K Data Analysis Use the circle graph for Exercises 34–36. 34. Find m∠AOB, m∠BOC, m∠COD, and m∠DOA. Classify each angle as acute, right, or obtuse. 35. What if...? Next year, the music store will use some of the shelves currently holding jazz music to double the space for rap. What will m∠COD and m∠BOC be next year? 36. Suppose a fifth type of music, salsa, is added. If the space is divided equally among the five types, what will be the angle measure for each type of music in the circle graph? 37. Critical Thinking Can an obtuse angle be congruent to an acute angle? Why or why not? 38. The measure of an obtuse angle is (5x + 45) °. What is the largest value for x? ___ › FH bisects ∠EFG.
Use the Angle Addition Postulate to explain 39. Write About It why m∠EFH = 1 __ 2 m∠EFG. 40. Multi-Step Use a protractor to draw a 70° angle. Then use a compass and straightedge to bisect the angle. What do you think will be the measure of each angle formed? Use a protractor to support your answer. 41. m∠UOW = 50°, and What is m∠VOY? ___ › OV bisects ∠UOW. 25° 65° 130° 155° 42. What is m∠UOX? 50° 115° 140° 165° 43. ___ › BD bisects ∠ABC, m∠ ABC = (4x + 5) °, and m∠ ABD = (3x - 1) °. What is the value of x? 2.2 3 3.5 7 44. If an angle is bisected and then 30° is added to the measure of the bisected angle, the result is the measure of a right angle. What is the measure of the original angle? 30° 60° 75° 120° 45. Short Response If an obtuse angle is bisected, are the resulting angles acute or obtuse? Explain. 26 26 Chapter 1 Foundations for Geometry �� CHALLENGE AND EXTEND 46. Find the measure of the angle formed by the hands of a clock when it is 7:00. 47. ___ › QS bisects ∠PQR, m∠PQR = (x2) °, and m∠PQS = (2x + 6) °. Find all the possible measures for ∠PQR. 48. For more precise measurements, a degree can be divided into 60 minutes, and each minute can be divided into 60 seconds. An angle measure of 42 degrees, 30 minutes, and 10 seconds is written as 42°30′10″. Subtract this angle measure from the measure 81°24′15″. 49. If 1 degree equals 60 minutes and 1 minute equals 60 seconds, how many seconds are in 2.25 degrees? 50. ∠ABC ≅ ∠DBC. m∠ABC = ( 3x __ 2 + 4) ° and m∠DBC = (2x - 27 1 __ 4 ) °. Is ∠ABD a straight angle? Explain. SPIRAL REVIEW 51. What number is 64% of 35? 52. What percent of 280 is 33.6? (Previous course) Sketch a figure that shows each of the following. (Lesson 1-1) 53. a line that contains _ AB and ___ › CB 54. two different lines that intersect _ MN 55. a plane and a ray that intersect only at Q Find the length of each segment. (Lesson 1-2) _ KL 57. _ JL 58. _ JK 56. Using Technology Segment and Angle Bisectors 1. Construct the bisector _ MN . of 2. Construct the bisector of ∠BAC. a. Draw _ MN and construct the midpoint B. a. Draw ∠BAC. b. Construct a point A not on the segment. ___ _ › ‹ c. Construct bisector MB AB and measure _ NB . and d. Drag M and N and observe MB and NB. b. Construct the angle bisector ___ › AD and measure ∠DAC and ∠DAB. c. Drag the angle and observe m∠DAB and m∠DAC. 1- 3 Measuring and Constructing Angles 27 27 �� 1-4 Pairs of Angles TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates .... Also G.2.B Objectives Identify adjacent, vertical, complementary, and supplementary angles. Find measures of pairs of angles. Vocabulary adjacent angles linear pair complementary angles supplementary angles vertical angles Who uses this? Scientists use properties of angle pairs to design fiber-optic cables. (See Example 4.) A fiber-optic cable is a strand of glass as thin as a human hair. Data can be transmitted over long distances by bouncing light off the inner walls of the cable. Many pairs of angles have special relationships. Some relationships are because of the measurements of the angles in the pair. Other relationships are because of the positions of the angles in the pair. Pairs of Angles Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common interior points. ∠1 and ∠2 are adjacent angles. A linear pair of angles is a pair of adjacent angles whose noncommon sides are opposite rays. ∠3 and ∠4 form a linear pair. E X A M P L E 1 Identifying Angle Pairs Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. A ∠1 and ∠2 ∠1 and ∠2 have a common vertex, B, a common  BC , and no common interior points. side, Therefore ∠1 and ∠2 are only adjacent angles. B ∠2 and ∠4 ∠2 and ∠4 share vertex, so ∠2 and ∠4 are not adjacent angles. ̶̶ BC but do not have a common C ∠1 and ∠3 ∠1 and ∠3 are adjacent angles. Their noncommon sides, are opposite rays, so ∠1 and ∠3 also form a linear pair.  BC and  BA , Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 1a. ∠5 and ∠6 1b. ∠7 and ∠SPU 1c. ∠7 and ∠8 28 28 Chapter 1 Foundations for Geometry �� Complementary and Supplementary Angles Complementary angles are two angles whose measures have a sum of 90°. ∠A and ∠B are complementary. Supplementary angles are two angles whose measures have a sum of 180°. ∠A and ∠C are supplementary. You can find the complement of an angle that measures x° by subtracting its measure from 90°, or (90 - x)°. You can find the supplement of an angle that measures x° by subtracting its measure from 180°, or (180 - x)°. E X A M P L E 2 Finding the Measures of Complements and Supplements Find the measure of each of the following. A complement of ∠M (90 - x)° 90° - 26.8° = 63.2° B supplement of ∠N (180 - x)° 180° - (2y + 20)° = 180° - 2y - 20 = (160 - 2y)° Find the measure of each of the following. 2a. complement of ∠E 2b. supplement of ∠F E X A M P L E 3 Using Complements and Supplements to Solve Problems An angle measures 3 degrees less than twice the measure of its complement. Find the measure of its complement. Step 1 Let m∠A = x°. Then ∠B, its complement, measures (90 - x)°. Step 2 Write and solve an equation. m∠A = 2m∠B - 3 x = 2 (90 - x) - 3 x = 180 - 2x -3 x = 177 - 2x + 2x ̶ + 2x ̶ 3x = 177 = 177_ 3x_ 3 3 x = 59 Substitute x for m∠A and 90 - x for m∠B. Distrib. Prop. Combine like terms. Add 2x to both sides. Simplify. Divide both sides by 3. Simplify. The measure of the complement, ∠B, is (90 - 59)° = 31°. 3. An angle’s measure is 12° more than 1__ 2 the measure of its supplement. Find the measure of the angle. 1- 4 Pairs of Angles 29 29 �� E X A M P L E 4 Problem-Solving Application Light passing through a fiber optic cable reflects off the walls in such a way that ∠1 ≅ ∠2. ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. If m∠1 = 38°, find m∠2, m∠3, and m∠4. Understand the Problem The answers are the measures of ∠2, ∠3, and ∠4. List the important information: • ∠1 ≅ ∠2 • ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. • m∠1 = 38° Make a Plan If ∠1 ≅ ∠2, then m∠1 = m∠2. If ∠3 and ∠1 are complementary, then m∠3 = (90 - 38) °. If ∠4 and ∠2 are complementary, then m∠4 = (90 - 38) °. Solve By the Transitive Property of Equality, if m∠1 = 38° and m∠1 = m∠2, then m∠2 = 38°. Since ∠3 and ∠1 are complementary, m∠3 = 52°. Similarly, since ∠2 and ∠4 are complementary, m∠4 = 52°. Look Back The answer makes sense because 38° + 52° = 90°, so ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. Thus m∠2 = 38°, m∠3 = 52°, and m∠4 = 52°. 4. What if...? Suppose m∠3 = 27.6°. Find m∠1, m∠2, and m∠4. Another angle pair relationship exists between two angles whose sides form two pairs of opposite rays. Vertical angles are two nonadjacent angles formed by two intersecting lines. ∠1 and ∠3 are vertical angles, as are ∠2 and ∠4. E X A M P L E 5 Identifying Vertical Angles Name one pair of vertical angles. Do they appear to have the same measure? Check by measuring with a protractor. ∠EDF and ∠GDH are vertical angles and appear to have the same measure. Check m∠EDF ≈ m∠GDH ≈ 135°. 5. Name another pair of vertical angles. Do they appear to have the same measure? Check by measuring with a protractor. 30 30 Chapter 1 Foundations for Geometry 4321Light1234�� THINK AND DISCUSS 1. Explain why any two right angles are supplementary. 2. Is it possible for a pair of vertical angles to also be adjacent? Explain. 3. GET ORGANIZED Copy and complete the graphic organizer below. In each box, draw a diagram and write a definition of the given angle pair. 1-4 Exercises Exercises KEYWORD: MG7 1-4 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An angle measures x°. What is the measure of its complement? What is the measure of its supplement? 2. ∠ABC and ∠CBD are adjacent angles. Which side do the angles have in common. 28 Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 3. ∠1 and ∠2 4. ∠1 and ∠3 5. ∠2 and ∠4 6. ∠2 and ∠ Find the measure of each of the following. p. 29 7. supplement of ∠A 8. complement of ∠A 9. supplement of ∠B 10. complement of ∠. 29 11. Multi-Step An angle’s measure is 6 degrees more than 3 times the measure of its complement. Find the measure of the angle. 30 12. Landscaping A sprinkler swings back and forth between A and B in such a way that ∠1 ≅ ∠2. ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. If m∠1 = 18.5°, find m∠2, m∠3, and m∠4 13. Name each pair of vertical angles. p. 30 1- 4 Pairs of Angles 31 31 �� Independent Practice For See Exercises Example 14–17 18–21 22 23 24 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S4 Application Practice p. S28 PRACTICE AND PROBLEM SOLVING Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 14. ∠1 and ∠4 15. ∠2 and ∠3 16. ∠3 and ∠4 17. ∠3 and ∠1 Given m∠A = 56.4° and m∠B = (2x - 4)°, find the measure of each of the following. 18. supplement of ∠A 20. supplement of ∠B 19. complement of ∠A 21. complement of ∠B 22. Multi-Step An angle’s measure is 3 times the measure of its complement. Find the measure of the angle and the measure of its complement. 23. Art In the stained glass pattern, ∠1 ≅ ∠2. ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. If m∠1 = 22.3°, find m∠2, m∠3, and m∠4. 24. Name the pairs of vertical angles. 25. Probability The angle measures 30°, 60°, 120°, and 150° are written on slips of paper. You choose two slips of paper at random. What is the probability that the angle measures are suppleme
ntary? Multi-Step ∠ABD and ∠BDE are supplementary. Find the measures of both angles. 26. m∠ABD = 5x°, m∠BDE = (17x - 18) ° 27. m∠ABD = (3x + 12) °, m∠BDE = (7x - 32) ° 28. m∠ABD = (12x - 12) °, m∠BDE = (3x + 48) ° Multi-Step ∠ABD and ∠BDC are complementary. Find the measures of both angles. 29. m∠ABD = (5y + 1) °, m∠BDC = (3y - 7) ° 30. m∠ABD = (4y + 5) °, m∠BDC = (4y + 8) ° 31. m∠ABD = (y - 30) °, m∠BDC = 2y° 32. Critical Thinking Explain why an angle that is supplementary to an acute angle must be an obtuse angle. 33. This problem will prepare you for the Multi-Step TAKS Prep on page 34. H is in the interior of ∠JAK. m∠JAH = (3x - 8) °, and m∠KAH = (x + 2) °. Draw a picture of each relationship. Then find the measure of each angle. a. ∠JAH and ∠KAH are complementary angles. b. ∠JAH and ∠KAH form a linear pair. c. ∠JAH and ∠KAH are congruent angles. 32 32 Chapter 1 Foundations for Geometry ��ge07sec01l0400a1234�� Determine whether each statement is true or false. If false, explain why. 34. If an angle is acute, then its complement must be greater than its supplement. 35. A pair of vertical angles may also form a linear pair. 36. If two angles are supplementary and congruent, the measure of each angle is 90°. 37. If a ray divides an angle into two complementary angles, then the original angle is a right angle. 38. Write About It Describe a situation in which two angles are both congruent and complementary. Explain. 39. What is the value of x in the diagram? 15 30 45 90 40. The ratio of the measures of two complementary angles is 1 : 2. What is the measure of the larger angle? (Hint: Let x and 2x represent the angle measures.) 30° 45° 60° 120° 41. m∠A = 3y, and m∠B = 2m∠A. Which value of y makes ∠A supplementary to ∠B? 10 18 20 36 42. The measures of two supplementary angles are in the ratio 7 : 5. Which value is the measure of the smaller angle? (Hint: Let 7x and 5x represent the angle measures.) 37.5 52.5 75 105 CHALLENGE AND EXTEND 43. How many pairs of vertical angles are in the diagram? 44. The supplement of an angle is 4 more than twice its complement. Find the measure of the angle. 45. An angle’s measure is twice the measure of its complement. The larger angle is how many degrees greater than the smaller angle? 46. The supplement of an angle is 36° less than twice the supplement of the complement of the angle. Find the measure of the supplement. SPIRAL REVIEW Solve each equation. Check your answer. (Previous course) 47. 4x + 10 = 42 49. 2 (y + 3) = 12 48. 5m - 9 = m + 4 50. - (d + 4) = 18 Y is between X and Z, XY = 3x + 1, YZ = 2x - 2, and XZ = 84. Find each of the following. (Lesson 1-2) 51. x 52. XY 53. YZ  XY bisects ∠WYZ. Given m∠WYX = 26°, find each of the following. (Lesson 1-3) 54. m∠XYZ 55. m∠WYZ 1- 4 Pairs of Angles 33 33 �� SECTION 1A Euclidean and Construction Tools Can You Dig It? A group of college and high school students participated in an archaeological dig. The team discovered four fossils. To organize their search, Sierra used a protractor and ruler to make a diagram of where different members of the group found fossils. She drew the locations based on the location of the campsite. The campsite is located at X on   XB . The four fossils were found at R, T, W, and M. 1. Are the locations of the campsite at X and the fossils at R and T collinear or noncollinear? 2. How is X related to ̶̶ RT ? If RX = 10x - 6 and XT = 3x + 8, what is the distance between the locations of the fossils at R and T? 3. ∠RXB and ∠BXT are right angles. � Find the measure of each angle formed by the locations of the fossils and the campsite. Then classify each angle by its measure. 4. Identify the special angle pairs shown in the diagram of the archaeological dig. � � �� 34 34 Chapter 1 Foundations for Geometry SECTION 1A Quiz for Lessons 1-1 Through 1-4 1-1 Understanding Points, Lines, and Planes Draw and label each of the following. 1. a segment with endpoints X and Y 2. a ray with endpoint M that passes through P 3. three coplanar lines intersecting at a point 4. two points and a line that lie in a plane Use the figure to name each of the following. 5. three coplanar points 6. two lines 7. a plane containing T, V, and X 8. a line containing V and Z 1-2 Measuring and Constructing Segments Find the length of each segment. ̶̶ SV 9. 10. ̶̶ TR 11. ̶̶ ST 12. The diagram represents a straight highway with three towns, Henri, Joaquin, and Kenard. Find the distance from Henri H to Joaquin J. 13. Sketch, draw, and construct a segment congruent to ̶̶ CD . 14. Q is the midpoint of ̶̶ PR , PQ = 2z, and PR = 8z - 12. Find z, PQ, and PR. 1-3 Measuring and Constructing Angles 15. Name all the angles in the diagram. Classify each angle by its measure. 16. m∠PVQ = 21° 19.  RS bisects ∠QRT, m∠QRS = (3x + 8) °, and m∠SRT = (9x - 4) °. Find m∠SRT. 20. Use a protractor and straightedge to draw a 130° angle. Then bisect the angle. 18. m∠PVS = 143° 17. m∠RVT = 96° 1-4 Pairs of Angles Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 21. ∠1 and ∠2 22. ∠4 and ∠5 23. ∠3 and ∠4 If m∠T = (5x - 10) °, find the measure of each of the following. 24. supplement of ∠T 25. complement of ∠T Ready to Go On? 35 35 �� 1-5 Using Formulas in Geometry TEKS G.8.A Congruence and the geometry of size: find areas of regular polygons, circles .... Also G.1.A, G.1.B Objective Apply formulas for perimeter, area, and circumference. Why learn this? Puzzles use geometric-shaped pieces. Formulas help determine the amount of materials needed. (See Exercise 6.) Vocabulary perimeter area base height diameter radius circumference pi The perimeter P of a plane figure is the sum of the side lengths of the figure. The area A of a plane figure is the number of nonoverlapping square units of a given size that exactly cover the figure. Perimeter and Area RECTANGLE SQUARE TRIANGLE P = 2ℓ + 2w or 2 (ℓ + w) A = ℓw P = 4s bh or bh ___ A = 1 __ 2 2 The base b can be any side of a triangle. The height h is a segment from a vertex that forms a right angle with a line containing the base. The height may be a side of the triangle or in the interior or the exterior of the triangle. E X A M P L E 1 Finding Perimeter and Area Find the perimeter and area of each figure. A rectangle in which ℓ = 17 cm B triangle in which a = 8, Perimeter is expressed in linear units, such as inches (in.) or meters (m). Area is expressed in square units, such as square centimeters ( cm 2 ). and w = 5 cm P = 2ℓ + 2w = 2 (17) + 2 (5) = 34 + 10 = 44 cm A = ℓw = (17) (5) = 85 cm 2 b = (x + 1) , c = 4x, and x + 1) + 4x = 5x + x + 1) (6) = 3x + 3 2 bh 1. Find the perimeter and area of a square with s = 3.5 in. 36 36 Chapter 1 Foundations for Geometry Project TitleGeometry 2007 Student EditionSpec Numberge07sec01l05002aCreated ByKrosscore Corporation�� E X A M P L E 2 Crafts Application The Texas Treasures quilt block includes 24 purple triangles. The base and height of each triangle are about 3 in. Find the approximate amount of fabric used to make the 24 triangles. The area of one triangle is (3)(3) = 4 1_ bh = 1_ in 2 . 2 2 A = 1_ 2 The total area of the 24 triangles is 24(4 1_ 2) = 108 in 2 . 2. Find the amount of fabric used to make the four rectangles. Each rectangle has a length of 6 1__ 2 in. and a width of 2 1__ 2 in. In a circle a diameter is a segment that passes through the center of the circle and whose endpoints are on the circle. A radius of a circle is a segment whose endpoints are the center of the circle and a point on the circle. The circumference of a circle is the distance around the circle. Circumference and Area of a Circle The circumference C of a circle is given by the formula C = πd or C = 2πr. The area A of a circle is given by the formula A = π r 2 . The ratio of a circle’s circumference to its diameter is the same for all circles. This ratio is represented by the Greek letter π (pi) . The value of π is irrational. Pi is often approximated as 3.14 or 22 __ Finding the Circumference and Area of a Circle Find the circumference and area of the circle. C = 2πr = 2π (3) = 6π ≈ 18.8 cm A = π r 2 = π (3) 2 = 9π ≈ 28.3 cm 2 3. Find the circumference and area of a circle with radius 14 m. THINK AND DISCUSS 1. Describe three different figures whose areas are each 16 in 2 . 2. GET ORGANIZED Copy and complete the graphic organizer. In each shape, write the formula for its area and perimeter. 1- 5 Using Formulas in Geometry 37 37 �� 1-5 Exercises Exercises KEYWORD: MG7 1-5 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Explain how the concepts of perimeter and circumference are related. 2. For a rectangle, length and width are sometimes used in place of __?__. (base and height or radius and diameter Find the perimeter and area of each figure. p. 36 3. 4. 5. 37 6. Manufacturing A puzzle contains a triangular piece with a base of 3 in. and a height of 4 in. A manufacturer wants to make 80 puzzles. Find the amount of wood used if each puzzle contains 20 triangular pieces. 37 Find the circumference and area of each circle. Use the π key on your calculator. Round to the nearest tenth. 7. 8. 9. PRACTICE AND PROBLEM SOLVING Find the perimeter and area of each figure. 10. 11. 12. Independent Practice For See Exercises Example 10–12 13 14–16 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S5 Application Practice p. S28 13. Crafts The quilt pattern includes 32 small triangles. Each has a base of 3 in. and a height of 1.5 in. Find the amount of fabric used to make the 32 triangles. Find the circumference and area of each circle with the given radius or diameter. Use the π key on your calculator. Rou
nd to the nearest tenth. 14. r = 12 m 15. d = 12.5 ft 16. d = 1 _ 2 mi Find the area of each of the following. 17. square whose sides are 9.1 yd in length 18. square whose sides are (x + 1) in length 19. triangle whose base is 5 1 __ 2 in. and whose height is 2 1 __ 4 in. 38 38 Chapter 1 Foundations for Geometry �� Given the area of each of the following figures, find each unknown measure. 20. The area of a triangle is 6.75 m 2 . If the base of the triangle is 3 m, what is the height of the triangle? 21. A rectangle has an area of 347.13 cm 2 . If the length is 20.3 cm, what is the width of the rectangle? 22. The area of a circle is 64π. Find the radius of the circle. 23. /////ERROR ANALYSIS///// Below are two statements about the area of the circle. Which is incorrect? Explain the error. r Equator Find the area of each circle. Leave answers in terms of π. 24. circle with a diameter of 28 m 25. circle with a radius of 3y 26. Geography The radius r of the earth at the equator is approximately 3964 mi. Find the distance around the earth at the equator. Use the π key on your calculator and round to the nearest mile. 27. Critical Thinking Explain how the formulas for the perimeter and area of a square may be derived from the corresponding formulas for a rectangle. 28. Find the perimeter and area of a rectangle whose length is (x + 1) and whose width is (x - 3) . Express your answer in terms of x. 29. Multi-Step If the height h of a triangle is 3 inches less than the length of the base b, and the area A of the triangle is 19 times the length of the base, find b and h. 30. This problem will prepare you for the Multi-Step TAKS Prep on page 58. A landscaper is to install edging around a garden. The edging costs $1.39 for each 24-inch-long strip. The landscaper estimates it will take 4 hours to install the edging. a. If the total cost is $120.30, what is the cost of the material purchased? b. What is the charge for labor? c. What is the area of the semicircle to the nearest tenth? d. What is the area of each triangle? e. What is the total area of the garden to the nearest foot? 1- 5 Using Formulas in Geometry 39 39 �� 31. Algebra The large rectangle has length a + b and width c + d. Therefore, its area is (a + b) (c + d) . a. Find the area of each of the four small rectangles in the figure. Then find the sum of these areas. Explain why this sum must be equal to the product (a + b)(c + d). b. Suppose b = d = 1. Write the area of the large rectangle as a product of its length and width. Then find the sum of the areas of the four small rectangles. Explain why this sum must be equal to the product (a + 1)(c + 1) . c. Suppose b = d = 1 and a = c. Write the area of the large rectangle as a product of its length and width. Then find the sum of the areas of the four small rectangles. Explain why this sum must be equal to the product (a + 1) 2 . 32. Sports The table shows the minimum and maximum dimensions for rectangular soccer fields used in international matches. Find the difference in area of the largest possible field and the smallest possible field. Length Width Minimum Maximum 100 m 64 m 110 m 75 m Find the value of each missing measure of a triangle. ft; A = 28 ft 2 33. b = 2 ft; h = 34. b = ft; h = 22.6 yd; A = 282.5 yd 2 Find the area of each rectangle with the given base and height. 35. 9.8 ft; 2.7 ft 36. 4 mi 960 ft; 440 ft 37. 3 yd 12 ft; 11 ft Find the perimeter of each rectangle with the given base and height. 38. 21.4 in.; 7.8 in. 39. 4 ft 6 in.; 6 in. 40. 2 yd 8 ft; 6 ft Find the diameter of the circle with the given measurement. Leave answers in terms of π. 41. C = 14 42. A = 100π 43. C = 50π 44. A skate park consists of a two adjacent rectangular regions as shown. Find the perimeter and area of the park. 45. Critical Thinking Explain how you would measure a triangular piece of paper if you wanted to find its area. 46. Write About It A student wrote in her journal, “To find the perimeter of a rectangle, add the length and width together and then double this value.” Does her method work? Explain. 47. Manda made a circular tabletop that has an area of 452 in2. Which is closest to the radius of the tabletop? 9 in. 12 in. 24 in. 72 in. 48. A piece of wire 48 m long is bent into the shape of a rectangle whose length is twice its width. Find the length of the rectangle. 8 m 16 m 24 m 32 m 40 40 Chapter 1 Foundations for Geometry �� 49. Which equation best represents the area A of the triangle? A = 2 x 2 + 4x A = 4x (x + 2 50. Ryan has a 30 ft piece of string. He wants to use the string to lay out the boundary of a new flower bed in his garden. Which of these shapes would use all the string? A circle with a radius of about 37.2 in. A rectangle with a length of 6 ft and a width of 5 ft A triangle with each side 9 ft long A square with each side 90 in. long Math History CHALLENGE AND EXTEND 51. A circle with a 6 in. diameter is stamped out of a rectangular piece of metal as shown. Find the area of the remaining piece of metal. Use the π key on your calculator and round to the nearest tenth. 52. a. Solve P = 2ℓ + 2w for w. �� b. Use your result from part a to find the width of a rectangle that has a perimeter of 9 ft and a length of 3 ft. 53. Find all possible areas of a rectangle whose sides are natural numbers and whose perimeter is 12. The Ahmes Papyrus is an ancient Egyptian source of information about mathematics. A page of the Ahmes Papyrus is about 1 foot wide and 18 feet long. Source: scholars.nus.edu.sg 54. Estimation The Ahmes Papyrus dates from approximately 1650 B.C.E. Lacking a precise value for π, the author assumed that the area of a circle with a diameter of 9 units had the same area as a square with a side length of 8 units. By what percent did the author overestimate or underestimate the actual area of the circle? 55. Multi-Step The width of a painting is 4 __ 5 the measure of the length of the painting. If the area is 320 in 2 , what are the length and width of the painting? SPIRAL REVIEW Determine the domain and range of each function. (Previous course)   ⎬ ⎨ (2, 4) , (-5, 8) , (-3, 4) 56.     ⎬ ⎨ (4, -2) , (-2, 8) , (16, 0) 57.   Name the geometric figure that each item suggests. (Lesson 1-1) 58. the wall of a classroom 59. the place where two walls meet 60. Marion has a piece of fabric that is 10 yd long. She wants to cut it into 2 pieces so that one piece is 4 times as long as the other. Find the lengths of the two pieces. (Lesson 1-2) 61. Suppose that A, B, and C are collinear points. B is the midpoint of of A is -8, and the coordinate of B is -2.5. What is the coordinate of C? (Lesson 1-2) 62. An angle’s measure is 9 degrees more than 2 times the measure of its supplement. Find the measure of the angle. (Lesson 1-4) 1- 5 Using Formulas in Geometry 41 41 _ AC . The coordinate �� Graphing in the Coordinate Plane Algebra The coordinate plane is used to name and locate points. Points in the coordinate plane are named by ordered pairs of the form (x, y) . The first number is the x-coordinate. The second number is the y-coordinate. The x-axis and y-axis intersect at the origin, forming right angles. The axes separate the coordinate plane into four regions, called quadrants, numbered with Roman numerals placed counterclockwise. See Skills Bank page S56 Examples 1 Name the coordinates of P. Starting at the origin (0, 0) , you count 1 unit to the right. Then count 3 units up. So the coordinates of P are (1, 3) . 2 Plot and label H (-2, -4) on a coordinate plane. Name the quadrant in which it is located. Start at the origin (0, 0) and move 2 units left. Then move 4 units down. Draw a dot and label it H. H is in Quadrant III. You can also use a coordinate plane to locate places on a map. Try This TAKS Grades 9–11 Obj. 6, 7 Name the coordinates of the point where the following streets intersect. 1. Chestnut and Plum 2. Magnolia and Chestnut 3. Oak and Hawthorn 4. Plum and Cedar Name the streets that intersect at the given points. 5. (-3, -1) 7. (1, 3) 6. (4, -1) 8. (-2, 1) 42 42 Chapter 1 Foundations for Geometry �� 1-6 Midpoint and Distance in the Coordinate Plane TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates .... Also G.7.A, G.7.C, G.8.C Objectives Develop and apply the formula for midpoint. Use the Distance Formula and the Pythagorean Theorem to find the distance between two points. Vocabulary coordinate plane leg hypotenuse Why learn this? You can use a coordinate plane to help you calculate distances. (See Example 5.) Major League baseball fields are laid out according to strict guidelines. Once you know the dimensions of a field, you can use a coordinate plane to find the distance between two of the bases. A coordinate plane is a plane that is divided into four regions by a horizontal line (x-axis) and a vertical line (y-axis) . The location, or coordinates, of a point are given by an ordered pair (x, y). Minute Maid Park, Houston You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints. Midpoint Formula _ AB with The midpoint M of endpoints A ( x 1 , y 1 ) and B ( x 2 , y 2 ) is found by To make it easier to picture the problem, plot the segment’s endpoints on a coordinate plane. Finding the Coordinates of a Midpoint ̶̶ CD M ( Find the coordinates of the midpoint of with endpoint
s C (-2, -1) and D (4, 21 + 2 -1, 1 _ ) 2 1. Find the coordinates of the midpoint of _ EF with endpoints E (-2, 3) and F (5, -3) . 1- 6 Midpoint and Distance in the Coordinate Plane 43 43 �� E X A M P L E 2 Finding the Coordinates of an Endpoint ̶̶ AB . A has coordinates (2, 2) , and M has coordinates M is the midpoint of (4, -3) . Find the coordinates of B. Step 1 Let the coordinates of B equal (x, y) . Step 2 Use the Midpoint Formula: (4, -3 = Step 3 Find the x-coordinate4) = 2 ( Set the coordinates equal. Multiply both sides by 2. Find the y-coordinate. -3 = 2 (-3 Simplify. - 2 - ̶̶̶̶ 2 ̶̶̶̶ 6 = x Subtract 2 from both sides. Simplify. -6 = 2 + y - 2 - ̶̶̶̶ -8 = y 2 ̶̶̶̶ The coordinates of B are (6, -8) . 2. S is the midpoint of _ RT . R has coordinates (-6, -1) , and S has coordinates (-1, 1) . Find the coordinates of T. The Ruler Postulate can be used to find the distance between two points on a number line. The Distance Formula is used to calculate the distance between two points in a coordinate plane. Distance Formula In a coordinate plane, the distance d between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is d = √  ( Using the Distance Formula Find AB and CD. Then determine if ̶̶ AB ≅ ̶̶ CD . Step 1 Find the coordinates of each point. A (0, 3) , B (5, 1) , C (-1, 1) , and D (-3, -4) Step 2 Use the Distance Formula. d = √  AB = √  ( 5 - 0) 2 + (1 - 3) 2 CD = √  ⎤ ⎦ ⎡ ⎣ 2 + (-4 - 1) 2 -3 - (-1) = √  5 2 + (-2) 2 = √  (-2) 2 + (-5) 2 = √  25 + 4 = √  29 = √  4 + 25 = √  29 Since AB = CD, _ AB ≅ _ CD . 3. Find EF and GH. Then determine if _ EF ≅ _ GH . 44 44 Chapter 1 Foundations for Geometry �� You can also use the Pythagorean Theorem to find the distance between two points in a coordinate plane. You will learn more about the Pythagorean Theorem in Chapter 5. In a right triangle, the two sides that form the right angle are the legs . The side across from the right angle that stretches from one leg to the other is the hypotenuse . In the diagram, a and b are the lengths of the shorter sides, or legs, of the right triangle. The longest side is called the hypotenuse and has length c. Theorem 1-6-1 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse Finding Distances in the Coordinate Plane Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from A to B. Method 1 Method 2 Use the Distance Formula. Substitute the values for the coordinates of A and B into the Distance Formula. AB = √  ( = √  ⎤ ⎦ ⎡ ⎣ 2 + (-2 - 3) 2 2 - (-2) = √  4 2 + (-5) 2 16 + 25 = √  = √  41 ≈ 6.4 Use the Pythagorean Theorem. Count the units for sides a and b. a = 4 and b = 5 = 16 + 25 = 41 c = √  41 c ≈ 6.4 Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from R to S. 4a. R (3, 2) and S (-3, -1) 4b. R (-4, 5) and S (2, -1) 1- 6 Midpoint and Distance in the Coordinate Plane 45 45 �� E X A M P L E 5 Sports Application The four bases on a baseball field form a square with 90 ft sides. When a player throws the ball from home plate to second base, what is the distance of the throw, to the nearest tenth? Set up the field on a coordinate plane so that home plate H is at the origin, first base F has coordinates (90, 0) , second base S has coordinates (90, 90) , and third base T has coordinates (0, 90) . The distance HS from home plate to second base is the length of the hypotenuse of a right triangle. HS = √  = √  ( 90 - 0) 2 + (90 - 0) 2 = √  90 2 + 90 2 = √  8100 + 8100 = √  16,200 ≈ 127.3 ft 5. The center of the pitching mound has coordinates (42.8, 42.8) . When a pitcher throws the ball from the center of the mound to home plate, what is the distance of the throw, to the nearest tenth? THINK AND DISCUSS 1. Can you exchange the coordinates ( x 1 , y 1 ) and ( x 2 , y 2 ) in the Midpoint Formula and still find the correct midpoint? Explain. 2. A right triangle has sides lengths of r, s, and t. Given that s 2 + t 2 = r 2 , which variables represent the lengths of the legs and which variable represents the length of the hypotenuse? 3. Do you always get the same result using the Distance Formula to find distance as you do when using the Pythagorean Theorem? Explain your answer. 4. Why do you think that most cities are laid out in a rectangular grid instead of a triangular or circular grid? 5. GET ORGANIZED Copy and complete the graphic organizer below. In each box, write a formula. Then make a sketch that will illustrate the formula. 46 46 Chapter 1 Foundations for Geometry H (0,0)F(90,0)S(90,90)T(0,90)ge07se_c01L06006a�� 1-6 Exercises Exercises KEYWORD: MG7 1-6 KEYWORD: MG7 Parent . 43 . 44 . 44 . 45 GUIDED PRACTICE 1. Vocabulary The right angle. (hypotenuse or leg) ? is the side of a right triangle that is directly across from the ̶̶̶̶ Find the coordinates of the midpoint of each segment. _ AB with endpoints A (4, -6) and B (-4, 2) _ CD with endpoints C (0, -8) and D (3, 0) 2. 3. 4. M is the midpoint of _ LN . L has coordinates (-3, -1) , and M has coordinates (0, 1) . Find the coordinates of N. 5. B is the midpoint of (-1 1 __ 2 , 1) . Find the coordinates of C. _ AC . A has coordinates (-3, 4) , and B has coordinates Multi-Step Find the length of the given segments and determine if they are congruent. _ JK and _ FG 6. _ JK and _ RS 7. Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 8. A (1, -2) and B (-4, -4) 9. X (-2, 7) and Y (-2, -8) 10. V (2, -1) and W (-4, 8. 46 11. Architecture The plan for a rectangular living room shows electrical wiring will be run in a straight line from the entrance E to a light L at the opposite corner of the room. What is the length of the wire to the nearest tenth? Independent Practice For See Exercises Example 12–13 14–15 16–17 18–20 21 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S5 Application Practice p. S28 PRACTICE AND PROBLEM SOLVING Find the coordinates of the midpoint of each segment. 12. _ XY with endpoints X (-3, -7) and Y (-1, 1) _ MN with endpoints M (12, -7) and N (-5, -2) _ QR . Q has coordinates (-3, 5) , and M has coordinates (7, -9) . 14. M is the midpoint of 13. Find the coordinates of R. 15. D is the midpoint of Find the coordinates of C. _ CE . E has coordinates (-3, -2) , and D has coordinates (2 1 __ 2 , 1) . Multi-Step Find the length of the given segments and determine if they are congruent. _ DE and _ DE and _ FG _ RS 16. 17. 1- 6 Midpoint and Distance in the Coordinate Plane 47 47 �� Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 18. U (0, 1) and V (-3, -9) 19. M (10, -1) and N (2, -5) 20. P (-10, 1) and Q (5, 5) 21. Consumer Application Televisions and computer screens are usually advertised based on the length of their diagonals. If the height of a computer screen is 11 in. and the width is 14 in., what is the length of the diagonal? Round to the nearest inch. 22. Multi-Step Use the Distance Formula to _ EF from shortest to longest. _ CD , and _ AB , order 23. Use the Pythagorean Theorem to find the distance from A to E. Round to the nearest hundredth. 24. X has coordinates (a, 3a) , and Y has coordinates (-5a, 0) . Find the coordinates of the midpoint of (XY). 25. Describe a shortcut for finding the midpoint of a segment when one of its endpoints has coordinates (a, b) and the other endpoint is the origin. On the map, each square of the grid represents 1 square mile. Find each distance to the nearest tenth of a mile. tenth of a mile. 26. 26. Find the distance along Highway 201 from Cedar City to Milltown. 27. 27. A car breaks down on Route 1, at the midpoint between Jefferson and Milltown. A tow truck is sent out from Jefferson. How far does the truck travel to reach the car? 28. History The Forbidden City in Beijing, China, is the world’s largest palace complex. Surrounded by a wall and a moat, the rectangular complex is 960 m long and 750 m wide. Find the distance, to the nearest meter, from one corner of the complex to the opposite corner. 29. Critical Thinking Give an example of a line segment with midpoint (0, 0) . The coordinates of the vertices of △ABC are A(1, 4) , B (-2, -1) , and C (-3, -2) . 30. Find the perimeter of △ABC to the nearest tenth. _ BC is √  2 , and b is the length of 31. The height h to side _ BC . What is the area of △ABC ? History The Forbidden City of Imperial China is replicated in Katy, Texas. The museum has 6000 miniature terra-cotta soldiers. Source: www.forbiddengardens.com 32. Write About It Explain why the Distance Formula is not needed to find the distance between two points that lie on a horizontal or a vertical line. 33. This problem will prepare you for the Multi-Step TAKS Prep on page 58. Tania uses a coordinate plane to map out plans for landscaping a rectangular patio area. On the plan, one square represents 2 feet. She plans to plant a tree at the midpoint of of the patio does she plant the tree? Round to the nearest tenth. _ AC . How far from each corner 48 48 Chapter 1 Foundations for Geometry �� 34. Which segment has a length closest to 4 units? _ EF _ GH _ JK _ LM 35. Find the distance, to the nearest tenth, between the midpoints of _ LM and _ JK . 1.8 3.6 4.0 5.3 36. What are the coordinates of the midpoint of a line segment that connects the points (7, -3) and (-5, 6) ? (2, 1 __ ) (1, 1 1 __ ) (6, -4 1 __ ) (2, 3) 2 2 2 37. A coordinate plane is p
laced over the map of a town. A library is located at (-5, 1) , and a museum is located at (3, 5) . What is the distance, to the nearest tenth, from the library to the museum? 4.5 5.7 6.3 8.9 CHALLENGE AND EXTEND 38. Use the diagram to find the following. _ AB , and R is the midpoint of a. P is the midpoint of _ BC . Find the coordinates of Q. b. Find the area of rectangle PBRQ. c. Find DB. Round to the nearest tenth. 39. The coordinates of X are (a - 5, -2a) . The coordinates of Y are (a + 1, 2a) . If the distance between X and Y is 10, find the value of a. 40. Find two points on the y-axis that are a distance of 5 units from (4, 2) . 41. Given ∠ACB is a right angle of △ABC, AC = x, and BC = y, find AB in terms of x and y. SPIRAL REVIEW Determine if the ordered pair (-1, 4) satisfies each function. (Previous course) 42. y = 3x - 1 44. g (x) = x 2 - x + 2 43. f (x) = 5 - x 2   BD bisects straight angle ABC, and Find the measure of each angle and classify it as acute, right, or obtuse. (Lesson 1-3)   BE bisects ∠CBD. 45. ∠ABD 46. ∠CBE 47. ∠ABE Find the area of each of the following. (Lesson 1-5) 48. square whose perimeter is 20 in. 49. triangle whose height is 2 ft and whose base is twice its height 50. rectangle whose length is x and whose width is (4x + 5) 1- 6 Midpoint and Distance in the Coordinate Plane 49 49 �� 1-7 Transformations in the Coordinate Plane TEKS G.5.C Geometric patterns: use properties of transformations ... to make connections between mathematics and the real world .... Also G.1.A Objectives Identify reflections, rotations, and translations. Graph transformations in the coordinate plane. Vocabulary transformation preimage image reflection rotation translation Who uses this? Artists use transformations to create decorative patterns. (See Example 4.) The Alhambra, a 13th-century palace in Grenada, Spain, is famous for the geometric patterns that cover its walls and floors. To create a variety of designs, the builders based the patterns on several different transformations. A transformation is a change in the position, size, or shape of a figure. The original figure is called the preimage . The resulting figure is called the image . A transformation maps the preimage to the image. Arrow notation (→) is used to describe a transformation, and primes (′) are used to label the image. Transformations REFLECTION ROTATION TRANSLATION A reflection (or flip) is a transformation across a line, called the line of reflection. Each point and its image are the same distance from the line of reflection. A rotation (or turn) is a transformation about a point P, called the center of rotation. Each point and its image are the same distance from P. A translation (or slide) is a transformation in which all the points of a figure move the same distance in the same direction. E X A M P L E 1 Identifying Transformations Identify the transformation. Then use arrow notation to describe the transformation. A The transformation cannot be a translation because each point and its image are not in the same position. The transformation is a reflection. △EFG → △E′F′G′ 50 50 Chapter 1 Foundations for Geometry �� Identify the transformation. Then use arrow notation to describe the transformation. B The transformation cannot be a reflection because each point and its image are not the same distance from a line of reflection. The transformation is a 90° rotation. RSTU → R′S′T′U′ Identify each transformation. Then use arrow notation to describe the transformation. 1a. 1b. E X A M P L E 2 Drawing and Identifying Transformations A figure has vertices at A (-1, 4) , B (-1, 1) , and C (3, 1) . After a transformation, the image of the figure has vertices at A′ (-1, -4) , B′ (-1, -1) , and C′ (3, -1) . Draw the preimage and image. Then identify the transformation. Plot the points. Then use a straightedge to connect the vertices. The transformation is a reflection across the x-axis because each point and its image are the same distance from the x-axis. 2. A figure has vertices at E (2, 0) , F (2, -1) , G (5, -1) , and H (5, 0) . After a transformation, the image of the figure has vertices at E′ (0, 2) , F′ (1, 2) , G′ (1, 5) , and H′ (0, 5) . Draw the preimage and image. Then identify the transformation. To find coordinates for the image of a figure in a translation, add a to the x-coordinates of the preimage and add b to the y-coordinates of the preimage. Translations can also be described by a rule such as (x, y) → (x + a, y + b Translations in the Coordinate Plane Find the coordinates for the image of △ABC after the translation (x, y) → (x + 3, y - 4) . Draw the image. Step 1 Find the coordinates of △ABC. The vertices of △ABC are A (-1, 1) , B (-3, 3) , and C (-4, 0) . 1- 7 Transformations in the Coordinate Plane 51 51 �� Step 2 Apply the rule to find the vertices of the image. A′ (-1 + 3, 1 - 4) = A′ (2, -3) B′ (-3 + 3, 3 - 4) = B′ (0, -1) C′ (-4 + 3, 0 - 4) = C′ (-1, -4) Step 3 Plot the points. Then finish drawing the image by using a straightedge to connect the vertices. 3. Find the coordinates for the image of JKLM after the translation (x, y) → (x - 2, y + 4) . Draw the image. E X A M P L E 4 Art History Application The pattern shown is similar to a pattern on a wall of the Alhambra. Write a rule for the translation of square 1 to square 2. Step 1 Choose 2 points Choose a point A on the preimage and a corresponding point A′ on the image. A has coordinates (3, 1) , and A′ has coordinates (1, 3) . Step 2 Translate To translate A to A′, 2 units are subtracted from the x-coordinate and 2 units are added to the y-coordinate. Therefore, the translation rule is (x, y) → (x - 2, y + 2) . 4. Use the diagram to write a rule for the translation of square 1 to square 3. THINK AND DISCUSS 1. Explain how to recognize a reflection when given a figure and its image. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, sketch an example of each transformation. 52 52 Chapter 1 Foundations for Geometry ��geo7sec01l07002a2A‘A13yx�� 1-7 Exercises Exercises KEYWORD: MG7 1-7 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Given the transformation △XYZ → △X′Y′Z′, name the preimage and image of the transformation. 2. The types of transformations of geometric figures in the coordinate plane can be described as a slide, a flip, or a turn. What are the other names used to identify these transformations Identify each transformation. Then use arrow notation to describe the transformation. p. 50 3. 4. 51 5. A figure has vertices at A (-3, 2) , B (-1, -1) , and C (-4, -2) . After a transformation, the image of the figure has vertices at A′ (3, 2) , B′ (1, -1) , and C′ (4, -2) . Draw the preimage and image. Then identify the transformation. 51 . 52 6. Multi-Step The coordinates of the vertices of △DEF are D (2, 3) , E (1, 1) , and F (4, 0) . Find the coordinates for the image of △DEF after the translation (x, y) → (x - 3, y - 2) . Draw the preimage and image. 7. Animation In an animated film, a simple scene can be created by translating a figure against a still background. Write a rule for the translation that maps the rocket from position 1 to position 2. PRACTICE AND PROBLEM SOLVING Identify each transformation. Then use arrow notation to describe the transformation. 8. 9. Independent Practice For See Exercises Example 8–9 10 11 12 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S5 Application Practice p. S28 10. A figure has vertices at J (-2, 3) , K (0, 3) , L (0, 1) , and M (-2, 1) . After a transformation, the image of the figure has vertices at J′ (2, 1) , K′ (4, 1) , L′ (4, -1) , and M′ (2, -1) . Draw the preimage and image. Then identify the transformation. 1- 7 Transformations in the Coordinate Plane 53 53 ��x– 44y4 – 412�� 11. Multi-Step The coordinates of the vertices of rectangle ABCD are A (-4, 1) , B (1, 1) , C (1, -2) , and D (-4, -2) . Find the coordinates for the image of rectangle ABCD after the translation (x, y) → (x + 3, y - 2) . Draw the preimage and the image. 12. Travel Write a rule for the translation that maps the descent of the hot air balloon. Which transformation is suggested by each of the following? 13. mountain range and its image on a lake 14. straight line path of a band marching down a street 15. wings of a butterfly Given points F (3, 5) , G (-1, 4) , and H (5, 0) , draw △FGH and its reflection across each of the following lines. 16. the x-axis 17. the y-axis 18. Find the vertices of one of the triangles on the graph. Then use arrow notation to write a rule for translating the other three triangles. A transformation maps A onto B and C onto D. 19. Name the image of A. 20. Name the preimage of B. 21. Name the image of C. 22. Name the preimage of D. 23. Find the coordinates for the image of △RST with vertices R (1, -4) , S (-1, -1) , and T (-5, 1) after the translation (x, y) → (x - 2, y - 8) . 24. Critical Thinking Consider the translations (x, y) → (x + 5, y + 3) and (x, y) → (x + 10, y + 5) . Compare the two translations. Graph each figure and its image after the given translation. _ MN with endpoints M (2, 8) and N (-3, 4) after the translation (x, y) → (x + 2, y - 5) _ KL with endpoints K (-1, 1) and L (3, -4) after the translation (x, y) → (x - 4, y + 3) 25. 26. 27. Write About It Given a triangle in the coordinate plane, explain how to draw its image after the translation (x, y) → (x + 1, y + 1) . 28. This problem will prepare you for the Multi-Step TAKS Prep on page 58. Greg wants to rearrange the triangular pattern of colored stones on his patio. What combination of transformations could he use to transform △CAE to the image on the coordinate plane?
54 54 Chapter 1 Foundations for Geometry xyge07sec01/07004a�� 29. Which type of transformation maps △XYZ to △X′Y′Z′? Reflection Rotation Translation Not here 30. △DEF has vertices at D (-4, 2) , E (-3, -3) , and F (1, 4) . Which of these points is a vertex of the image of △DEF after the translation (x, y) → (x - 2, y + 1) ? (-2, 1) (3, 3) (-5, -2) (-6, -1) 31. Consider the translation (1, 4) → (-2, 3) . What number was added to the x-coordinate? -3 -1 1 7 32. Consider the translation (-5, -7) → (-2, -1) . What number was added to the y-coordinate? -3 3 6 8 CHALLENGE AND EXTEND 33. △RST with vertices R (-2, -2) , S (-3, 1) , and T (1, 1) is translated by (x, y) → (x - 1, y + 3) . Then the image, △R′S′T ′, is translated by (x, y) → (x + 4, y - 1) , resulting in △R "S"T ". a. Find the coordinates for the vertices of △R "S"T ". b. Write a rule for a single translation that maps △RST to △R"S"T ". 34. Find the angle through which the minute hand of a clock rotates over a period of 12 minutes. 35. A triangle has vertices A (1, 0) , B (5, 0) , and C (2, 2) . The triangle is rotated 90° counterclockwise about the origin. Draw and label the image of the triangle. Determine the coordinates for the reflection image of any point A (x, y) across the given line. 36. x-axis 37. y-axis SPIRAL REVIEW Use factoring to find the zeros of each function. (Previous course) 39. y = x 2 + 3x - 18 38. y = x 2 + 12x + 35 40. y = x 2 - 18x + 81 41. y = x 2 - 3x + 2 Given m∠A = 76.1°, find the measure of each of the following. (Lesson 1-4) 42. supplement of ∠A 43. complement of ∠A Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. (Lesson 1-6) 44. (2, 3) and (4, 6) 46. (-3, 7) and (-6, -2) 45. (-1, 4) and (0, 8) 47. (5, 1) and (-1, 3) 1- 7 Transformations in the Coordinate Plane 55 55 �� 1-7 Explore Transformations A transformation is a movement of a figure from its original position (preimage) to a new position (image). In this lab, you will use geometry software to perform transformations and explore their properties. Use with Lesson 1-7 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.2.B KEYWORD: MG7 Lab1 Activity 1 1 Construct a triangle using the segment tool. Use the text tool to label the vertices A, B, and C. 2 Select points A and B in that order. Choose Mark Vector from the Transform menu. 3 Select △ABC by clicking on all three segments of the triangle. 4 Choose Translate from the Transform menu, using Marked as the translation vector. What do you notice about the relationship between your preimage and its image? 5 What happens when you drag a vertex or a side of △ABC? Try This For Problems 1 and 2 choose New Sketch from the File menu. 1. Construct a triangle and a segment outside the triangle. Mark this segment as a translation vector as you did in Step 2 of Activity 1. Use Step 4 of Activity 1 to translate the triangle. What happens when you drag an endpoint of the new segment? 2. Instead of translating by a marked vector, use Rectangular as the translation vector and translate by a horizontal distance of 1 cm and a vertical distance of 2 cm. Compare this method with the marked vector method. What happens when you drag a side or vertex of the triangle? 3. Select the angles and sides of the preimage and image triangles. Use the tools in the Measure menu to measure length, angle measure, perimeter, and area. What do you think is true about these two figures? 56 56 Chapter 1 Foundations for Geometry Activity 2 1 Construct a triangle. Label the vertices G, H, and I. 2 Select point H and choose Mark Center from the Transform menu. 3 Select ∠GHI by selecting points G, H, and I in that order. Choose Mark Angle from the Transform menu. 4 Select the entire triangle △GHI by dragging a selection box around the figure. 5 Choose Rotate from the Transform menu, using Marked Angle as the angle of rotation. 6 What happens when you drag a vertex or a side of △GHI? Try This For Problems 4–6 choose New Sketch from the File menu. 4. Instead of selecting an angle of the triangle as the rotation angle, draw a new angle outside of the triangle. Mark this angle. Mark ∠GHI as Center and rotate the triangle. What happens when you drag one of the points that form the rotation angle? 5. Construct △QRS, a new rotation angle, and a point P not on the triangle. Mark P as the center and mark the angle. Rotate the triangle. What happens when you drag P outside, inside, or on the preimage triangle? 6. Instead of rotating by a marked angle, use Fixed Angle as the rotation method and rotate by a fixed angle measure of 30°. Compare this method with the marked angle method. 7. Using the fixed angle method of rotation, can you find an angle measure that will result in an image figure that exactly covers the preimage figure? 1- 7 Technology Lab 57 57 SECTION 1B Coordinate and Transformation Tools Pave the Way Julia wants to use L-shaped paving stones to pave a patio. Two stones will cover a 12 in. by 18 in. rectangle. 1. She drew diagram ABCDEF to represent the patio. Find the area and perimeter of the patio. How many paving stones would Julia need to purchase to pave the patio? If each stone costs $2.25, what is the total cost of the stones for the patio? Describe how you calculated your answer. 2. Julia plans to place a fountain at the ̶̶ AF . How far is the fountain midpoint of from B, C, E, and F ? Round to the nearest tenth. 3. Julia used a pair of paving stones to create another pattern for the patio. Describe the transformation she used to create the pattern. If she uses just one transformation, how many other patterns can she create using two stones? Draw all the possible combinations. Describe the transformation used to create each pattern. 58 58 Chapter 1 Foundations for Geometry �� SECTION 1B Quiz for Lessons 1-5 Through 1-7 1-5 Using Formulas in Geometry Find the perimeter and area of each figure. 1. 3. 2. 4. 5. Find the circumference and area of a circle with a radius of 6 m. Use the π key on your calculator and round to the nearest tenth. 1-6 Midpoint and Distance in the Coordinate Plane 6. Find the coordinates for the midpoint of ̶̶ XY with endpoints X (-4, 6) and Y (3, 8) . 7. J is the midpoint of ̶̶ HK , H has coordinates (6, -2) , and J has coordinates (9, 3) . Find the coordinates of K. 8. Using the Distance Formula, find QR and ST to the nearest tenth. Then determine if ̶̶ QR ≅ ̶̶ ST . 9. Using the Distance Formula and the Pythagorean Theorem, find the distance, to the nearest tenth, from F (4, 3) to G (-3, -2) . 1-7 Transformations in the Coordinate Plane Identify the transformation. Then use arrow notation to describe the transformation. 10. 11. 12. A graphic designer used the translation (x, y) → (x - 3, y + 2) to transform square HJKL. Find the coordinates and graph the image of square HJKL. 13. A figure has vertices at X (1, 1) , Y (3, 1) , and Z (3, 4) . After a transformation, the image of the figure has vertices at X′ (-1, -1) , Y′ (-3, -1) , and Z′ (-3, -4) . Graph the preimage and image. Then identify the transformation. Ready to Go On? 5959 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary acute angle . . . . . . . . . . . . . . . . . . 21 diameter . . . . . . . . . . . . . . . . . . . . 37 plane . . . . . . . . . . . . . . . . . . . . . . . . 6 adjacent angles . . . . . . . . . . . . . . 28 distance . . . . . . . . . . . . . . . . . . . . . 13 point . . . . . . . . . . . . . . . . . . . . . . . . 6 angle . . . . . . . . . . . . . . . . . . . . . . . 20 endpoint . . . . . . . . . . . . . . . . . . . . . 7 postulate . . . . . . . . . . . . . . . . . . . . . 7 angle bisector . . . . . . . . . . . . . . . 23 exterior of an angle . . . . . . . . . . 20 preimage . . . . . . . . . . . . . . . . . . . . 50 area . . . . . . . . . . . . . . . . . . . . . . . . 36 height . . . . . . . . . . . . . . . . . . . . . . 36 radius . . . . . . . . . . . . . . . . . . . . . . . 37 base . . . . . . . . . . . . . . . . . . . . . . . . 36 hypotenuse . . . . . . . . . . . . . . . . . 45 ray . . . . . . . . . . . . . . . . . . . . . . . . . . 7 between . . . . . . . . . . . . . . . . . . . . . 14 image . . . . . . . . . . . . . . . . . . . . . . . 50 reflection . . . . . . . . . . . . . . . . . . . 50 bisect . . . . . . . . . . . . . . . . . . . . . . . 15 interior of an angle . . . . . . . . . . 20 right angle . . . . . . . . . . . . . . . . . . 21 circumference . . . . . . . . . . . . . . . 37 leg . . . . . . . . . . . . . . . . . . . . . . . . . . 45 rotation . . . . . . . . . . . . . . . . . . . . . 50 collinear . . . . . . . . . . . . . . . . . . . . . 6 length . . . . . . . . . . . . . . . . . . . . . . 13 segment . . . . . . . . . . . . . . . . . . . . . 7 complementary angles . . . . . . . 29 line . . . . . . . . . . . . . . . . . . . . . . . . . . 6 segment bisector . . . . . . . . . . . . 16 congruent angles . . . . . . . . . . . . 22 linear pair . . . . . . . . . . . . . . . . . . . 28 straight angle . . . . . . . . . . . . . . . . 21 congruent segments . . . . . . . . . 13 measure . . . . . . . . . . . . . . . . . . . . 20 supplementary angles . . . . . . . . 29 construction . . . . . . . . . . . . . . . . 14 midpoint . . . . . . . . . . . . . . . . . . . . 15 transformation . . . . . . . . . . . . . . 50 coordinate . . . . . . . . . . . . . . . . . . 13 obtuse angle . . . . . . . . . . . . . . . . . 21 translation . . . . . . . . . . . . . . . . . . 50 coordinate plane . . . . . . . . . . . . 43 opposite rays . . . . . . . . . . . . . . . . . 7 undefined term . . . . . . . . . . . . . . . 6 coplanar . . . . . . . . . . . . . . . . . . . . . 6 perimeter .
. . . . . . . . . . . . . . . . . . 36 vertex . . . . . . . . . . . . . . . . . . . . . . . 20 degree . . . . . . . . . . . . . . . . . . . . . . 20 pi . . . . . . . . . . . . . . . . . . . . . . . . . . 37 vertical angles . . . . . . . . . . . . . . . 30 Complete the sentences below with vocabulary words from the list above. 1. A(n) ? divides an angle into two congruent angles. ̶̶̶̶̶̶ 2. ? are two angles whose measures have a sum of 90°. ̶̶̶̶̶̶ 3. The length of the longest side of a right triangle is called the ? . ̶̶̶̶̶̶ 1-1 Understanding Points, Lines, and Planes (pp. 6–11) TEKS G.1.A, G.7.A E X A M P L E S ■ Name the common endpoint of SR and ST .  ST are opposite rays with common  SR and endpoint S. 60 60 Chapter 1 Foundations for Geometry EXERCISES Name each of the following. 4. four coplanar points 5. line containing B and C 6. plane that contains A, G, and E �� ■ Draw and label three coplanar lines intersecting in one point. Draw and label each of the following. 7. line containing P and Q 8. pair of opposite rays both containing C 9.  CD intersecting plane P at B 1-2 Measuring and Constructing Segments (pp. 13–19) TEKS G.2.A, G.2.B, G.3.B, G.7.C E X A M P L E S ■ Find the length of XY = ⎜-2 - 1⎟ = ⎜-3⎟ = 3 ̶ XY . ■ S is between R and T. Find RT. RT = RS + ST 3x + 2 = 5x - 6 + 2x 3x + 2 = 7x - 6 x = 2 RT = 3 (2) + 2 = 8 EXERCISES Find each length. 10. JL 11. HK 12. Y is between X and Z, XY = 13.8, and XZ = 21.4. Find YZ. 13. Q is between P and R. Find PR. 14. U is the midpoint of ̶ TV , TU = 3x + 4, and UV = 5x - 2. Find TU, UV, and TV. 15. E is the midpoint of ̶ DF , DE = 9x, and EF = 4x + 10. Find DE, EF, and DF. 1-3 Measuring and Constructing Angles (pp. 20–27) TEKS G.1.A, G.1.B, G.2.A, G.2.B, G.3.B E X A M P L E S EXERCISES ■ Classify each angle as acute, right, or obtuse. 16. Classify each angle as acute, right, or obtuse. ∠ABC acute; ∠CBD acute; ∠ABD obtuse; ∠DBE acute; ∠CBE obtuse ■ ̶ KM bisects ∠JKL, m∠JKM = (3x + 4) °, and m∠MKL = (6x - 5) °. Find m∠JKL. 3x + 4 = 6x - 5 Def. of ∠ bisector 3x + 9 = 6x Add 5 to both sides. 9 = 3x Subtract 3x from both sides. x = 3 Divide both sides by 3. 17. m∠HJL = 116°. Find m∠HJK. 18.  NP bisects ∠MNQ, m∠MNP = (6x - 12) °, and m∠PNQ = (4x + 8) °. Find m∠MNQ. m∠JKL = 3x + 4 + 6x - 5 = 9x -1 = 9 (3) - 1 = 26° Study Guide: Review 61 61 �� 1-4 Pairs of Angles (pp. 28–33) TEKS G.1.A, G.2.B E X A M P L E S EXERCISES ■ Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. ∠1 and ∠2 are only adjacent. ∠2 and ∠4 are not adjacent. ∠2 and ∠3 are adjacent and form a linear pair. ∠1 and ∠4 are adjacent and form a linear pair. ■ Find the measure of the complement and supplement of each angle. 90 - 67.3 = 22.7° 180 - 67.3 = 112.7° 90 - (3x - 8) = (98 - 3x) ° 180 - (3x - 8) = (188 - 3x) ° Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 19. ∠1 and ∠2 20. ∠3 and ∠4 21. ∠2 and ∠5 Find the measure of the complement and supplement of each angle. 22. 23. 24. An angle measures 5 degrees more than 4 times its complement. Find the measure of the angle. 1-5 Using Formulas in Geometry (pp. 36–41) TEKS G.1.A, G.1.B, G.8.A E X A M P L E S EXERCISES ■ Find the perimeter and area of the triangle. P = 2x + 3x + 5 + 10 = 5x + 15 A = 1 _ (3x + 5) (2x) 2 Find the perimeter and area of each figure. 25. 26. = 3 x 2 + 5x 27. 28. ■ Find the circumference and area of the circle to the nearest tenth. C = 2π r = 2π (11) = 22π ≈ 69.1 cm A = π r 2 = π (11) 2 = 121π ≈ 380.1 cm 2 Find the circumference and area of each circle to the nearest tenth. 29. 30. 31. The area of a triangle is 102 m 2 . The base of the triangle is 17 m. What is the height of the triangle? 62 62 Chapter 1 Foundations for Geometry �� 1-6 Midpoint and Distance in the Coordinate Plane (pp. 43–49) TEKS G.1.A, G.7.A, E X A M P L E S EXERCISES G.7.C, G.8.C ̶ AB . Find the missing coordinates Y is the midpoint of of each point. 32. A (3, 2) ; B (-1, 4) ; Y ( 33. A (5, 0) ; B ( 34 (-2, 3) ) ; B (-4, 4) ; Y (-2, 3) Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 35. X (-2, 4) and Y (6, 1) 36. H (0, 3) and K (-2, -4) 37. L (-4, 2) and M (3, -2) ■ X is the midpoint of ̶ CD . C has coordinates (-4, 1) , and X has coordinates (3, -2) . Find the coordinates of D. (3, -2) = ( ) 1 + y -4 + x _ _ , 2 2 -2 = 3 = 6 = -4 + x -4 = 1 + y 10 = x -5 = y The coordinates of D are (10, -5) . ■ Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from (1, 6) to (4, 2) . c2 = a 2 + b 2 d = √  4 - (1) 2 + 2 - (6) 2 3 2 + (-4) 2 = √  = √  9 + 16 = √  25 = 5.0 = 3 2 + 4 2 = 9 + 16 = 25 c = √  25 = 5.0 1-7 Transformations in the Coordinate Plane (pp. 50–55) TEKS G.1.A, G.5.C E X A M P L E S EXERCISES ■ Identify the transformation. Then use arrow notation to describe the transformation. Identify each transformation. Then use arrow notation to describe the transformation. The transformation is a reflection. △ABC → △A′B′C′ ■ The coordinates of the vertices of rectangle HJKL are H (2, -1) , J (5, -1) , K (5, -3) , and L (2, -3) . Find the coordinates of the image of rectangle HJKL after the translation (x, y) → (x - 4, y + 1) . H′ = (2 - 4, -1 + 1) = H′ (-2, 0) J′ = (5 - 4, -1 + 1) = J′ (1, 0) K′ = (5 - 4, -3 + 1) = K′ (1, -2) L′ = (2 - 4, -3 + 1) = L′ (-2, -2) 38. 39. 40. The coordinates for the vertices of △XYZ are X (-5, -4) , Y (-3, -1) , and Z (-2, -2) . Find the coordinates for the image of △XYZ after the translation (x, y) → (x + 4, y + 5) . Study Guide: Review 63 63 �� 1. Draw and label plane N containing two lines that intersect at B. Use the figure to name each of the following. 2. four noncoplanar points 3. line containing B and E 4. The coordinate of A is -3, and the coordinate of B is 0.5. Find AB. 5. E, F, and G represent mile markers along a straight highway. Find EF. 6. J is the midpoint of ̶ HK . Find HJ, JK, and HK. Classify each angle by its measure. 7. m∠LMP = 70° 8. m∠QMN = 90° 9. m∠PMN = 125° 10.  TV bisects ∠RTS. If the m∠RTV = (16x - 6) ° and m∠VTS = (13x + 9) °, what is the m∠RTV? 11. An angle’s measure is 5 degrees less than 3 times the measure of its supplement. Find the measure of the angle and its supplement. Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 12. ∠2 and ∠3 13. ∠4 and ∠5 14. ∠1 and ∠4 15. Find the perimeter and area of a rectangle with b = 8 ft and h = 4 ft. Find the circumference and area of each circle to the nearest tenth. 16. r = 15 m 17. d = 25 ft 18. d = 2.8 cm 19. Find the midpoint of the segment with endpoints (-4, 6) and (3, 2) . 20. M is the midpoint of ̶ LN . M has coordinates (-5, 1) , and L has coordinates (2, 4) . Find the coordinates of N. 21. Given A (-5, 1) , B (-1, 3) , C (1, 4) , and D (4, 1) , is ̶ AB ≅ ̶ CD ? Explain. Identify each transformation. Then use arrow notation to describe the transformation. 22. 23. 24. A designer used the translation (x, y) → (x + 3, y - 3) to transform a triangular-shaped pin ABC. Find the coordinates and draw the image of △ABC. 64 64 Chapter 1 Foundations for Geometry �� FOCUS ON SAT The SAT has three sections: Math, Critical Reading, and Writing. Your SAT scores show how you compare with other students. It can be used by colleges to determine admission and to award merit-based financial aid. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. On SAT multiple-choice questions, you receive one point for each correct answer, but you lose a fraction of a point for each incorrect response. Guess only when you can eliminate at least one of the answer choices. 1. Points D, E, F, and G are on a line, in that order. If DE = 2, FG = 5, and DF = 6, what is the value of EG (DG) ? (A) 13 (B) 18 (C) 19 (D) 42 (E) 99 4. What is the area of the square? (A) 16 (B) 25 (C) 32 (D) 36 (E) 41 2.  QS bisects ∠PQR, m∠PQR = (4x + 2) °, and m∠SQR = (3x - 6) °. What is the value of x? 5. If ∠BFD and ∠AFC are right angles and m∠CFD = 72°, what is the value of x? (A) 1 (B) 4 (C) 7 (D) 10 (E) 19 Note: Figure not drawn to scale. 3. A rectangular garden is enclosed by a brick border. The total length of bricks used to enclose the garden is 42 meters. If the length of the garden is twice the width, what is the area of the garden? (A) 18 (B) 36 (C) 72 (D) 90 (E) 108 (A) 7 meters (B) 14 meters (C) 42 meters (D) 42 square meters (E) 98 square meters College Entrance Exam Practice 65 65 �� Multiple Choice: Work Backward When you do not know how to solve a multiple-choice test item, use the answer choices and work the question backward. Plug in the answer choices to see which choice makes the question true. T is the midpoint of ̶ RC , RT = 12x - 8, and TC = 28. What is the value of x? -4 2 3 28 Since T is the midpoint of ̶ RC , then RT = RC, or 12x - 8 = 28. Find what value of x makes the left side of the equation equal 28. Try choice A: If x = -4, then 12x - 8 = 12 (-4) - 8 = -56. This choice is not correct because length is always a positive number. Try choice B: If x = 2, then 12x - 8 = 12 (2) - 8 = 16. Since 16 ≠ 28, choice B is not the answer. Try choice C: If x = 3, then 12x - 8 = 12 (3) - 8 = 28. Since 28 = 28, the correct answer is C, 3. Joel used 6400 feet of fencing to make a rectangular horse pen. The width of the pen is 4 times as long as the length. What is the length of the horse pen? 25 feet 480 feet 640 feet 1600 feet Use the formula P = 2ℓ + 2w. P = 6400 and w = 4ℓ. You can work backward to determine which answer choice is the mo
st reasonable. Try choice J: Use mental math. If ℓ = 1600, then 4ℓ = 6400. This choice is not reasonable because the perimeter of the pen would then be far greater than 6400 feet. Try choice F: Use mental math. If ℓ = 25, then 4ℓ = 100. This choice is incorrect because the perimeter of the pen is 6400 ft, which is far greater than 2 (25) + 2 (100) . Try choice H: If ℓ = 640, then 4ℓ = 2560. When you substitute these values into the perimeter formula, it makes a true statement. The correct answer is H, 640 ft. 66 66 Chapter 1 Foundations for Geometry �� Read each test item and answer the questions that follow. �� Item A The measure of an angle is 3 times as great as that of its complement. Which value is the measure of the smaller angle? 22.5° 27.5° 63.5° 67.5° 1. Are there any definitions that you can use to solve this problem? If so, what are they? 2. Describe how to work backward to find the correct answer. When you work a test question backward start with choice C. The choices are usually listed in order from least to greatest. If choice C is incorrect because it is too low, you do not need to plug in the smaller numbers. Item D △QRS has vertices at Q (3, 5) , R (3, 9) , and S (7, 5) . Which of these points is a vertex of the image of △QRS after the translation (x, y) → (x - 7, y - 6) ? Item B In a town’s annual relay marathon race, the second runner of each team starts at mile marker 4 and runs to the halfway point of the 26-mile marathon. At that point the second runner passes the relay baton to the third runner of the team. How many total miles does the second runner of each team run? 4 miles 6.5 miles 9 miles 13 miles (-4, 3) (0, 0) (4, 1) (4, -3) 7. Explain how to use mental math to find an answer that is NOT reasonable. 8. Describe, by working backward, how you can determine the correct answer. 3. Which answer choice should you plug in first? Why? 4. Describe, by working backward, how you know that choices F and G are not correct. Item E  TS bisects ∠PTR. If m∠PTS = (9x + 2) ° and m∠STR = (x + 18) °, what is the value of x? Item C Consider the translation (-2, 8) → (8, -4) . What number was added to the x-coordinate? -12 -6 4 10 -10 0 2 20 5. Which answer choice should you plug in first? Why? 6. Explain how to work the test question backward to determine the correct answer. 9. Explain how to use mental math to find an answer that is NOT reasonable. 10. Describe how to use the answer choices to work backward to find which answer is reasonable. TAKS Tackler 67 67 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTER 1 Multiple Choice Use the diagram for Items 1–3. Use the diagram for Items 8–10. 1. Which points are collinear? A, B, and C B, C, and D A, B, and E B, D, and E 2. What is another name for plane R? Plane C Plane AB Plane ACE Plane BDE 3. Use your protractor to find the approximate measure of ∠ABD. 123° 117° 77° 63° 4. S is between R and T. The distance between R and T is 4 times the distance between S and T. If RS = 18, what is RT? 24 22.5 14.4 6 5. A ray bisects a straight angle into two congruent angles. Which term describes each of the congruent angles that are formed? Acute Obtuse Right Straight 6. Which expression states that ̶ AB is congruent ̶ CD ? to AB ≅ CD AB = CD ̶ AB = ̶ AB ≅ ̶ CD ̶ CD 7. The measure of an angle is 35°. What is the measure of its complement? 35° 45° 55° 145° 68 68 Chapter 1 Foundations for Geometry 8. Which of these angles is adjacent to ∠MQN? ∠QMN ∠NPQ ∠QNP ∠PQN 9. What is the area of △NQP? 3.7 square meters 7.4 square meters 6.8 square meters 13.6 square meters 10. Which of the following pairs of angles are complementary? ∠MNQ and ∠QNP ∠NQP and ∠QPN ∠MNP and ∠QNP ∠QMN and ∠NPQ 11. K is the midpoint of ̶ JL . J has coordinates (2, -1) , and K has coordinates (-4, 3) . What are the coordinates of L? (3, -2) (1, -1) (-1, 1) (-10, 7) 12. A circle with a diameter of 10 inches has a circumference equal to the perimeter of a square. To the nearest tenth, what is the length of each side of the square? 2.5 inches 3.9 inches 5.6 inches 7.9 inches 13. The map coordinates of a campground are (1, 4) , and the coordinates of a fishing pier are (4, 7) . Each unit on the map represents 1 kilometer. If Alejandro walks in a straight line from the campground to the pier, how many kilometers, to the nearest tenth, will he walk? 3.5 kilometers 6.0 kilometers 4.2 kilometers 12.1 kilometers �� For many types of geometry problems, it may be helpful to draw a diagram and label it with the information given in the problem. This method is a good way of organizing the information and helping you decide how to solve the problem. 14. m∠R is 57°. What is the measure of its supplement? 33° 43° 123° 133° 15. What rule would you use to translate a triangle STANDARDIZED TEST PREP Short Response 23. △ABC has vertices A (-2, 0) , B (0, 0) , and C (0, 3) . The image of △ABC has vertices A′(1, -4), B′ (3, -4) , and C′ (3, -1) . a. Draw △ABC and its image △A′B′C′ on a coordinate plane. b. Write a rule for the transformation of △ABC using arrow notation. 24. You are given the measure of ∠4. You also know the following angles are supplementary: ∠1 and ∠2, ∠2 and ∠3, and ∠1 and ∠4. 4 units to the right? (x, y) → (x + 4, y) (x, y) → (x - 4, y) (x, y) → (x, y + 4) (x, y) → (x, y - 4) 16. If ̶ WZ bisects ∠XWY, which of the following statements is true? m∠XWZ > m∠YWZ m∠XWZ < m∠YWZ m∠XWZ = m∠YWZ m∠XWZ ≅ m∠YWZ 17. The x- and y-axes separate the coordinate plane into four regions, called quadrants. If (c, d) is a point that is not on the axes, such that c < 0 and d < 0, which quadrant would contain point (c, d) ? I III II IV Gridded Response 18. The measure of ∠1 is 4 times the measure of its supplement. What is the measure, in degrees, of ∠1? 19. The exits for Market St. and Finch St. are 3.5 miles apart on a straight highway. The exit for King St. is at the midpoint between these two exits. How many miles apart are the King St. and Finch St. exits? 20. R has coordinates (-4, 9) . S has coordinates (4, -6) . What is RS? 21. If ∠A is a supplement of ∠B and is a right angle, then what is m∠B in degrees? 22. ∠C and ∠D are complementary. m∠C is 4 times m∠D. What is m∠C? Explain how you can determine the measures of ∠1, ∠2, and ∠3. 25. Marian is making a circular tablecloth from a rectangular piece of fabric that measures 6 yards by 4 yards. What is the area of the largest circular piece that can be cut from the fabric? Leave your answer in terms of π. Show your work or explain in words how you found your answer. Extended Response 26. Demara is creating a design using a computer illustration program. She begins by drawing the rectangle shown on the coordinate grid. a. Demara translates rectangle PQRS using the rule (x, y) → (x - 4, y - 6) . On a copy of the coordinate grid, draw this translation and label each vertex. b. Describe one way that Demara could have moved rectangle PQRS to the same position in part a using a reflection and then a translation. c. On the same coordinate grid, Demara reflects rectangle PQRS across the x-axis. She draws a figure with vertices at (1, -3) , (3, -3) , (3, -5) , and (1, -5) . Did Demara reflect rectangle PQRS correctly? Explain your answer. Cumulative Assessment, Chapter 1 69 69 �� Geometric Reasoning 2A Inductive and Deductive Reasoning 2-1 Using Inductive Reasoning to Make Conjectures 2-2 Conditional Statements 2-3 Using Deductive Reasoning to Verify Conjectures Lab Solve Logic Puzzles 2-4 Biconditional Statements and Definitions 2B Mathematical Proof 2-5 Algebraic Proof 2-6 Geometric Proof Lab Design Plans for Proofs 2-7 Flowchart and Paragraph Proofs Ext Introduction to Symbolic Logic KEYWORD: MG7 ChProj A corn maze from the 7A Ranch near Hondo 70 70 Chapter 2 Vocabulary Match each term on the left with a definition on the right. 1. angle A. a straight path that has no thickness and extends forever 2. line 3. midpoint 4. plane 5. segment B. a figure formed by two rays with a common endpoint C. a flat surface that has no thickness and extends forever D. a part of a line between two points E. names a location and has no size F. a point that divides a segment into two congruent segments Angle Relationships Select the best description for each labeled angle pair. 6. 7. 8. linear pair or vertical angles adjacent angles or vertical angles supplementary angles or complementary angles Classify Real Numbers Tell if each number is a natural number, a whole number, an integer, or a rational number. Give all the names that apply. 9. 6 11. –3 12. 5.2 10. –0.8 13. 3_ 8 Points, Lines, and Planes Name each of the following. 15. a point 16. a line 17. a ray 18. a segment 19. a plane 14. 0 � � � � � � � Solve One-Step Equations Solve. 20. 8 + x = 5 23. p - 7 = 9 21. 6y = -12 24. z_ 5 = 5 22. 9 = 6s 25. 8.4 = -1.2r Geometric Reasoning 71 71 �� Key Vocabulary/Vocabulario conjecture conjetura counterexample contraejemplo deductive reasoning razonamiento deductivo inductive reasoning razonamiento inductivo polygon proof polígono demostración quadrilateral cuadrilátero theorem triangle teorema triángulo Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word counterexample is made up of two words: counter and example. In this case, counter means “against.” What is a counterexample to the statement “All numbers are positive”? 2. The root of the word inductive is ducere, which means “to lead.” When you are inducted into a club, you are “led into” membership. When you use inductive reasoning in math, you start with specific examples. What do you think inductive reasoning leads you to? 3. In Greek, the word poly means “many,” and the word gon means “angle.” How can you use these meanings to understand the term polygon ? Geometry TEKS G.1.A Geometric structure* d
evelop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems Les. 2-1 Les. 2-2 Les. 2-3 2-3 Geo. Lab Les. 2-4 Les. 2-5 Les. 2-6 2-6 Geo. Lab Les. 2-7 Ext. ★ ★ ★ G.2.B Geometric structure* make conjectures ... and determine ★ ★ ★ the validity of the conjectures, ... G.3.A Geometric structure* determine the validity of a conditional statement, its converse, inverse, and contrapositive ★ ★ G.3.B Geometric structure* construct and justify statements ★ ★ ★ ★ ★ about geometric figures and their properties G.3.C Geometric structure* use logical reasoning to prove ★ ★ ★ ★ ★ ★ ★ statements are true and find counterexamples to disprove statements that are false G.3.D Geometric structure* use inductive reasoning to ★ formulate a conjecture G.3.E Geometric structure* use deductive reasoning to prove a ★ ★ ★ ★ ★ statement G.4.A Geometric structure* select an appropriate representation ... in order to solve problems G.5.B Geometric patterns* use numeric and geometric patterns ★ to make generalizations about geometric properties ... ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 72 72 Chapter 2 Reading Strategy: Read and Interpret a Diagram A diagram is an informational tool. To correctly read a diagram, you must know what you can and cannot assume based on what you see in it. ✔ Collinear points ✔ Betweenness of points ✔ Coplanar points ✘ Measures of segments ✘ Measures of angles ✘ Congruent segments ✔ Straight angles and lines ✘ Congruent angles ✔ Adjacent angles ✔ Linear pairs of angles ✔ Vertical angles ✘ Right angles If a diagram includes labeled information, such as an angle measure or a right angle mark, treat this information as given. ✔ Points A, B, and C are collinear. ✘ ∠CBD is acute. ✔ Points A, B, C, and D are coplanar. ✔ B is between A and C. ✔   AC is a line. ✔ ∠ABD and ∠CBD are adjacent angles. ✔ ∠ABD and ∠CBD form a linear pair. ✘ ∠ABD is obtuse. ̶̶ BC ̶̶ AB ≅ ✘ Try This List what you can and cannot assume from each diagram. 1. 2. Geometric Reasoning 73 73 �� 2-1 Using Inductive Reasoning to Make Conjectures TEKS G.3.D Geometric structure: use inductive reasoning to formulate a conjecture. Also G.2.B, G.5.B Objectives Use inductive reasoning to identify patterns and make conjectures. Find counterexamples to disprove conjectures. Who uses this? Biologists use inductive reasoning to develop theories about migration patterns. Vocabulary inductive reasoning conjecture counterexample Biologists studying the migration patterns of California gray whales developed two theories about the whales’ route across Monterey Bay. The whales either swam directly across the bay or followed the shoreline. E X A M P L E 1 Identifying a Pattern Find the next item in each pattern. A Monday, Wednesday, Friday, … Alternating days of the week make up the pattern. The next day is Sunday. B 3, 6, 9, 12, 15, … Multiples of 3 make up the pattern. The next multiple is 18. C ←, ↖, ↑, … In this pattern, the figure rotates 45° clockwise each time. The next figure is ↗. 1. Find the next item in the pattern 0.4, 0.04, 0.004, … When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductive reasoning to draw a conclusion from a pattern. A statement you believe to be true based on inductive reasoning is called a conjecture . E X A M P L E 2 Making a Conjecture Complete each conjecture. A The product of an even number and an odd number is ? . ̶̶̶ List some examples and look for a pattern. (2) (3) = 6 The product of an even number and an odd number is even. (4) (3) = 12 (2) (5) = 10 (4) (5) = 20 74 74 Chapter 2 Geometric Reasoning Complete each conjecture. B The number of segments formed by n collinear points is ? . ̶̶̶ Draw a segment. Mark points on the segment, and count the number of individual segments formed. Be sure to include overlapping segments. Points Segments = 10 The number of segments formed by n collinear points is the sum of the whole numbers less than n. 2. Complete the conjecture: The product of two odd numbers is ? . ̶̶̶ E X A M P L E 3 Biology Application To learn about the migration behavior of California gray whales, biologists observed whales along two routes. For seven days they counted the numbers of whales seen along each route. Make a conjecture based on the data. Numbers of Whales Each Day Direct Route Shore Route More whales were seen along the shore route each day. The data supports the conjecture that most California gray whales migrate along the shoreline. 3. Make a conjecture about the lengths of male and female whales based on the data. Average Whale Lengths Length of Female (ft) Length of Male (ft) 49 47 51 45 50 44 48 46 51 48 47 48 To show that a conjecture is always true, you must prove it. To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample . A counterexample can be a drawing, a statement, or a number. Inductive Reasoning 1. Look for a pattern 2. Make a conjecture. 3. Prove the conjecture or find a counterexample. 2- 1 Using Inductive Reasoning to Make Conjectures 75 75 E X A M P L E 4 Finding a Counterexample Show that each conjecture is false by finding a counterexample. A For all positive numbers n, 1 _ n ≤ n. Pick positive values for n and substitute them into the equation to see if the conjecture holds. Let n = 1. Since 1 _ n = 1 and 1 ≤ 1, the conjecture holds. Let n = 2. Since 1 _ n = 1 _ and 1 _ ≤ 2, the conjecture holds. 2 2 . Since 1 _ n = 1 _ Let and 2 ≰ 1 _ 2 , the conjecture is false. n = 1 _ 2 is a counterexample. B For any three points in a plane, there are three different lines that contain two of the points. Draw three collinear points. If the three points are collinear, the conjecture is false. C The temperature in Abilene, Texas, never exceeds 100°F during the spring months (March, April, and May). Monthly High Temperatures (°F) in Abilene, Texas Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 88 89 97 99 107 109 110 107 106 103 92 89 The temperature in May was 107°F, so the conjecture is false. Show that each conjecture is false by finding a counterexample. 4a. For any real number x, x 2 ≥ x. 4b. Supplementary angles are adjacent. 4c. The radius of every planet in the solar system is less than 50,000 km. Planets’ Diameters (km) Mercury Venus Earth Mars Jupiter Saturn Uranus Nepture Pluto 4880 12,100 12,800 6790 143,000 121,000 51,100 49,500 2300 THINK AND DISCUSS 1. Can you prove a conjecture by giving one example in which the conjecture is true? Explain your reasoning. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the steps of the inductive reasoning process. 76 76 Chapter 2 Geometric Reasoning �� 2-1 Exercises Exercises KEYWORD: MG7 2-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain why a conjecture may be true or false. 74 Find the next item in each pattern. , 3 _ , 2 _ 3. 1 _ 5 4 3 2. March, May, July, … , … 4 Complete each conjecture. p. 74 5. The product of two even numbers is ? . ̶̶̶ 6. A rule in terms of n for the sum of the first n odd positive integers is ? . ̶̶̶ . Biology A laboratory culture contains 150 bacteria. After twenty minutes, the p. 75 culture contains 300 bacteria. After one hour, the culture contains 1200 bacteria. Make a conjecture about the rate at which the bacteria increases. 76 Show that each conjecture is false by finding a counterexample. 8. Kennedy is the youngest U.S. president to be inaugurated. 9. Three points on a plane always form a triangle. 10. For any real number x, if x 2 ≥ 1, then x ≥ 1. President Washington T. Roosevelt Truman Kennedy Clinton Age at Inauguration 57 42 60 43 46 Independent Practice For See Exercises Example 11–13 14–15 16 17–19 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S6 Application Practice p. S29 PRACTICE AND PROBLEM SOLVING Find the next item in each pattern. 11. 8 A.M., 11 A.M., 2 P.M., … 12. 75, 64, 53, … 13. △, □, , … Complete each conjecture. 14. A rule in terms of n for the sum of the first n even positive integers is ? . ̶̶̶ 15. The number of nonoverlapping segments formed by n collinear points is ? . ̶̶̶ 16. Industrial Arts About 5% of the students at Lubbock High School usually participate in the robotics competition. There are 526 students in the school this year. Make a conjecture about the number of students who will participate in the robotics competition this year. Show that each conjecture is false by finding a counterexample. 17. If 1 - y > 0, then 0 < y < 1. 18. For any real number x, x 3 ≥ x 2 . 19. Every pair of supplementary angles includes one obtuse angle. Make a conjecture about each pattern. Write the next two items. , 1 _ , 1 _ 21. 1 _ 4 8 2 20. 2, 4, 16, … , … 22. –3, 6, –9, 12, … 23. Draw a square of dots. Make a conjecture about the number of dots needed to increase the size of the square from n × n to (n + 1) × (n + 1) . 2- 1 Using Inductive Reasoning to Make Conjectures 77 77 �� Math History Goldbach first stated his conjecture in a letter to Leonhard Euler in 1742. Euler, a Swiss mathematician who published over 800 papers, replied, “I consider [the conjecture] a theorem which is quite true, although I cannot demonstrate it.” Determine if each conjecture is true. If not, write or draw a counterexample. 24. Points X, Y, and Z are coplanar. 25. If n is an integer, then –n is positive. 26. In a triangle with one right angle, two of the sides are congruent. 27. If  BD bisects ∠ABC, then m∠ABD = m∠CBD. 28. Estimation The Westside High School band is selling coupon books to raise money for a trip. The table shows the amount of money raised for the first four days of the sale. If the pattern continues, estimate the amount of money ra
ised during the sixth day. Day Money Raised ($) 1 2 3 4 146.25 195.75 246.25 295.50 29. Write each fraction in the pattern 1 _ 11 description of the fraction pattern and the resulting decimal pattern. , 2 _ 11 , 3 _ 11 , … as a repeating decimal. Then write a 30. Math History Remember that a prime number is a whole number greater than 1 that has exactly two factors, itself and 1. Goldbach’s conjecture states that every even number greater than 2 can be written as the sum of two primes. For example, 4 = 2 + 2. Write the next five even numbers as the sum of two primes. 31. The pattern 1, 1, 2, 3, 5, 8, 13, 21, … is known as the Fibonacci sequence. Find the next three terms in the sequence and write a conjecture for the pattern. 32. Look at a monthly calendar and pick any three squares in a row—across, down, or diagonal. Make a conjecture about the number in the middle. 33. Make a conjecture about the value of 2n - 1 when n is an integer. 34. Critical Thinking The turnaround date for migrating gray whales occurs when the number of northbound whales exceeds the number of southbound whales. Make a conjecture about the turnaround date, based on the table below. What factors might affect the validity of your conjecture in the future? Migration Direction of Gray Whales Feb. 16 Feb. 17 Feb. 18 Feb. 19 Feb. 20 Feb. 21 Feb. 22 Southbound Northbound 35. Write About It Explain why a true conjecture about even numbers does not necessarily hold for all numbers. Give an example to support your answer. 36. This problem will prepare you for the Multi-Step TAKS Prep on page 102. a. For how many hours did the Mock Turtle do lessons on the third day? b. On what day did the Mock Turtle do 1 hour of lessons? “And how many hours a day did you do lessons?” said Alice, in a hurry to change the subject. “Ten hours the first day,” said the Mock Turtle: “nine the next, and so on.” 78 78 Chapter 2 Geometric Reasoning �� 37. Which of the following conjectures is false? If x is odd, then x + 1 is even. The sum of two odd numbers is even. The difference of two even numbers is positive. If x is positive, then –x is negative. 38. A student conjectures that if x is a prime number, then x + 1 is not prime. Which of the following is a counterexample? x = 11 x = 6 x = 3 x = 2 39. The class of 2004 holds a reunion each year. In 2005, 87.5% of the 120 graduates attended. In 2006, 90 students went, and in 2007, 75 students went. About how many students do you predict will go to the reunion in 2010? 12 15 24 30 CHALLENGE AND EXTEND 40. Multi-Step Make a table of values for the rule x 2 + x + 11 when x is an integer from 1 to 8. Make a conjecture about the type of number generated by the rule. Continue your table. What value of x generates a counterexample? 41. Political Science Presidential elections are held every four years. U.S. senators are elected to 6-year terms, but only 1 __ 3 of the Senate is up for election every two years. If 1 __ 3 of the Senate is elected during a presidential election year, how many years must pass before these same senate seats are up for election during another presidential election year? 42. Physical Fitness Rob is training for the President’s Challenge physical fitness program. During his first week of training, Rob does 15 sit-ups each day. He will add 20 sit-ups to his daily routine each week. His goal is to reach 150 sit-ups per day. a. Make a table of the number of sit-ups Rob does each week from week 1 through week 10. b. During which week will Rob reach his goal? c. Write a conjecture for the number of sit-ups Rob does during week n. ̶̶ AB and is the ̶̶ BC . Compare m∠CAB and m∠CBA ̶̶ AB . Then construct point C so that it is not on ̶̶ AC and 43. Construction Draw same distance from A and B. Construct and compare AC and CB. Make a conjecture. SPIRAL REVIEW Determine if the given point is a solution to y = 3x - 5. (Previous course) 44. (1, 8) 45. (-2, -11) 46. (3, 4) 47. (-3.5, 0.5) Find the perimeter or circumference of each of the following. Leave answers in terms of x. (Lesson 1-5) 48. a square whose area is x 2 49. a rectangle with dimensions x and 4x - 3 50. a triangle with side lengths of x + 2 51. a circle whose area is 9π x 2 A triangle has vertices (-1, -1) , (0, 1) , and (4, 0) . Find the coordinates for the vertices of the image of the triangle after each transformation. (Lesson 1-7) 52. (x, y) → (x, y + 2) 53. (x, y) → (x + 4, y - 1) 2- 1 Using Inductive Reasoning to Make Conjectures 79 79 Venn Diagrams Number Theory Recall that in a Venn diagram, ovals are used to represent each set. The ovals can overlap if the sets share common elements. The real number system contains an infinite number of subsets. The following chart shows some of them. Other examples of subsets are even numbers, multiples of 3, and numbers less than 6. See Skills Bank pages S53 and S81 Set Description Natural numbers The counting numbers Examples 1, 2, 3, 4, 5, … Whole numbers The set of natural numbers and 0 0, 1, 2, 3, 4, … Integers The set of whole numbers and their opposites …, -2, -1, 0, 1, 2, … Rational numbers The set of numbers that can be written as a ratio of integers - 3 _ 4 , 5, -2, 0.5, 0 Irrational numbers The set of numbers that cannot be written as a ratio of integers π, √  10 , 8 + √  2 Example Draw a Venn diagram to show the relationship between the set of even numbers and the set of natural numbers. The set of even numbers includes all numbers that are divisible by 2. This includes natural numbers such as 2, 4, and 6. But even numbers such as –4 and –10 are not natural numbers. So the set of even numbers includes some, but not all, elements in the set of natural numbers. Similarly, the set of natural numbers includes some, but not all, even numbers. Draw a rectangle to represent all real numbers. Draw overlapping ovals to represent the sets of even and natural numbers. You may write individual elements in each region. Try This TAKS Grades 9–11 Obj. 1, 10 Draw a Venn diagram to show the relationship between the given sets. 1. natural numbers, whole numbers whole numbers 2. odd numbers, 3. irrational numbers, integers 80 80 Chapter 2 Geometric Reasoning �� 2-2 Conditional Statements TEKS G.3.A Geometric structure: determine the validity of a conditional statement, its converse, inverse, and contrapositive. Also G.3.C Objectives Identify, write, and analyze the truth value of conditional statements. Why learn this? To identify a species of butterfly, you must know what characteristics one butterfly species has that another does not. Write the inverse, converse, and contrapositive of a conditional statement. Vocabulary conditional statement hypothesis conclusion truth value negation converse inverse contrapositive logically equivalent statements It is thought that the viceroy butterfly mimics the bad-tasting monarch butterfly to avoid being eaten by birds. By comparing the appearance of the two butterfly species, you can make the following conjecture: If a butterfly has a curved black line on its hind wing, then it is a viceroy. Conditional Statements DEFINITION SYMBOLS VENN DIAGRAM A conditional statement is a statement that can be written in the form “if p, then q.” The hypothesis is the part p of a conditional statement following the word if. p → q The conclusion is the part q of a conditional statement following the word then. By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion. E X A M P L E 1 Identifying the Parts of a Conditional Statement “If p, then q” can also be written as “if p, q,” “q, if p,” “p implies q,” and “p only if q.” Identify the hypothesis and conclusion of each conditional. A If a butterfly has a curved black line on its hind wing, then it is a viceroy. Hypothesis: A butterfly has a curved black line on its hind wing. Conclusion: The butterfly is a Viceroy. B A number is an integer if it is a natural number. Hypothesis: A number is a natural number. Conclusion: The number is an integer. 1. Identify the hypothesis and conclusion of the statement “A number is divisible by 3 if it is divisible by 6.” Many sentences without the words if and then can be written as conditionals. To do so, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other. 2- 2 Conditional Statements 81 81 �� E X A M P L E 2 Writing a Conditional Statement Write a conditional statement from each of the following. A The midpoint M of a segment bisects the segment. The midpoint M of a segment bisects the segment. Conditional: If M is the midpoint of a segment, and conclusion. Identify the hypothesis then M bisects the segment. B The inner oval represents the hypothesis, and the outer oval represents the conclusion. Conditional: If an animal is a tarantula, then it is a spider. 2. Write a conditional statement from the sentence “Two angles that are complementary are acute.” A conditional statement has a truth value of either true (T) or false (F). It is false only when the hypothesis is true and the conclusion is false. Consider the conditional “If I get paid, I will take you to the movie.” If I don’t get paid, I haven’t broken my promise. So the statement is still true. To show that a conditional statement is false, you need to find only one counterexample where the hypothesis is true and the conclusion is false. E X A M P L E 3 Analyzing the Truth Value of a Conditional Statement Determine if each conditional is true. If false, give a counterexample. A If you live in El Paso, then you live in Texas. When the hypothesis is true, the conclusion is also true because El Paso is in Texas. So the conditional is true. B If an angle is obtuse, then it has a measure of 100°. You can draw an obtuse angle whose measure is not 100°. In this case, the hypothesis is true, but the conclusion is false. Since you can find a counterexample, the conditional is false
. C If an odd number is divisible by 2, then 8 is a perfect square. An odd number is never divisible by 2, so the hypothesis is false. The number 8 is not a perfect square, so the conclusion is false. However, the conditional is true because the hypothesis is false. 3. Determine if the conditional “If a number is odd, then it is divisible by 3” is true. If false, give a counterexample. If the hypothesis is false, the conditional statement is true, regardless of the truth value of the conclusion. The negation of statement p is “not p,” written as ∼p. The negation of the statement “M is the midpoint of The negation of a true statement is false, and the negation of a false statement is true. Negations are used to write related conditional statements. ̶̶ AB ” is “M is not the midpoint of ̶̶ AB .” 82 82 Chapter 2 Geometric Reasoning �� Related Conditionals DEFINITION SYMBOLS A conditional is a statement that can be written in the form “If p, then q.” The converse is the statement formed by exchanging the hypothesis and conclusion. The inverse is the statement formed by negating the hypothesis and the conclusion. The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion. p → q q → p ∼p → ∼q ∼q → ∼p E X A M P L E 4 Biology Application Moth The logical equivalence of a conditional and its contrapositive is known as the Law of Contrapositive. Write the converse, inverse, and contrapositive of the conditional statement. Use the photos to find the truth value of each. If an insect is a butterfly, then it has four wings. If an insect is a butterfly, then it has four wings. Converse: If an insect has four wings, then it is a butterfly. A moth also is an insect with four wings. So the converse is false. Inverse: If an insect is not a butterfly, then it does not have four wings. A moth is not a butterfly, but it has four wings. So the inverse is false. Contrapositive: If an insect does not have four wings, then it is not Butterfly a butterfly. Butterflies must have four wings. So the contrapositive is true. 4. Write the converse, inverse, and contrapositive of the conditional statement “If an animal is a cat, then it has four paws.” Find the truth value of each. In the example above, the conditional statement and its contrapositive are both true, and the converse and inverse are both false. Related conditional statements that have the same truth value are called logically equivalent statements . A conditional and its contrapositive are logically equivalent, and so are the converse and inverse. Statement Example Truth Value Conditional If m∠A = 95°, then ∠A is obtuse. Converse Inverse If ∠A is obtuse, then m∠A = 95°. If m∠A ≠ 95°, then ∠A is not obtuse. Contrapositive If ∠A is not obtuse, then m∠A ≠ 95°. T F F T However, the converse of a true conditional is not necessarily false. All four related conditionals can be true, or all four can be false, depending on the statement. 2- 2 Conditional Statements 83 83 THINK AND DISCUSS 1. If a conditional statement is false, what are the truth values of its hypothesis and conclusion? 2. What is the truth value of a conditional whose hypothesis is false? 3. Can a conditional statement and its converse be logically equivalent? Support your answer with an example. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the definition and give an example. 2-2 Exercises Exercises KEYWORD: MG7 2-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. The ? of a conditional statement is formed by exchanging the hypothesis ̶̶̶̶ and conclusion. (converse, inverse, or contrapositive) 2. A conditional and its contrapositive are value. (logically equivalent or converses) ? because they have the same truth ̶̶̶̶ Identify the hypothesis and conclusion of each conditional. p. 81 3. If a person is at least 16 years old, then the person can drive a car. 4. A figure is a parallelogram if it is a rectangle. 5. The statement a - b < a implies that b is a positive number Write a conditional statement from each of the following. p. 82 6. Eighteen-year-olds are eligible to vote. 2 7 when 0 < a < b. 8 Determine if each conditional is true. If false, give a counterexample. p. 82 9. If three points form the vertices of a triangle, then they lie in the same plane. 10. If x > y, then ⎜x⎟ > ⎜y⎟ . 11. If the season is spring, then the month is March 12. Travel Write the converse, inverse, and contrapositive of the following conditional p. 83 statement. Find the truth value of each. If Brielle drives at exactly 30 mi/h, then she travels 10 mi in 20 min. 84 84 Chapter 2 Geometric Reasoning �� Independent Practice For See Exercises Example 13–15 16–18 19–21 22–23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S6 Application Practice p. S29 PRACTICE AND PROBLEM SOLVING Identify the hypothesis and conclusion of each conditional. 13. If an animal is a tabby, then it is a cat. 14. Four angles are formed if two lines intersect. 15. If 8 ounces of cereal cost $2.99, then 16 ounces of cereal cost $5.98. Write a conditional statement from each sentence. 16. You should monitor the heart rate of a patient who is ill. 17. After three strikes, the batter is out. 18. Congruent segments have equal measures. Determine if each conditional is true. If false, give a counterexample. 19. If you subtract -2 from -6, then the result is -4. 20. If two planes intersect, then they intersect in exactly one point. 21. If a cat is a bird, then today is Friday. Write the converse, inverse, and contrapositive of each conditional statement. Find the truth value of each. 22. Probability If the probability of an event is 0.1, then the event is unlikely to occur. 23. Meteorology If freezing rain is falling, then the air temperature is 32°F or less. (Hint: The freezing point of water is 32°F.) Find the truth value of each statement. 24. E lies in plane R. 25.  CD lies in plane F. 26. C, E, and D are coplanar. 27. Plane F contains  ED . 28. B and E are collinear. 29.  BC contains F and R. Draw a Venn diagram. 30. All integers are rational numbers. 31. All natural numbers are real. 32. All rectangles are quadrilaterals. 33. Plane is an undefined term. Write a conditional statement from each Venn diagram. 34. 35. 36. 37. This problem will prepare you for the Multi-Step TAKS Prep on page 102. a. Identify the hypothesis and conclusion in the Duchess’s statement. b. Rewrite the Duchess’s claim as a conditional statement. “Tut, tut, child!” said the Duchess. “Everything’s got a moral, if only you can find it.” And she squeezed herself up closer to Alice’s side as she spoke. 2- 2 Conditional Statements 85 85 �� Find a counterexample to show that the converse of each conditional is false. 38. If x = -5, then x 2 = 25. 39. If two angles are vertical angles, then they are congruent. 40. If two angles are adjacent, then they share a vertex. 41. If you use sunscreen, then you will not get sunburned. Geology Geology Mohs’ scale is used to identify minerals. A mineral with a higher number is harder than a mineral with a lower number. Mohs’ Scale Hardness Mineral Diamond is four times as hard as the next mineral on Mohs’ scale, corundum (ruby and sapphire). Use the table and the statements below for Exercises 42–47. Write each conditional and find its truth value. p: calcite q: not apatite r: a hardness of 3 s: a hardness less than 5 42. p → r 45. q → p 43. s → q 46. r → q 44. q → s 47. p → s 48. Critical Thinking Consider the conditional “If two angles are congruent, then they have the same measure.” Write the converse, inverse, and contrapositive and find the truth value of each. Use the related conditionals to draw a Venn diagram that represents the relationship between congruent angles and their measures 10 Talc Gypsum Calcite Fluorite Apatite Orthoclase Quartz Topaz Corundum Diamond 49. Write About It When is a conditional statement false? Explain why a true conditional statement can have a hypothesis that is false. 50. What is the inverse of “If it is Saturday, then it is the weekend”? If it is the weekend, then it is Saturday. If it is not Saturday, then it is the weekend. If it is not Saturday, then it is not the weekend. If it is not the weekend, then it is not Saturday. 51. Let a represent “Two lines are parallel to the same line,” and let b represent “The two lines are parallel.” Which symbolic statement represents the conditional “If two lines are NOT parallel, then they are parallel to the same line”? a → b b → a ∼b → a b → ∼a 52. Which statement is a counterexample for the conditional statement “If f (x) = √  25 - x 2 , then f (x) is positive”? 53. Which statement has the same truth value as its converse? If a triangle has a right angle, its side lengths are 3 centimeters, 4 centimeters, and 5 centimeters. If an angle measures 104°, then the angle is obtuse. If a number is an integer, then it is a natural number. If an angle measures 90°, then it is an acute angle. 86 86 Chapter 2 Geometric Reasoning CHALLENGE AND EXTEND For each Venn diagram, write two statements beginning with Some, All, or No. 54. 55. 56. Given: If a figure is a square, then it is a rectangle. Figure A is not a rectangle. Conclusion: Figure A is not a square. a. Draw a Venn diagram to represent the given conditional statement. Use the Venn diagram to explain why the conclusion is valid. b. Write the contrapositive of the given conditional. How can you use the contrapositive to justify the conclusion? 57. Multi-Step How many true conditionals can you write using the statements below? r : n is a natural number. q: n is a whole number. p: n is an integer. SPIRAL REVIEW Write a rule to describe each relationship. (Previous course) 58. x -8 y -5 4 7 7 9 10 12 59. x -3 -1 y -5 -1 0 1 4 9 60. x -2 0 y -9 -4 4 6 6 11 Deter
mine whether each statement is true or false. If false, explain why. (Lesson 1-4) 61. If two angles are complementary and congruent, then the measure of each is 45°. 62. A pair of acute angles can be supplementary. 63. A linear pair of angles is also a pair of supplementary angles. Find the next item in each pattern. (Lesson 2-1) 64. 1, 13, 131, 1313, … , 2 _ , 2 _ 65. 2, 2 _ 27 9 3 , … 66. x: What high school math classes did you take? A: I took three years of math: Pre-Algebra, Algebra, and Geometry. KEYWORD: MG7 Career Q: What training do you need to be a desktop publisher? A: Most of my training was done on the job. The computer science and typing classes I took in high school have been helpful. Q: How do you use math? A: Part of my job is to make sure all the text, charts, and photographs are formatted to fit the layout of each page. I have to manipulate things by comparing ratios, calculating areas, and using estimation. Stephanie Poulin Desktop Publisher Daily Reporter Q: What future plans do you have? A: My goal is to start my own business as a freelance graphic artist. 2- 2 Conditional Statements 87 87 �� 2-3 Using Deductive Reasoning to Verify Conjectures TEKS G.3.E Geometric structure: use deductive reasoning to prove a statement. Also G.2.B, G.3.B, G.3.C Objective Apply the Law of Detachment and the Law of Syllogism in logical reasoning. Vocabulary deductive reasoning Why learn this? You can use inductive and deductive reasoning to decide whether a common myth is accurate. You learned in Lesson 2-1 that one counterexample is enough to disprove a conjecture. But to prove that a conjecture is true, you must use deductive reasoning. Deductive reasoning is the process of using logic to draw conclusions from given facts, definitions, and properties. E X A M P L E 1 Media Application Urban legends and modern myths spread quickly through the media. Many Web sites and television shows are dedicated to confirming or disproving such myths. Is each conclusion a result of inductive or deductive reasoning? A There is a myth that toilets and sinks drain in opposite directions in the Southern and Northern Hemispheres. However, if you were to observe sinks draining in the two hemispheres, you would see that this myth is false. Since the conclusion is based on a pattern of observation, it is a result of inductive reasoning. B There is a myth that you should not touch a baby bird that has fallen from its nest because the mother bird will disown the baby if she detects human scent. However, biologists have shown that birds cannot detect human scent. Therefore, the myth cannot be true. The conclusion is based on logical reasoning from scientific research. It is a result of deductive reasoning. 1. There is a myth that an eelskin wallet will demagnetize credit cards because the skin of the electric eels used to make the wallet holds an electric charge. However, eelskin products are not made from electric eels. Therefore, the myth cannot be true. Is this conclusion a result of inductive or deductive reasoning? In deductive reasoning, if the given facts are true and you apply the correct logic, then the conclusion must be true. The Law of Detachment is one valid form of deductive reasoning. 88 88 Chapter 2 Geometric Reasoning Law of Detachment If p → q is a true statement and p is true, then q is true. E X A M P L E 2 Verifying Conjectures by Using the Law of Detachment Determine if each conjecture is valid by the Law of Detachment. A Given: If two segments are congruent, then they have the same length. ̶̶ XY . ̶̶ AB ≅ Conjecture: AB = XY Identify the hypothesis and conclusion in the given conditional. If two segments are congruent, then they have the same length. The given statement conditional. By the Law of Detachment AB = XY. The conjecture is valid. ̶̶ XY matches the hypothesis of a true ̶̶ AB ≅ B Given: If you are tardy 3 times, you must go to detention. Shea is in detention. Conjecture: Shea was tardy at least 3 times. Identify the hypothesis and conclusion in the given conditional. If you are tardy 3 times, you must go to detention. The given statement “Shea is in detention” matches the conclusion of a true conditional. But this does not mean the hypothesis is true. Shea could be in detention for another reason. The conjecture is not valid. 2. Determine if the conjecture is valid by the Law of Detachment. Given: If a student passes his classes, the student is eligible to play sports. Ramon passed his classes. Conjecture: Ramon is eligible to play sports. Another valid form of deductive reasoning is the Law of Syllogism. It allows you to draw conclusions from two conditional statements when the conclusion of one is the hypothesis of the other. Law of Syllogism If p → q and q → r are true statements, then p → r is a true statement. E X A M P L E 3 Verifying Conjectures by Using the Law of Syllogism Determine if each conjecture is valid by the Law of Syllogism. A Given: If m∠ A < 90°, then ∠ A is acute. If ∠ A is acute, then it is not a right angle. Conjecture: If m∠ A < 90°, then it is not a right angle. Let p, q, and r represent the following. p: The measure of an angle is less than 90°. q: The angle is acute. r : The angle is not a right angle. You are given that p → q and q → r . Since q is the conclusion of the first conditional and the hypothesis of the second conditional, you can conclude that p → r . The conjecture is valid by the Law of Syllogism. 2- 3 Using Deductive Reasoning to Verify Conjectures 89 89 It is possible to arrive at a true conclusion by applying invalid logical reasoning, as in Example 3B. Determine if each conjecture is valid by the Law of Syllogism. B Given: If a number is divisible by 4, then it is divisible by 2. If a number is even, then it is divisible by 2. Conjecture: If a number is divisible by 4, then it is even. Let x, y, and z represent the following. x : A number is divisible by 4. y : A number is divisible by 2. z : A number is even. You are given that x → y and z → y. The Law of Syllogism cannot be used to draw a conclusion since y is the conclusion of both conditionals. Even though the conjecture x → z is true, the logic used to draw the conclusion is not valid. 3. Determine if the conjecture is valid by the Law of Syllogism. Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal. Conjecture: If an animal is a dog, then it has hair. E X A M P L E 4 Applying the Laws of Deductive Reasoning Draw a conclusion from the given information. A Given: If a team wins 10 games, then they play in the finals. If a team plays in the finals, then they travel to Boston. The Ravens won 10 games. Conclusion: The Ravens will travel to Boston. B Given: If two angles form a linear pair, then they are adjacent. If two angles are adjacent, then they share a side. ∠1 and ∠2 form a linear pair. Conclusion: ∠1 and ∠2 share a side. 4. Draw a conclusion from the given information. Given: If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is not a quadrilateral. Polygon P is a triangle. THINK AND DISCUSS 1. Could “A square has exactly two sides” be the conclusion of a valid argument? If so, what do you know about the truth value of the given information? 2. Explain why writing conditional statements as symbols might help you evaluate the validity of an argument. 3. GET ORGANIZED Copy and complete the graphic organizer. Write each law in your own words and give an example of each. 90 90 Chapter 2 Geometric Reasoning �� 2-3 Exercises Exercises KEYWORD: MG7 2-3 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain how deductive reasoning differs from inductive reasoning Does each conclusion use inductive or deductive reasoning? p. 88 2. At Bell High School, students must take Biology before they take Chemistry. Sam is in Chemistry, so Marcia concludes that he has taken Biology. 3. A detective learns that his main suspect was out of town the day of the crime. He concludes that the suspect is innocent Determine if each conjecture is valid by the Law of Detachment. p. 89 4. Given: If you want to go on a field trip, you must have a signed permission slip. Zola has a signed permission slip. Conjecture: Zola wants to go on a field trip. 5. Given: If the side lengths of a rectangle are 3 ft and 4 ft, then its area is 12 ft 2 . A rectangle has side lengths of 3 ft and 4 ft. Conjecture: The area of the rectangle is 12 ft Determine if each conjecture is valid by the Law of Syllogism. p. 89 6. Given: If you fly from Texas to California, you travel from the central to the Pacific time zone. If you travel from the central to the Pacific time zone, then you gain two hours. Conjecture: If you fly from Texas to California, you gain two hours. 7. Given: If a figure is a square, then the figure is a rectangle. If a figure is a square, then it is a parallelogram. Conjecture: If a figure is a parallelogram, then it is a rectangle. 90 8. Draw a conclusion from the given information. Given: If you leave your car lights on overnight, then your car battery will drain. If your battery is drained, your car might not start. Alex left his car lights on last night. Independent Practice Does each conclusion use inductive or deductive reasoning? PRACTICE AND PROBLEM SOLVING For See Exercises Example 9–10 11 12 13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S6 Application Practice p. S29 9. The sum of the angle measures of a triangle is 180°. Two angles of a triangle measure 40° and 60°, so Kandy concludes that the third angle measures 80°. 10. All of the students in Henry’s Geometry class are juniors. Alexander takes Geometry, but has another teacher. Henry concludes that Alexander is also a junior. 11. Determine if the conjecture is valid by the Law of Detachment. Given: If one integer is odd and another integer is even, their product is even. The product of two integers is 24. C
onjecture: One of the two integers is odd. 2- 3 Using Deductive Reasoning to Verify Conjectures 91 91 � 12. Science Determine if the conjecture is valid by the Law of Syllogism. Given: If an element is an alkali metal, then it reacts with water. If an element is in the first column of the periodic table, then it is an alkali metal. Conjecture: If an element is in the first column of the periodic table, then it reacts with water. 13. Draw a conclusion from the given information. Given: If Dakota watches the news, she is informed about current events. If Dakota knows about current events, she gets better grades in Social Studies. Dakota watches the news. 14. Technology Joseph downloads a file in 18 minutes with a dial-up modem. How long would it take to download the file with a Cheetah-Net cable modem? Recreation Recreation Use the true statements below for Exercises 15–18. Determine whether each conclusion is valid. The Top Thrill Dragster is 420 feet tall and includes a 400-foot vertical drop. It twists 270° as it drops. It is one of 16 roller coasters at Cedar Point amusement park. I. The Top Thrill Dragster is at Cedar Point amusement park in Sandusky, OH. II. Carter and Mary go to Cedar Point. III. The Top Thrill Dragster roller coaster reaches speeds of 120 mi/h. IV. When Carter goes to an amusement park, he rides all the roller coasters. 15. Carter went to Sandusky, OH. 16. Mary rode the Top Thrill Dragster. 17. Carter rode a roller coaster that travels 120 mi/h. 18. Mary rode a roller coaster that travels 120 mi/h. 19. Critical Thinking Is the argument below a valid application of the Law of Syllogism? Is the conclusion true? Explain your answers. If 3 - x < 5, then x < -2. If x < -2, then -5x > 10. Thus, if 3 - x < 5, then -5x > 10. 20. /////ERROR ANALYSIS///// Below are two conclusions. Which is incorrect? Explain the error. If two angles are complementary, their measures add to 90°. If an angle measures 90°, then it is a right angle. ∠A and ∠B are complementary. 21. Write About It Write one example of a real-life logical argument that uses the Law of Detachment and one that uses the Law of Syllogism. Explain why the conclusions are valid. 22. This problem will prepare you for the Multi-Step TAKS Prep on page 102. When Alice meets the Pigeon in Wonderland, the Pigeon thinks she is a serpent. The Pigeon reasons that serpents eat eggs, and Alice confirms that she has eaten eggs. a. Write “Serpents eat eggs” as a conditional statement. b. Is the Pigeon’s conclusion that Alice is a serpent valid? Explain your reasoning. 92 92 Chapter 2 Geometric Reasoning �� 23. The Supershots scored over 75 points in each of ten straight games. The newspaper predicts that they will score more than 75 points tonight. Which form of reasoning is this conclusion based on? Deductive reasoning, because the conclusion is based on logic Deductive reasoning, because the conclusion is based on a pattern Inductive reasoning, because the conclusion is based on logic Inductive reasoning, because the conclusion is based on a pattern 24.  HF bisects ∠EHG. Which conclusion is NOT valid? E, F, and G are coplanar. ∠EHF ≅ ∠FHG ̶̶ EF ≅ ̶̶ FG m∠EHF = m∠FHG 25. Gridded Response If Whitney plays a low G on her piano, the frequency of the note is 24.50 hertz. The frequency of a note doubles with each octave. What is the frequency in hertz of a G note that is 3 octaves above low G? CHALLENGE AND EXTEND 26. Political Science To be eligible to hold the office of the president of the United States, a person must be at least 35 years old, be a natural-born U.S. citizen, and have been a U.S. resident for at least 14 years. Given this information, what conclusion, if any, can be drawn from the statements below? Explain your reasoning. Andre is not eligible to be the president of the United States. Andre has lived in the United States for 16 years. 27. Multi-Step Consider the two conditional statements below. If you live in San Diego, then you live in California. If you live in California, then you live in the United States. a. Draw a conclusion from the given conditional statements. b. Write the contrapositive of each conditional statement. c. Draw a conclusion from the two contrapositives. d. How does the conclusion in part a relate to the conclusion in part c? 28. If Cassie goes to the skate park, Hanna and Amy will go. If Hanna or Amy goes to the skate park, then Marc will go. If Marc goes to the skate park, then Dallas will go. If only two of the five people went to the skate park, who were they? SPIRAL REVIEW Simplify each expression. (Previous course) 29. 2 (x + 5) 30. (4y + 6) - (3y - 5) 31. (3c + 4c) + 2 (-7c + 7) Find the coordinates of the midpoint of the segment connecting each pair of points. (Lesson 1-6) 32. (1, 2) and (4, 5) 33. (-3, 6) and (0, 1) 34. (-2.5, 9) and (2.5, -3) Identify the hypothesis and conclusion of each conditional statement. (Lesson 2-2) 35. If the fire alarm rings, then everyone should exit the building. 36. If two different lines intersect, then they intersect at exactly one point. 37. The statement ̶̶ AB ≅ ̶̶ CD implies that AB = CD. 2- 3 Using Deductive Reasoning to Verify Conjectures 93 93 �� 2-3 Solve Logic Puzzles In Lesson 2-3, you used deductive reasoning to analyze the truth values of conditional statements. Now you will learn some methods for diagramming conditional statements to help you solve logic puzzles. Use with Lesson 2-3 TEKS G.4.A Geometric structure: select an appropriate representation ... in order to solve problems Activity 1 Bonnie, Cally, Daphne, and Fiona own a bird, cat, dog, and fish. No girl has a type of pet that begins with the same letter as her name. Bonnie is allergic to animal fur. Daphne feeds Fiona’s bird when Fiona is away. Make a table to determine who owns which animal. 1 Since no girl has a type of pet that starts with the same letter as her name, place an X in each box along the diagonal of the table. 2 Bonnie cannot have a cat or dog because of her allergy. So she must own the fish, and no other girl can have the fish. Bird Cat Dog Fish Bird Cat Dog Fish Bonnie × Cally Daphne Fiona × × × Bonnie × Cally Daphne Fiona × × × × ✓ × × × 3 Fiona owns the bird, so place a check in 4 Therefore, Daphne owns the cat, and Cally Fiona’s row, in the bird column. Place an X in the remaining boxes in the same column and row. owns the dog. Bird Cat Dog Fish Bird Cat Dog Fish Bonnie Cally Daphne Fiona × × × ✓ × × × × × × ✓ × × × Bonnie Cally Daphne Fiona × × × ✓ × × ✓ × × ✓ × × ✓ × × × Try This 1. After figuring out that Fiona owns the bird in Step 3, why can you place an X in every other box in that row and column? 2. Ally, Emily, Misha, and Tracy go to a dance with Danny, Frank, Jude, and Kian. Ally and Frank are siblings. Jude and Kian are roommates. Misha does not know Kian. Emily goes with Kian’s roommate. Tracy goes with Ally’s brother. Who went to the dance with whom? Ally Emily Misha Tracy 94 94 Chapter 2 Geometric Reasoning Danny Frank Jude Kian Activity 2 A farmer has a goat, a wolf, and a cabbage. He wants to transport all three from one side of a river to the other. He has a boat, but it has only enough room for the farmer and one thing. The wolf will eat the goat if they are left alone together, and the goat will eat the cabbage if they are left alone. How can the farmer get everything to the other side of the river? You can use a network to solve this kind of puzzle. A network is a diagram of vertices and edges, also known as a graph. An edge is a curve or a segment that joins two vertices of the graph. A vertex is a point on the graph. 1 Let F represent the farmer, W represent the wolf, G represent the goat, and C represent the cabbage. Use an ordered pair to represent what is on each side of the river. The first ordered pair is (FWGC, —), and the desired result is (—, FWGC). 2 Draw a vertex and label it with the first ordered pair. Then draw an edge and vertex for each possible trip the farmer could make across the river. If at any point a path results in an unworkable combination of things, no more edges can be drawn from that vertex. 3 From each workable vertex, continue to draw edges and vertices that represent the next trip across the river. When you get to a vertex for (—, FWGC), the network is complete. 4 Use the network to write out the solution in words. Try This 3. What combinations are unworkable? Why? 4. How many solutions are there to the farmer’s transport problem? How many steps does each solution take? 5. What is the advantage of drawing a complete solution network rather than working out one solution with a diagram? 6. Madeline has two measuring cups—a 1-cup measuring cup and a 3__ 4 -cup measuring cup. Neither cup has any markings on it. How can Madeline get exactly 1 __ 2 cup of flour in the larger measuring cup? Complete the network below to solve the problem. 2- 3 Geometry Lab 95 95 �� 2-4 Biconditional Statements and Definitions TEKS G.3.A Geometric structure: determine the validity of a conditional statement, its converse, inverse, and contrapositive. Also G.3.B Objective Write and analyze biconditional statements. Vocabulary biconditional statement definition polygon triangle quadrilateral Who uses this? A gardener can plan the color of the hydrangeas she plants by checking the pH of the soil. The pH of a solution is a measure of the concentration of hydronium ions in the solution. If a solution has a pH less than 7, it is an acid. Also, if a solution is an acid, it has a pH less than 7. When you combine a conditional statement and its converse, you create a biconditional statement. A biconditional statement is a statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, t
hen p.” The biconditional “p if and only if q” can also be written as “p iff q” or p ↔ q. So you can define an acid with the following biconditional statement: A solution is an acid if and only if it has a pH less than 7. E X A M P L E 1 Identifying the Conditionals within a Biconditional Statement Write the conditional statement and converse within each biconditional. A Two angles are congruent if and only if their measures are equal. Let p and q represent the following. p : Two angles are congruent. q : Two angle measures are equal. The two parts of the biconditional p ↔ q are p → q and q → p. Conditional: If two angles are congruent, then their measures are equal. Converse: If two angle measures are equal, then the angles are congruent. B A solution is a base ↔ it has a pH greater than 7. Let x and y represent the following. x : A solution is a base. y : A solution has a pH greater than 7. The two parts of the biconditional x ↔ y are x → y and y → x. Conditional: If a solution is a base, then it has a pH greater than 7. Converse: If a solution has a pH greater than 7, then it is a base. Write the conditional statement and converse within each biconditional. 1a. An angle is acute iff its measure is greater than 0° and less than 90°. 1b. Cho is a member if and only if he has paid the $5 dues. 96 96 Chapter 2 Geometric Reasoning ��meansandProject TitleGeometry 2007 Student EditionSpec Numberge07sec01l04002aCreated ByKrosscore CorporationCreation Date10/07/2004 E X A M P L E 2 Writing a Biconditional Statement For each conditional, write the converse and a biconditional statement. A If 2x + 5 = 11, then x = 3. Converse: If x = 3, then 2x + 5 = 11. Biconditional: 2x + 5 = 11 if and only if x = 3. B If a point is a midpoint, then it divides the segment into two congruent segments. Converse: If a point divides a segment into two congruent segments, then the point is a midpoint. Biconditional: A point is a midpoint if and only if it divides the segment into two congruent segments. For each conditional, write the converse and a biconditional statement. 2a. If the date is July 4th, then it is Independence Day. 2b. If points lie on the same line, then they are collinear. For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false. E X A M P L E 3 Analyzing the Truth Value of a Biconditional Statement Determine if each biconditional is true. If false, give a counterexample. A A square has a side length of 5 if and only if it has an area of 25. Conditional: If a square has a side length of 5, then it has an area of 25. The conditional is true. Converse: If a square has an area of 25, then it has a side length of 5. The converse is true. Since the conditional and its converse are true, the biconditional is true. B The number n is a positive integer ↔ 2n is a natural number. Conditional: If n is a positive integer, then 2n is a natural number. The conditional is true. Converse: If 2n is a natural number, then n is a positive integer. The converse is false. If 2n = 1, then n = 1 __ 2 , which is not an integer. Because the converse is false, the biconditional is false. Determine if each biconditional is true. If false, give a counterexample. 3a. An angle is a right angle iff its measure is 90°. 3b. y = -5 ↔ y 2 = 25 In geometry, biconditional statements are used to write definitions. A definition is a statement that describes a mathematical object and can be written as a true biconditional. Most definitions in the glossary are not written as biconditional statements, but they can be. The “if and only if” is implied. 2- 4 Biconditional Statements and Definitions 97 97 In the glossary, a polygon is defined as a closed plane figure formed by three or more line segments. Each segment intersects exactly two other segments only at their endpoints, and no two segments with a common endpoint are collinear. Polygons Not Polygons A triangle is defined as a three-sided polygon, and a quadrilateral is a four-sided polygon. A good, precise definition can be used forward and backward. For example, if a figure is a quadrilateral, then it is a four-sided polygon. If a figure is a four-sided polygon, then it is a quadrilateral. To make sure a definition is precise, it helps to write it as a biconditional statement. E X A M P L E 4 Writing Definitions as Biconditional Statements Write each definition as a biconditional. A A triangle is a three-sided polygon. Think of definitions as being reversible. Postulates, however, are not necessarily true when reversed. A figure is a triangle if and only if it is a three-sided polygon. B A segment bisector is a ray, segment, or line that divides a segment into two congruent segments. A ray, segment, or line is a segment bisector if and only if it divides a segment into two congruent segments. Write each definition as a biconditional. 4a. A quadrilateral is a four-sided polygon. 4b. The measure of a straight angle is 180°. THINK AND DISCUSS 1. How do you determine if a biconditional statement is true or false? 2. Compare a triangle and a quadrilateral. 3. GET ORGANIZED Copy and complete the graphic organizer. Use the definition of a polygon to write a conditional, converse, and biconditional in the appropriate boxes. 98 98 Chapter 2 Geometric Reasoning �� 2-4 Exercises Exercises KEYWORD: MG7 2-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary How is a biconditional statement different from a conditional statement Write the conditional statement and converse within each biconditional. p. 96 2. Perry can paint the entire living room if and only if he has enough paint. 3. Your medicine will be ready by 5 P.M. if and only if you drop your prescription off by 8 A.M For each conditional, write the converse and a biconditional statement. p. 97 4. If a student is a sophomore, then the student is in the tenth grade. 5. If two segments have the same length, then they are congruent. 97 Multi-Step Determine if each biconditional is true. If false, give a counterexample. 6. xy = 0 ↔ x = 0 or y = 0. 7. A figure is a quadrilateral if and only if it is a polygon Write each definition as a biconditional. p. 98 8. Parallel lines are two coplanar lines that never intersect. 9. A hummingbird is a tiny, brightly colored bird with narrow wings, a slender bill, and a long tongue. Independent Practice Write the conditional statement and converse within each biconditional. PRACTICE AND PROBLEM SOLVING For See Exercises Example 10–12 13–15 16–17 18–19 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S6 Application Practice p. S29 10. Three points are coplanar if and only if they lie in the same plane. 11. A parallelogram is a rectangle if and only if it has four right angles. 12. A lunar eclipse occurs if and only if Earth is between the Sun and the Moon. For each conditional, write the converse and a biconditional statement. 13. If today is Saturday or Sunday, then it is the weekend. 14. If Greg has the fastest time, then he wins the race. 15. If a triangle contains a right angle, then it is a right triangle. Multi-Step Determine if each biconditional is true. If false, give a counterexample. 16. Felipe is a swimmer if and only if he is an athlete. 17. The number 2n is even if and only if n is an integer. Write each definition as a biconditional. 18. A circle is the set of all points in a plane that are a fixed distance from a given point. 19. A catcher is a baseball player who is positioned behind home plate and who catches throws from the pitcher. 2- 4 Biconditional Statements and Definitions 99 99 Algebra Determine if a true biconditional can be written from each conditional statement. If not, give a counterexample. 20. If a = b, then ⎜a⎟ = ⎜b⎟ . x + 8 = 12. 21. If 3x - 2 = 13, then 4 _ 5 23. If x > 0, then x 2 > 0. 22. If y 2 = 64, then 3y = 24. Use the diagrams to write a definition for each figure. 24. 25. Biology 26. Biology White blood cells are cells that defend the body against invading organisms by engulfing them or by releasing chemicals called antibodies. Write the definition of a white blood cell as a biconditional statement. White blood cells live less than a few weeks. A drop of blood can contain anywhere from 7000 to 25,000 white blood cells. Explain why the given statement is not a definition. 27. An automobile is a vehicle that moves along the ground. 28. A calculator is a machine that performs computations with numbers. 29. An angle is a geometric object formed by two rays. Chemistry Use the table for Exercises 30–32. Determine if a true biconditional statement can be written from each conditional. 30. If a solution has a pH of 4, then it is tomato juice. 31. If a solution is bleach, then its pH is 13. 32. If a solution has a pH greater than 7, then it is not battery acid. pH 0 4 6 8 13 14 Examples Battery Acid Acid rain, tomato juice Saliva Sea water Bleach, oven cleaner Drain cleaner Complete each statement to form a true biconditional. 33. The circumference of a circle is 10π if and only if its radius is ? . ̶̶̶ 34. Four points in a plane form a ? if and only if no three of them are collinear. ̶̶̶ 35. Critical Thinking Write the definition of a biconditional statement as a biconditional statement. Use the conditional and converse within the statement to explain why your biconditional is true. 36. Write About It Use the definition of an angle bisector to explain what is meant by the statement “A good definition is reversible.” 37. This problem will prepare you for the Multi-Step TAKS Prep on page 102. a. Write “I say what I mean” and “I mean what I say” as conditionals. b. Explain why the biconditional statement implied by Alice is false. “Then you should say what you mean,” the March Hare went on. “I do,” Alice hastily replied; “at least—at least I mean what I say—that’s the
same thing, you know.” 100 100 Chapter 2 Geometric Reasoning �� 38. Which is a counterexample for the biconditional “An angle measures 80° if and only if the angle is acute”? m∠S = 60° m∠S = 115° m∠S = 90° m∠S = 360° 39. Which biconditional is equivalent to the spelling phrase “I before E except after C”? The letter I comes before E if and only if I follows C. The letter E comes before I if and only if E follows C. The letter E comes before I if and only if E comes before C. The letter I comes before E if and only if I comes before C. 40. Which conditional statement can be used to write a true biconditional? If a number is divisible by 4, then it is even. If a ratio compares two quantities measured in different units, the ratio is a rate. If two angles are supplementary, then they are adjacent. If an angle is right, then it is not acute. 41. Short Response Write the two conditional statements that make up the biconditional “You will get a traffic ticket if and only if you are speeding.” Is the biconditional true or false? Explain your answer. CHALLENGE AND EXTEND 42. Critical Thinking Describe what the Venn diagram of a true biconditional statement looks like. How does this support the idea that a definition can be written as a true biconditional? 43. Consider the conditional “If an angle measures 105°, then the angle is obtuse.” a. Write the inverse of the conditional statement. b. Write the converse of the inverse. c. How is the converse of the inverse related to the original conditional? d. What is the truth value of the biconditional statement formed by the inverse of the original conditional and the converse of the inverse? Explain. 44. Suppose A, B, C, and D are coplanar, and A, B, and C are not collinear. What is the truth value of the biconditional formed from the true conditional “If m∠ABD + m∠DBC = m∠ABC, then D is in the interior of ∠ABC”? Explain. 45. Find a counterexample for “n is divisible by 4 if and only if n 2 is even.” SPIRAL REVIEW Describe how the graph of each function differs from the graph of the parent function y = x 2 . (Previous course) 46 47. y = -2 x 2 - 1 48. y = (x - 2) (x + 2) A transformation maps S onto T and X onto Y. Name each of the following. (Lesson 1-7) 49. the image of S 50. the image of X 51. the preimage of T Determine if each conjecture is true. If not, give a counterexample. (Lesson 2-1) 52. If n ≥ 0, then n _ 2 54. The vertices of the image of a figure under the translation (x, y) → (x + 0, y + 0) 53. If x is prime, then x + 2 is also prime. > 0. have the same coordinates as the preimage. 2- 4 Biconditional Statements and Definitions 101 101 SECTION 2A Inductive and Deductive Reasoning Rhyme or Reason Alice’s Adventures in Wonderland originated as a story told by Charles Lutwidge Dodgson (Lewis Carroll) to three young traveling companions. The story is famous for its wordplay and logical absurdities. 1. When Alice first meets the Cheshire Cat, she asks what sort of people live in Wonderland. The Cat explains that everyone in Wonderland is mad. What conjecture might the Cat make since Alice, too, is in Wonderland? 2. “I don’t much care where—” said Alice. “Then it doesn’t matter which way you go,” said the Cat. “—so long as I get somewhere,” Alice added as an explanation. “Oh, you’re sure to do that,” said the Cat, “if you only walk long enough.” Write the conditional statement implied by the Cat’s response to Alice. 3. “Well, then,” the Cat went on, “you see a dog growls when it’s angry, and wags its tail when it’s pleased. Now I growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad.” Is the Cat’s conclusion valid by the Law of Detachment or the Law of Syllogism? Explain your reasoning. 4. “You might just as well say,” added the Dormouse, who seemed to be talking in his sleep, “that ‘I breathe when I sleep’ is the same thing as ‘I sleep when I breathe’!” Write a biconditional statement from the Dormouse’s example. Explain why the biconditional statement is false. 102 102 Chapter 2 Geometric Reasoning Quiz for Lessons 2-1 Through 2-4 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in each pattern. 1. 1, 10, 18, 25. July, May, March, … 3. 1 _ 4 2 8 5. A biologist recorded the following data about the weight of male lions in a wildlife park in Africa. Use the table to make a conjecture about the average weight of a male lion. , ... 6. Complete the conjecture “The sum of two negative numbers is ? .” ̶̶̶̶ 7. Show that the conjecture “If an even number is divided by 2, then the result is an even number” is false by finding a counterexample. SECTION 2A 4. ∣, , , ... ID Number Weight (lb) A1902SM A1904SM A1920SM A1956SM A1974SM 387.2 420.5 440.6 398.7 415.0 2-2 Conditional Statements 8. Identify the hypothesis and conclusion of the conditional statement “An angle is obtuse if its measure is 107°.” Write a conditional statement from each of the following. 9. A whole number is 10. an integer. 11. The diagonals of a square are congruent. Determine if each conditional is true. If false, give a counterexample. 12. If an angle is acute, then it has a measure of 30°. 13. If 9x - 11 = 2x + 3, then x = 2. 14. Write the converse, inverse, and contrapositive of the statement “If a number is even, then it is divisible by 4.” Find the truth value of each. 2-3 Using Deductive Reasoning to Verify Conjectures 15. Determine if the following conjecture is valid by the Law of Detachment. Given: If Sue finishes her science project, she can go to the movie. Sue goes to the movie. Conjecture: Sue finished her science project. 16. Use the Law of Syllogism to draw a conclusion from the given information. Given: If one angle of a triangle is 90°, then the triangle is a right triangle. If a triangle is a right triangle, then its acute angle measures are complementary. 2-4 Biconditional Statements and Definitions 17. For the conditional “If two angles are supplementary, the sum of their measures is 180°,” write the converse and a biconditional statement. 18. Determine if the biconditional “ √  x = 4 if and only if x = 16” is true. If false, give a counterexample. Ready to Go On? 103 103 �� 2-5 Algebraic Proof TEKS G.3.E Geometric structure: use deductive reasoning to prove a statement. Also G.3.B, G.3.C Objectives Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence. Vocabulary proof The Distributive Property states that a (b + c) = ab + ac. Who uses this? Game designers and animators solve equations to simulate motion. (See Example 2.) A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. If you’ve ever solved an equation in Algebra, then you’ve already done a proof ! An algebraic proof uses algebraic properties such as the properties of equality and the Distributive Property. Properties of Equality Properties of Equality Addition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a - c = b - c. Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality If a = b, then ac = bc. If a = b and c ≠ 0, then a __ c = b __ c . a = a Symmetric Property of Equality If a = b, then b = a. Transitive Property of Equality If a = b and b = c, then a = c. Substitution Property of Equality If a = b, then b can be substituted for a in any expression. As you learned in Lesson 2-3, if you start with a true statement and each logical step is valid, then your conclusion is valid. An important part of writing a proof is giving justifications to show that every step is valid. For each justification, you can use a definition, postulate, property, or a piece of information that is given. E X A M P L E 1 Solving an Equation in Algebra Solve the equation -5 = 3n + 1. Write a justification for each step. -5 = 3n + 1 - 1 - 1 ̶̶̶̶̶ ̶̶̶ -6 = 3n _ _ -6 3n 3 3 -2 = n n = -2 = Given equation Subtraction Property of Equality Simplify. Division Property of Equality Simplify. Symmetric Property of Equality 1. Solve the equation 1 _ 2 t = -7. Write a justification for each step. 104 104 Chapter 2 Geometric Reasoning E X A M P L E 2 Problem-Solving Application To simulate the motion of an object in a computer game, the designer uses the formula sr = 3.6p to find the number of pixels the object must travel during each second of animation. In the formula, s is the desired speed of the object in kilometers per hour, r is the scale of pixels per meter, and p is the number of pixels traveled per second. The graphics in a game are based on a scale of 6 pixels per meter. The designer wants to simulate a vehicle moving at 75 km/h. How many pixels must the vehicle travel each second? Solve the equation for p and justify each step. Understand the Problem The answer will be the number of pixels traveled per second. List the important information: • sr = 3.6p • p: pixels traveled per second • s = 75 km/h • r = 6 pixels per meter Make a Plan Substitute the given information into the formula and solve. Solve sr = 3.6p (75) (6) = 3.6p 450 = 3.6p _ _ 3.6p 450 3.6 3.6 125 = p = Given equation Substitution Property of Equality Simplify. Division Property of Equality Simplify. p = 125 pixels Symmetric Property of Equality Look Back Check your answer by substituting it back into the original formula. sr = 3.6p (75) (6) = 3.6 (125) 450 = 450 ✓ AB represents the ̶̶ length of AB , so you can think of AB as a variable representing a number. 2. What is the temperature in degrees Celsius C when it is 86°F? Solve the equation C = 5 _ (F - 32) for C and justify each step. 9 Like algebra, geometry also uses numbers, variables, and operations. For example, segment lengths and angle measures are numbers. So you can use these same properties of equality to write algebraic proofs in geometry. 2- 5 Algebraic Proof 105 105 1234�� E
X A M P L E 3 Solving an Equation in Geometry Write a justification for each step. KM = KL + LM 5x - 4 = (x + 3) + (2x - 1) 5x - 4 = 3x + 2 2x - 4 = 2 2x = 6 x = 3 Segment Addition Postulate Substitution Property of Equality Simplify. Subtraction Property of Equality Addition Property of Equality Division Property of Equality 3. Write a justification for each step. m∠ABC = m∠ABD + m∠DBC 8x ° = (3x + 5) ° + (6x - 16) ° 8x = 9x - 11 -x = -11 x = 11 You learned in Chapter 1 that segments with equal lengths are congruent and that angles with equal measures are congruent. So the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. Properties of Congruence SYMBOLS EXAMPLE Reflexive Property of Congruence figure A ≅ figure A (Reflex. Prop. of ≅) Symmetric Property of Congruence ̶̶ EF ≅ ̶̶ EF If figure A ≅ figure B, then figure B ≅ figure A. (Sym. Prop. of ≅) If ∠1 ≅ ∠2, then ∠2 ≅ ∠1. Transitive Property of Congruence If figure A ≅ figure B and figure B ≅ figure C, then figure A ≅ figure C. (Trans. Prop. of ≅) ̶̶ PQ ≅ ̶̶ PQ ≅ ̶̶ RS and ̶̶ TU . If then ̶̶ RS ≅ ̶̶ TU , E X A M P L E 4 Identifying Properties of Equality and Congruence Identify the property that justifies each statement. Numbers are equal (=) and figures are congruent (≅) . A m∠1 = m∠1 B ̶̶ XY ≅ ̶̶ VW , so ̶̶ VW ≅ ̶̶ XY . C ∠ABC ≅ ∠ABC Reflex. Prop. of = Sym. Prop. of ≅ Reflex. Prop. of ≅ D ∠1 ≅ ∠2, and ∠2 ≅ ∠3. So ∠1 ≅ ∠3. Trans. Prop. of ≅ Identify the property that justifies each statement. 4a. DE = GH, so GH = DE. 4c. 0 = a, and a = x. So 0 = x. 4b. 94° = 94° 4d. ∠A ≅ ∠Y, so ∠Y ≅ ∠A. 106 106 Chapter 2 Geometric Reasoning �� THINK AND DISCUSS 1. Tell what property you would use to solve the equation k _ 6 2. Explain when to use a congruence symbol instead of an equal sign. = 3.5. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write an example of the property, using the correct symbol. 2-5 Exercises Exercises KEYWORD: MG7 2-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Write the definition of proof in your own words Multi-Step Solve each equation. Write a justification for each step. p. 104 2. y + 1 = 5 3. t - 3.2 = -8.3 4. 2p - 30 = -4p + 6 n = 3 _ 6. 1 _ 4 2 5. x + 3 _ -2 = 8 7. 0 = 2 (r - 3. Nutrition Amy’s favorite breakfast cereal has 102 Calories per serving. The equation p. 105 C = 9f + 90 relates the grams of fat f in one serving to the Calories C in one serving. How many grams of fat are in one serving of the cereal? Solve the equation for f and justify each step. 9. Movie Rentals The equation C = $5.75 + $0.89m relates the number of movie rentals m to the monthly cost C of a movie club membership. How many movies did Elias rent this month if his membership cost $11.98? Solve the equation for m and justify each step Write a justification for each step. p. 106 10. 11. AB = BC 5y + 6 = 2y + 21 3y + 6 = 21 3y = 15 y = 5 PQ + QR = PR 3n + 25 = 9n -5 25 = 6n -5 30 = 6n . 106 12. ̶̶ AB ≅ ̶̶ AB Identify the property that justifies each statement. 14. x = y, so y = x. 13. m∠1 = m∠2, and m∠2 = m∠4. So m∠1 = m∠4. ̶̶ PR . So ̶̶ YZ , and ̶̶ YZ ≅ ̶̶ ST ≅ ̶̶ ST ≅ ̶̶ PR . 15. 2- 5 Algebraic Proof 107 107 �� Independent Practice Multi-Step Solve each equation. Write a justification for each step. PRACTICE AND PROBLEM SOLVING For See Exercises Example 16. 5x - 3 = 4 (x + 2) 17. 1.6 = 3.2n 19. - (h + 3) = 72 20. 9y + 17 = -19 - 2 = -10 18. z _ 3 21. 1 _ (p - 16) = 13 2 16–21 22 23–24 25–28 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S7 Application Practice p. S29 22. Ecology The equation T = 0.03c + 0.05b relates the numbers of cans c and bottles b collected in a recycling rally to the total dollars T raised. How many cans were collected if $147 was raised and 150 bottles were collected? Solve the equation for c and justify each step. Write a justification for each step. 23. m∠XYZ = m∠2 + m∠3 4n - 6 = 58 + (2n - 12) 4n - 6 = 2n + 46 2n - 6 = 46 2n = 52 n = 26 24. m∠WYV = m∠1 + m∠2 5n = 3 (n - 2) + 58 5n = 3n - 6 + 58 5n = 3n + 52 2n = 52 n = 26 Identify the property that justifies each statement. 25. ̶̶ KL ≅ ̶̶ PR , so ̶̶ PR ≅ ̶̶ KL . 26. 412 = 412 27. If a = b and b = 0, then a = 0. 28. figure A ≅ figure A 29. Estimation Round the numbers in the equation 2 (3.1x - 0.87) = 94.36 to the nearest whole number and estimate the solution. Then solve the equation, justifying each step. Compare your estimate to the exact solution. Use the indicated property to complete each statement. 30. Reflexive Property of Equality: 3x - 1 = ? ̶̶̶ 31. Transitive Property of Congruence: If ∠ A ≅ ∠ X and ∠ X ≅ ∠T, then 32. Symmetric Property of Congruence: If ̶̶ BC ≅ ̶̶ NP , then ? . ̶̶̶ ? . ̶̶̶ 33. Recreation The north campground is midway between the Northpoint Overlook and the waterfall. Use the midpoint formula to find the values of x and y, and justify each step. 34. Business A computer repair technician charges $35 for each job plus $21 per hour of labor and 110% of the cost of parts. The total charge for a 3-hour job was $169.50. What was the cost of parts for this job? Write and solve an equation and justify each step in the solution. 35. Finance Morgan spent a total of $1,733.65 on her car last year. She spent $92.50 on registration, $79.96 on maintenance, and $983 on insurance. She spent the remaining money on gas. She drove a total of 10,820 miles. a. How much on average did the gas cost per mile? Write and solve an equation and justify each step in the solution. b. What if…? Suppose Morgan’s car averages 32 miles per gallon of gas. How much on average did Morgan pay for a gallon of gas? 36. Critical Thinking Use the definition of segment congruence and the properties of equality to show that all three properties of congruence are true for segments. 108 108 Chapter 2 Geometric Reasoning (1, y)NorthpointOverlook Northcampground(3, 5)(x, 1)WaterfallFinal file 2/25/05Campground mapHolt Rinehart WinstonGeometry 2007 Texasge07sec02l05003a Karen Minot(415)883-6560�� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 126. Recall from Algebra 1 that the Multiplication and Division Properties of Inequality tell you to reverse the inequality sign when multiplying or dividing by a negative number. a. Solve the inequality x + 15 ≤ 63 and write a justification for each step. b. Solve the inequality -2x > 36 and write a justification for each step. 38. Write About It Compare the conclusion of a deductive proof and a conjecture based on inductive reasoning. 39. Which could NOT be used to justify the statement ̶̶ AB ≅ ̶̶ CD ? Definition of congruence Symmetric Property of Congruence Reflexive Property of Congruence Transitive Property of Congruence 40. A club membership costs $35 plus $3 each time t the member uses the pool. Which equation represents the total cost C of the membership? C + 35 = 3t C = 35 + 3t 35 = C + 3t 41. Which statement is true by the Reflexive Property of Equality? ̶̶ CD = ̶̶ RT ≅ x = 35 ̶̶ CD ̶̶ TR C = 35t + 3 CD = CD 42. Gridded Response In the triangle, m∠1 + m∠2 + m∠3 = 180°. If m∠3 = 2m∠1 and m∠1 = m∠2, find m∠3 in degrees. CHALLENGE AND EXTEND 43. In the gate, PA = QB, QB = RA, and PA = 18 in. Find PR, and justify each step. 44. Critical Thinking Explain why there is no Addition Property of Congruence. 45. Algebra Justify each step in the solution of the inequality 7 - 3x > 19. SPIRAL REVIEW 46. The members of a high school band have saved $600 for a trip. They deposit the money in a savings account. What additional information is needed to find the amount of interest the account earns during a 3-month period? (Previous course) Use a compass and straightedge to construct each of the following. (Lesson 1-2) 47. ̶̶ JK congruent to ̶̶̶ MN 48. a segment bisector of ̶̶ JK Identify whether each conclusion uses inductive or deductive reasoning. (Lesson 2-3) 49. A triangle is obtuse if one of its angles is obtuse. Jacob draws a triangle with two acute angles and one obtuse angle. He concludes that the triangle is obtuse. 50. Tonya studied 3 hours for each of her last two geometry tests. She got an A on both tests. She concludes that she will get an A on the next test if she studies for 3 hours. 2- 5 Algebraic Proof 109 109 �� 2-6 Geometric Proof TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Also G.3.B, G.3.C, G.3.E Objectives Write two-column proofs. Prove geometric theorems by using deductive reasoning. Vocabulary theorem two-column proof Who uses this? To persuade your parents to increase your allowance, your argument must be presented logically and precisely. When writing a geometric proof, you use deductive reasoning to create a chain of logical steps that move from the hypothesis to the conclusion of the conjecture you are proving. By proving that the conclusion is true, you have proven that the original conjecture is true When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them. E X A M P L E 1 Writing Justifications When a justification is based on more than the previous step, you can note this after the reason, as in Example 1 Step 5. Write a justification for each step, given that ∠ A and ∠B are complementary and ∠ A ≅ ∠C. 1. ∠ A and ∠B are complementary. 2. m∠ A + m∠B = 90° 3. ∠A ≅ ∠C 4. m∠ A = m∠C 5. m∠C + m∠B = 90° 6. ∠C and ∠B are complementary. Given information Def. of comp.  Given information Def. of ≅  Subst. Prop. of = Def. of comp.  Steps 2, 4 1. Write a justification for each step, given that B is the midpoint ̶̶ AC and ̶̶ AB ≅ ̶̶ EF . of 1. B is the midpoint of 2.
3. 4. ̶̶ AB ≅ ̶̶ AB ≅ ̶̶ BC ≅ ̶̶ BC ̶̶ EF ̶̶ EF ̶̶ AC . A theorem is any statement that you can prove. Once you have proven a theorem, you can use it as a reason in later proofs. Theorem THEOREM HYPOTHESIS CONCLUSION 2-6-1 Linear Pair Theorem If two angles form a linear pair, then they are supplementary. ∠A and ∠B form a linear pair. ∠A and ∠B are supplementary. 110 110 Chapter 2 Geometric Reasoning �� Theorem THEOREM HYPOTHESIS CONCLUSION 2-6-2 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. ∠1 and ∠2 are supplementary. ∠2 and ∠3 are supplementary. ∠1 ≅ ∠3 A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a two-column proof , you list the steps of the proof in the left column. You write the matching reason for each step in the right column. E X A M P L E 2 Completing a Two-Column Proof Fill in the blanks to complete a two-column proof of the Linear Pair Theorem. Given: ∠1 and ∠2 form a linear pair. Prove: ∠1 and ∠2 are supplementary. Proof: Statements Reasons 1. ∠1 and ∠2 form a linear pair. 1. Given 2.  BA and  BC form a line. 3. m∠ABC = 180° 4. a. 5. b. ? ̶̶̶̶̶̶ ? ̶̶̶̶̶̶ 6. ∠1 and ∠2 are supplementary. 2. Def. of lin. pair 3. Def. of straight ∠ 4. ∠ Add. Post. 5. Subst. Steps 3, 4 6. c. ? ̶̶̶̶̶̶ Since there is no other substitution property, the Substitution Property of Equality is often written as “Substitution” or “Subst.” Use the existing statements and reasons in the proof to fill in the blanks. a. m∠1 + m∠2 = m∠ABC b. m∠1 + m∠2 = 180° c. Def. of supp.  The ∠ Add. Post. is given as the reason. Substitute 180° for m∠ABC. The measures of supp.  add to 180° by def. 2. Fill in the blanks to complete a two-column proof of one case of the Congruent Supplements Theorem. Given: ∠1 and ∠2 are supplementary, and ∠2 and ∠3 are supplementary. Prove: ∠1 ≅ ∠3 Proof: Statements Reasons 1. a. ? ̶̶̶̶̶̶ 2. m∠1 + m∠2 = 180° m∠2 + m∠3 = 180° 3. b. ? ̶̶̶̶̶̶ 4. m∠2 = m∠2 5. m∠1 = m∠3 6. d. ? ̶̶̶̶̶̶ 1. Given 2. Def. of supp.  3. Subst. 4. Reflex. Prop. of = 5. c. ? ̶̶̶̶̶̶ 6. Def. of ≅  2- 6 Geometric Proof 111 111 �� Before you start writing a proof, you should plan out your logic. Sometimes you will be given a plan for a more challenging proof. This plan will detail the major steps of the proof for you. Theorems THEOREM HYPOTHESIS CONCLUSION 2-6-3 Right Angle Congruence Theorem All right angles are congruent. 2-6-4 Congruent Complements Theorem If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. ∠A and ∠B are right angles. ∠A ≅ ∠B ∠1 and ∠2 are complementary. ∠2 and ∠3 are complementary. ∠1 ≅ ∠3 E X A M P L E 3 Writing a Two-Column Proof from a Plan If a diagram for a proof is not provided, draw your own and mark the given information on it. But do not mark the information in the Prove statement on it. Use the given plan to write a two-column proof of the Right Angle Congruence Theorem. Given: ∠1 and ∠2 are right angles. Prove: ∠1 ≅ ∠2 Plan: Use the definition of a right angle to write the measure of each angle. Then use the Transitive Property and the definition of congruent angles. Proof: Statements Reasons 1. ∠1 and ∠2 are right angles. 1. Given 2. m∠1 = 90°, m∠2 = 90° 3. m∠1 = m∠2 4. ∠1 ≅ ∠2 2. Def. of rt. ∠ 3. Trans. Prop. of = 4. Def. of ≅  3. Use the given plan to write a two-column proof of one case of the Congruent Complements Theorem. Given: ∠1 and ∠2 are complementary, and ∠2 and ∠3 are complementary. Prove: ∠1 ≅ ∠3 Plan: The measures of complementary angles add to 90° by definition. Use substitution to show that the sums of both pairs are equal. Use the Subtraction Property and the definition of congruent angles to conclude that ∠1 ≅ ∠3. The Proof Process 1. Write the conjecture to be proven. 2. Draw a diagram to represent the hypothesis of the conjecture. 3. State the given information and mark it on the diagram. 4. State the conclusion of the conjecture in terms of the diagram. 5. Plan your argument and prove the conjecture. 112 112 Chapter 2 Geometric Reasoning �� THINK AND DISCUSS 1. Which step in a proof should match the Prove statement? 2. Why is it important to include every logical step in a proof? 3. List four things you can use to justify a step in a proof. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the steps of the proof process. 2-6 Exercises Exercises KEYWORD: MG7 2-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In a two-column proof, you list the ? in the left column and the ̶̶̶̶ the right column. (statements or reasons) ? in ̶̶̶̶ . Write a justification for each step, given that m∠A = 60° and m∠B = 2m∠A. 2. A ? is a statement you can prove. (postulate or theorem) ̶̶̶̶ p. 110 1. m∠A = 60°, m∠B = 2m∠A 2. m∠B = 2 (60°) 3. m∠B = 120° 4. m∠A + m∠ B = 60° + 120° 5. m∠A + m∠B = 180° 6. ∠A and ∠B are supplementary. Fill in the blanks to complete the two-column proof. p. 111 Given: ∠2 ≅ ∠3 Prove: ∠1 and ∠3 are supplementary. Proof: Statements Reasons 1. ∠2 ≅ ∠3 2. m∠2 = m∠3 3. b. ? ̶̶̶̶̶̶ 1. Given 2. a. ? ̶̶̶̶̶̶ 3. Lin. Pair Thm. 4. m∠1 + m∠2 = 180° 4. Def. of supp.  5. m∠1 + m∠3 = 180° 6. d. ? ̶̶̶̶̶̶ 5. c. ? Steps 2, 4 ̶̶̶̶̶ 6. Def. of supp. 112 5. Use the given plan to write a two-column proof. Given: X is the midpoint of ̶̶ YB Prove: ̶̶ AX ≅ ̶̶ AY , and Y is the midpoint of ̶̶ XB . Plan: By the definition of midpoint, Use the Transitive Property to conclude that ̶̶ AX ≅ ̶̶ XY , and ̶̶ AX ≅ ̶̶ XY ≅ ̶̶ YB . ̶̶ YB . 2- 6 Geometric Proof 113 113 �� Independent Practice For See Exercises Example 6 7–8 9–10 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S7 Application Practice p. S29 PRACTICE AND PROBLEM SOLVING 6. Write a justification for each step, given that  BX bisects ∠ABC and m∠XBC = 45°. 1.  BX bisects ∠ABC. 2. ∠ABX ≅ ∠XBC 3. m∠ABX = m∠XBC 4. m∠XBC = 45° 5. m∠ABX = 45° 6. m∠ABX + m∠XBC = m∠ABC 7. 45° + 45° = m∠ABC 8. 90° = m∠ABC 9. ∠ABC is a right angle. Fill in the blanks to complete each two-column proof. 7. Given: ∠1 and∠2 are supplementary, and ∠3 and ∠4 are supplementary. ∠2 ≅ ∠3 Prove: ∠1 ≅ ∠4 Proof: Statements Reasons 1. ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. 1. Given 2. a. ? ̶̶̶̶̶ 3. m∠1 + m∠2 = m∠3 + m∠4 4. ∠2 ≅ ∠3 5. m∠2 = m∠3 6. c. ? ̶̶̶̶̶ 7. ∠1 ≅ ∠4 2. Def. of supp.  3. b. ? ̶̶̶̶̶ 4. Given 5. Def. of ≅  6. Subtr. Prop. of = Steps 3, 5 7. d. ? ̶̶̶̶̶ 8. Given: ∠BAC is a right angle. ∠2 ≅ ∠3 Prove: ∠1 and ∠3 are complementary. Proof: Statements Reasons 1. ∠BAC is a right angle. 1. Given 2. m∠BAC = 90° 3. b. ? ̶̶̶̶̶ 2. a. ? ̶̶̶̶̶ 3. ∠ Add. Post. 4. m∠1 + m∠2 = 90° 4. Subst. Steps 2, 3 5. ∠2 ≅ ∠3 6. c. ? ̶̶̶̶̶ 7. m∠1 + m∠3 = 90° 8. e. ? ̶̶̶̶̶ 5. Given 6. Def. of ≅  7. d. ? Steps 4, 6 ̶̶̶̶̶ 8. Def. of comp.  Use the given plan to write a two-column proof. 9. Given: Prove: ̶̶ BE ≅ ̶̶ AB ≅ ̶̶ CE , ̶̶ CD ̶̶ DE ≅ ̶̶ AE Plan: Use the definition of congruent segments to write the given information in terms of lengths. Then use the Segment Addition Postulate to show that AB = CD and thus ̶̶ AB ≅ ̶̶ CD . 114 114 Chapter 2 Geometric Reasoning �� Use the given plan to write a two-column proof. 10. Given: ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. ∠3 ≅ ∠4 Prove: ∠1 ≅ ∠2 Plan: Since ∠1 and ∠3 are complementary and ∠2 and ∠4 are complementary, both pairs of angle measures add to 90°. Use substitution to show that the sums of both pairs are equal. Since ∠3 ≅ ∠4, their measures are equal. Use the Subtraction Property of Equality and the definition of congruent angles to conclude that ∠1 ≅ ∠2. Engineering Find each angle measure. Find each angle measure. 11. 11. m∠1 12. m∠2 13. m∠3 The Bluff Dale Bridge, constructed by Texas builder William Flinn in 1891, is the oldest known cable-stayed bridge in the United States. 14. Engineering The Bluff Dale Bridge is 140 feet long and spans the Paluxy River in Bluff Dale, Texas. If ∠1 ≅ ∠2, which theorem can you use to conclude that ∠3 ≅ ∠4? 15. Critical Thinking Explain why there are two cases to consider when proving the Congruent Supplements Theorem and the Congruent Complements Theorem. Tell whether each statement is sometimes, always, or never true. 16. An angle and its complement are congruent. 17. A pair of right angles forms a linear pair. 18. An angle and its complement form a right angle. 19. A linear pair of angles is complementary. Algebra Find the value of each variable. 20. 21. 22. � � � � 23. Write About It How are a theorem and a postulate alike? How are they different? 24. This problem will prepare you for the Multi-Step TAKS Prep on page 126. Sometimes you may be asked to write a proof without a specific statement of the Given and Prove information being provided for you. For each of the following situations, use the triangle to write a Given and Prove statement. a. The segment connecting the midpoints of two sides of a triangle is half as long as the third side. b. The acute angles of a right triangle are complementary. c. In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. 2- 6 Geometric Proof 115 115 �� 25. Which theorem justifies the conclusion that ∠1 ≅ ∠4? Linear Pair Theorem Congruent Supplements Theorem Congruent Complements Theorem Right Angle Congruence Theorem 26. What can be concluded from the statement m∠1 + m∠2 = 180°? ∠1 and ∠2 are congruent. ∠1 and ∠2 are supplementary. ∠1 and ∠2 are complementary. ∠1 and ∠2 form a linear pair. 27. Given: Two angles are complementary. The measure of one angle is 10° less than the measure of the other angle. Conclusion: The measures of the angles are 85° and 95°. Which statement is true? T
he conclusion is correct because 85° is 10° less than 95°. The conclusion is verified by the first statement given. The conclusion is invalid because the angles are not congruent. The conclusion is contradicted by the first statement given. CHALLENGE AND EXTEND 28. Write a two-column proof. Given: m∠LAN = 30°, m∠1 = 15° Prove:  AM bisects ∠LAN. Multi-Step Find the value of the variable and the measure of each angle. 29. 30. SPIRAL REVIEW The table shows the number of tires replaced by a repair company during one week, classified by the mileage on the tires when they were replaced. Use the table for Exercises 31 and 32. (Previous course) 31. What percent of the tires had mileage between 40,000 and 49,999 when replaced? 32. If the company replaces twice as many tires next week, about how many tires would you expect to have lasted between 80,000 and 89,999 miles? Mileage on Replaced Tires Mileage Tires 40,000–49,999 50,000–59,999 60,000–69,999 70,000–79,999 80,000–89,999 60 82 54 40 14 Sketch a figure that shows each of the following. (Lesson 1-1) 33. Through any two collinear points, there is more than one plane containing them. 34. A pair of opposite rays forms a line. Identify the property that justifies each statement. (Lesson 2-5) 35. ̶̶ JK ≅ ̶̶ KL , so ̶̶ KL ≅ ̶̶ JK . 36. If m = n and n = p, then m = p. 116 116 Chapter 2 Geometric Reasoning �� 2-6 Use with Lesson 2-6 Activity Design Plans for Proofs Sometimes the most challenging part of writing a proof is planning the logical steps that will take you from the Given statement to the Prove statement. Like working a jigsaw puzzle, you can start with any piece. Write down everything you know from the Given statement. If you don’t see the connection right away, start with the Prove statement and work backward. Then connect the pieces into a logical order. TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system... Also G.3.B, G.3.C, G.3.E Prove the Common Angles Theorem. Given: ∠AXB ≅ ∠CXD Prove: ∠AXC ≅ ∠BXD 1 Start by considering the difference in the Given and Prove statements. How does ∠AXB compare to ∠AXC? How does ∠CXD compare to ∠BXD? In both cases, ∠BXC is combined with the first angle to get the second angle. 2 The situation involves combining adjacent angle measures, so list any definitions, properties, postulates, and theorems that might be helpful. Definition of congruent angles, Angle Addition Postulate, properties of equality, and Reflexive, Symmetric, and Transitive Properties of Congruence 3 Start with what you are given and what you are trying to prove and then work toward the middle. ∠AXB ≅ ∠CXD m∠AXB = m∠CXD ??? m∠AXC = m∠BXD ∠AXC ≅ ∠BXD The first reason will be “Given.” Def. of ≅  ??? Def. of ≅  The last statement will be the Prove statement. 4 Based on Step 1, ∠BXC is the missing piece in the middle of the logical flow. So write down what you know about ∠BXC. ∠BXC ≅ ∠BXC m∠BXC = m∠BXC Reflex. Prop. of ≅ Reflex. Prop. of = 5 Now you can see that the Angle Addition Postulate needs to be used to complete the proof. m∠AXB + m∠BXC = m∠AXC m∠BXC + m∠CXD = m∠BXD ∠ Add. Post. ∠ Add. Post. 6 Reorder the pieces above to write a two-column proof of the Common Angles Theorem. Try This 1. Describe how a plan for a proof differs from the actual proof. 2. Write a plan and a two-column proof. BD bisects ∠ABC. Given: Prove: 2m∠1 = m∠ABC 3. Write a plan and a two-column proof. Given: ∠LXN is a right angle. Prove: ∠1 and ∠2 are complementary. 2-6 Geometry Lab 117 117 �� 2-7 Flowchart and Paragraph Proofs TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Also G.2.B, G.3.C, G.3.E Objectives Write flowchart and paragraph proofs. Prove geometric theorems by using deductive reasoning. Vocabulary flowchart proof paragraph proof Why learn this? Flowcharts make it easy to see how the steps of a process are linked together. A second style of proof is a flowchart proof , which uses boxes and arrows to show the structure of the proof. The steps in a flowchart proof move from left to right or from top to bottom, shown by the arrows connecting each box. The justification for each step is written below the box. Theorem 2-7-1 Common Segments Theorem THEOREM HYPOTHESIS CONCLUSION Given collinear points A, B, C, and D arranged ̶̶ CD , then as shown, if ̶̶ AC ≅ ̶̶ AB ≅ ̶̶ BD . ̶̶ AB ≅ ̶̶ CD ̶̶ AC ≅ ̶̶ BD E X A M P L E 1 Reading a Flowchart Proof Use the given flowchart proof to write a two-column proof of the Common Segments Theorem. Given: Prove: ̶̶ AB ≅ ̶̶ AC ≅ ̶̶ CD ̶̶ BD Flowchart proof: �� Two-column proof: Statements Reasons ̶̶ AB ≅ ̶̶ CD 1. 2. AB = CD 3. BC = BC 4. AB + BC = BC + CD 1. Given 2. Def. of ≅ segs. 3. Reflex. Prop. of = 4. Add. Prop. of = 5. AB + BC = AC, BC + CD = BD 5. Seg. Add. Post. 6. AC = BD ̶̶ BD ̶̶ AC ≅ 7. 6. Subst. 7. Def. of ≅ segs. 118 118 Chapter 2 Geometric Reasoning �� 1. Use the given flowchart proof to write a two-column proof. Given: RS = UV, ST = TU Prove: ̶̶ RT ≅ ̶̶ TV Flowchart proof: E X A M P L E 2 Writing a Flowchart Proof Use the given two-column proof to write a flowchart proof of the Converse of the Common Segments Theorem. Given: Prove: ̶̶ AC ≅ ̶̶ AB ≅ ̶̶ BD ̶̶ CD Two-column proof: Statements Reasons Like the converse of a conditional statement, the converse of a theorem is found by switching the hypothesis and conclusion. ̶̶ AC ≅ ̶̶ BD 1. 2. AC = BD 1. Given 2. Def. of ≅ segs. 3. AB + BC = AC, BC + CD = BD 3. Seg. Add. Post. 4. AB + BC = BC + CD 5. BC = BC 6. AB = CD ̶̶ CD ̶̶ AB ≅ 7. 4. Subst. Steps 2, 3 5. Reflex. Prop. of = 6. Subtr. Prop. of = 7. Def. of ≅ segs. Flowchart proof: 2. Use the given two-column proof to write a flowchart proof. Given: ∠2 ≅ ∠4 Prove: m∠1 = m∠3 Two-column proof: Statements Reasons 1. ∠2 ≅ ∠4 2. ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. 3. ∠1 ≅ ∠3 4. m∠1 = m∠3 1. Given 2. Lin. Pair Thm. 3. ≅ Supps. Thm. 4. Def. of ≅  2-7 Flowchart and Paragraph Proofs 119 119 �� A paragraph proof is a style of proof that presents the steps of the proof and their matching reasons as sentences in a paragraph. Although this style of proof is less formal than a two-column proof, you still must include every step. Theorems THEOREM HYPOTHESIS CONCLUSION 2-7-2 Vertical Angles Theorem Vertical angles are congruent. ∠A and ∠B are vertical angles. ∠A ≅ ∠B 2-7-3 If two congruent angles are supplementary, then each angle is a right angle. (≅  supp. → rt. ) ∠1 ≅ ∠2 ∠1 and ∠2 are supplementary. ∠1 and ∠2 are right angles. E X A M P L E 3 Reading a Paragraph Proof Use the given paragraph proof to write a two-column proof of the Vertical Angles Theorem. Given: ∠1 and ∠3 are vertical angles. Prove: ∠1 ≅ ∠3 � � � Paragraph proof: ∠1 and ∠3 are vertical angles, so they are formed by intersecting lines. Therefore ∠1 and ∠2 are a linear pair, and ∠2 and ∠3 are a linear pair. By the Linear Pair Theorem, ∠1 and ∠2 �� are supplementary, and ∠2 and ∠3 are supplementary. So by the �� Congruent Supplements Theorem, ∠1 ≅ ∠3. �� Two-column proof: Statements Reasons 1. ∠1 and ∠3 are vertical angles. 1. Given 2. ∠1 and ∠3 are formed by intersecting lines. 2. Def. of vert.  3. ∠1 and ∠2 are a linear pair. ∠2 and ∠3 are a linear pair. 4. ∠1 and ∠2 are supplementary. ∠2 and ∠3 are supplementary. 5. ∠1 ≅ ∠3 3. Def. of lin. pair 4. Lin. Pair Thm. 5. ≅ Supps. Thm. 3. Use the given paragraph proof to write a two-column proof. Given: ∠WXY is a right angle. ∠1 ≅ ∠3 Prove: ∠1 and ∠2 are complementary. Paragraph proof: Since ∠WXY is a right angle, m∠WXY = 90° by the definition of a right angle. By the Angle Addition Postulate, m∠WXY = m∠2 + m∠3. By substitution, m∠2 + m∠3 = 90°. Since ∠1 ≅ ∠3, m∠1 = m∠3 by the definition of congruent angles. Using substitution, m∠2 + m∠1 = 90°. Thus by the definition of complementary angles, ∠1 and ∠2 are complementary. 120 120 Chapter 2 Geometric Reasoning �� Writing a Proof When I have to write a proof and I don’t see how to start, I look at what I’m supposed to be proving and see if it makes sense. If it does, I ask myself why. Sometimes this helps me to see what the reasons in the proof might be. If all else fails, I just start writing down everything I know based on the diagram and the given statement. By brainstorming like this, I can usually figure out the steps of the proof. You can even write each thing on a separate piece of paper and arrange the pieces of paper like a flowchart. Claire Jeffords Riverbend High School E X A M P L E 4 Writing a Paragraph Proof Use the given two-column proof to write a paragraph proof of Theorem 2-7-3. Given: ∠1 and ∠2 are supplementary. ∠1 ≅ ∠2 Prove: ∠1 and ∠2 are right angles. Two-column proof: Statements Reasons 1. ∠1 and ∠2 are supplementary. 1. Given ∠1 ≅ ∠2 2. m∠1 + m∠2 = 180° 3. m∠1 = m∠2 4. m∠1 + m∠1 = 180° 5. 2m∠1 = 180° 6. m∠1 = 90° 7. m∠2 = 90° 2. Def. of supp.  3. Def. of ≅  Step 1 4. Subst. Steps 2, 3 5. Simplification 6. Div. Prop. of = 7. Trans. Prop. of = Steps 3, 6 8. ∠1 and ∠2 are right angles. 8. Def. of rt. ∠ Paragraph proof: ∠1 and ∠2 are supplementary, so m∠1 + m∠2 = 180° by the definition of supplementary angles. They are also congruent, so their measures are equal by the definition of congruent angles. By substitution, m∠1 + m∠1 = 180°, so m∠1 = 90° by the Di
vision Property of Equality. Because m∠1 = m∠2, m∠2 = 90° by the Transitive Property of Equality. So both are right angles by the definition of a right angle. 4. Use the given two-column proof to write a paragraph proof. Given: ∠1 ≅ ∠4 Prove: ∠2 ≅ ∠3 Two-column proof: Statements Reasons 1. ∠1 ≅ ∠4 1. Given 2. ∠1 ≅ ∠2, ∠3 ≅ ∠4 2. Vert.  Thm. 3. ∠2 ≅ ∠4 4. ∠2 ≅ ∠3 3. Trans. Prop. of ≅ Steps 1, 2 4. Trans. Prop. of ≅ Steps 2, 3 2-7 Flowchart and Paragraph Proofs 121 121 �� THINK AND DISCUSS 1. Explain why there might be more than one correct way to write a proof. 2. Describe the steps you take when writing a proof. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the proof style in your own words. 2-7 Exercises Exercises KEYWORD: MG7 2-7 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In a ? proof, the logical order is represented by arrows that connect each step. ̶̶̶̶ (flowchart or paragraph) 2. The steps and reasons of a (flowchart or paragraph) ? proof are written out in sentences. ̶̶̶̶ . Use the given flowchart proof to write p. 118 a two-column proof. Given: ∠1 ≅ ∠2 Prove: ∠1 and ∠2 are right angles. Flowchart proof. Use the given two-column proof to write p. 119 a flowchart proof. Given: ∠2 and ∠4 are supplementary. Prove: m∠2 = m∠3 Two-column proof: Statements Reasons 1. ∠2 and ∠4 are supplementary. 1. Given 2. ∠3 and ∠4 are supplementary. 2. Lin. Pair Thm. 3. ∠2 ≅ ∠3 4. m∠2 = m∠3 3. ≅ Supps. Thm. Steps 1, 2 4. Def. of ≅  122 122 Chapter 2 Geometric Reasoning �� . Use the given paragraph proof to write a two-column proof. p. 120 Given: ∠2 ≅ ∠4 Prove: ∠1 ≅ ∠3 Paragraph proof: By the Vertical Angles Theorem, ∠1 ≅ ∠2, and ∠3 ≅ ∠4. It is given that ∠2 ≅ ∠4. By the Transitive Property of Congruence, ∠1 ≅ ∠4, and thus ∠1 ≅ ∠3. Use the given two-column proof to write a paragraph proof. p. 121 Given: Prove:  BD bisects ∠ABC.  BG bisects ∠FBH. Two-column proof: Statements Reasons 1.  BD bisects ∠ABC. 1. Given 2. ∠1 ≅ ∠2 2. Def. of ∠ bisector 3. ∠1 ≅ ∠4, ∠2 ≅ ∠3 3. Vert.  Thm. 4. ∠4 ≅ ∠2 5. ∠4 ≅ ∠3 4. Trans. Prop. of ≅ Steps 2, 3 5. Trans. Prop. of ≅ Steps 3, 4 6.  BG bisects ∠FBH. 6. Def. of ∠ bisector Independent Practice For See Exercises Example 7 8 9 10 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S7 Application Practice p. S29 PRACTICE AND PROBLEM SOLVING 7. Use the given flowchart proof to write a two-column proof. Given: B is the midpoint of ̶̶ AC . AD = EC Prove: DB = BE Flowchart proof: 8. Use the given two-column proof to write a flowchart proof. Given: ∠3 is a right angle. Prove: ∠4 is a right angle. Two-column proof: Statements Reasons 1. ∠3 is a right angle. 2. m∠3 = 90° 1. Given 2. Def. of rt. ∠ 3. ∠3 and ∠4 are supplementary. 3. Lin. Pair Thm. 4. m∠3 + m∠4 = 180° 5. 90° + m∠4 = 180° 6. m∠4 = 90° 7. ∠4 is a right angle. 4. Def. of supp.  5. Subst. Steps 2, 4 6. Subtr. Prop. of = 7. Def. of rt. ∠ 2-7 Flowchart and Paragraph Proofs 123 123 �� 9. Use the given paragraph proof to write a two-column proof. Given: ∠1 ≅ ∠4 Prove: ∠2 and ∠3 are supplementary. Paragraph proof: ∠4 and ∠3 form a linear pair, so they are supplementary by the Linear Pair Theorem. Therefore, m∠4 + m∠3 = 180°. Also, ∠1 and ∠2 are vertical angles, so ∠1 ≅ ∠2 by the Vertical Angles Theorem. It is given that ∠1 ≅ ∠4. So by the Transitive Property of Congruence, ∠4 ≅ ∠2, and by the definition of congruent angles, m∠4 = m∠2. By substitution, m∠2 + m∠3 = 180°, so ∠2 and ∠3 are supplementary by the definition of supplementary angles. 10. Use the given two-column proof to write a paragraph proof. Given: ∠1 and ∠2 are complementary. Prove: ∠2 and ∠3 are complementary. Two-column proof: Statements Reasons 1. ∠1 and ∠2 are complementary. 1. Given 2. m∠1 + m∠2 = 90° 3. ∠1 ≅ ∠3 4. m∠1 = m∠3 5. m∠3 + m∠2 = 90° 2. Def. of comp.  3. Vert.  Thm. 4. Def. of ≅  5. Subst. Steps 2, 4 6. ∠2 and ∠3 are complementary. 6. Def. of comp.  Find each measure and name the theorem that justifies your answer. 11. AB 12. m∠2 13. m∠3 Algebra Find the value of each variable. 14. 15. 16. 17. /////ERROR ANALYSIS///// Below are two drawings for the given proof. Which is incorrect? Explain the error. ̶̶ BC Given: Prove: ∠A ≅ ∠C ̶̶ AB ≅ 18. This problem will prepare you for the Multi-Step TAKS Prep on page 126. Rearrange the pieces to create a flowchart proof. 124 124 Chapter 2 Geometric Reasoning ��x � ��x � ��y�� 19. Critical Thinking Two lines intersect, and one of the angles formed is a right angle. Explain why all four angles are congruent. 20. Write About It Which style of proof do you find easiest to write? to read? 21. Which pair of angles in the diagram must be congruent? ∠1 and ∠5 ∠3 and ∠4 ∠5 and ∠8 None of the above 22. What is the measure of ∠2? 38° 52° 128° 142° 23. Which statement is NOT true if ∠2 and ∠6 are supplementary? m∠2 + m∠6 = 180° ∠2 and ∠3 are supplementary. ∠1 and ∠6 are supplementary. m∠1 + m∠4 = 180° CHALLENGE AND EXTEND 24. Textiles Use the woven pattern to write a flowchart proof. Given: ∠1 ≅ ∠3 Prove: m∠4 + m∠5 = m∠6 25. Write a two-column proof. Given: ∠AOC ≅ ∠BOD Prove: ∠AOB ≅ ∠COD 26. Write a paragraph proof. Given: ∠2 and ∠5 are right angles. m∠1 + m∠2 + m∠3 = m∠4 + m∠5 + m∠6 Prove: ∠1 ≅ ∠4 27. Multi-Step Find the value of each variable and the measures of all four angles. SPIRAL REVIEW Solve each system of equations. Check your solution. (Previous course) ⎧ 7x - y = -33 29. ⎨ 3x + y = -7 ⎩ ⎧ 28. ⎨ ⎩ y = -6x + 18 y = 2x + 14 ⎧ ⎨ ⎩ 30. 2x + y = 8 -x + 3y = 10 Use a protractor to draw an angle with each of the following measures. (Lesson 1-3) 31. 125° 32. 38° 33. 94° 34. 175° For each conditional, write the converse and a biconditional statement. (Lesson 2-4) 35. If a positive integer has more than two factors, then it is a composite number. 36. If a quadrilateral is a trapezoid, then it has exactly one pair of parallel sides. 2-7 Flowchart and Paragraph Proofs 125 125 �� SECTION 2B Mathematical Proof Intersection Inspection According to the U.S. Department of Transportation, it is ideal for two intersecting streets to form four 90° angles. If this is not possible, roadways should meet at an angle of 75° or greater for maximum safety and visibility. 1. Write a compound inequality to represent the range of measures an angle in an intersection should have. 2. Suppose that an angle in an intersection meets the guidelines specified by the U.S. Department of Transportation. Find the range of measures for the adjacent angle in the intersection. The intersection of West Elm Street and Lamar Boulevard has a history of car accidents. The Southland neighborhood association is circulating a petition to have the city reconstruct the intersection. A surveyor measured the intersection, and one of the angles measures 145°. 3. Given that m∠2 = 145°, write a two-column proof to show that m∠1 and m∠3 are less than 75°. 4. Write a paragraph proof to justify the argument that the intersection of West Elm Street and Lamar Boulevard should be reconstructed. 126 126 Chapter 2 Geometric Reasoning �� SECTION 2B Quiz for Lessons 2-5 Through 2-7 2-5 Algebraic Proof Solve each equation. Write a justification for each step. 1. m - 8 = 13 2. 4y - 1 = 27 3. - x _ 3 = 2 Identify the property that justifies each statement. 4. m∠XYZ = m∠PQR, so m∠PQR = m∠XYZ. ̶̶ AB ≅ ̶̶ AB 5. 6. ∠4 ≅ ∠A, and ∠A ≅ ∠1. So ∠4 ≅ ∠1. 7. k = 7, and m = 7. So k = m. 2-6 Geometric Proof 8. Fill in the blanks to complete the two-column proof. Given: m∠1 + m∠3 = 180° Prove: ∠1 ≅ ∠4 Proof: Statements Reasons 1. m∠1 + m∠3 = 180° 2. b. ? ̶̶̶̶̶̶ 1. a. ? ̶̶̶̶̶̶ 2. Def. of supp.  3. ∠3 and ∠4 are supplementary. 3. Lin. Pair Thm. 4. ∠3 ≅ ∠3 5. d. ? ̶̶̶̶̶̶ 4. c. ? ̶̶̶̶̶ 5. ≅ Supps. Thm. 9. Use the given plan to write a two-column proof of the Symmetric Property of ̶̶ EF ̶̶ AB Congruence. ̶̶ AB ≅ Given: ̶̶ EF ≅ Prove: Plan: Use the definition of congruent segments to write of equality. Then use the Symmetric Property of Equality to show that EF = AB. So ̶̶ AB by the definition of congruent segments. ̶̶ EF as a statement ̶̶ AB ≅ ̶̶ EF ≅ 2-7 Flowchart and Paragraph Proofs Use the given two-column proof to write the following. Given: ∠1 ≅ ∠3 Prove: ∠2 ≅ ∠4 Proof: Statements Reasons 1. ∠1 ≅ ∠3 1. Given 2. ∠1 ≅ ∠2, ∠3 ≅ ∠4 2. Vert.  Thm. 3. ∠2 ≅ ∠3 4. ∠2 ≅ ∠4 3. Trans. Prop. of ≅ 4. Trans. Prop. of ≅ 10. a flowchart proof 11. a paragraph proof Ready to Go On? 127 127 �� EXTENSION EXTENSION Introduction to Symbolic Logic Objectives Analyze the truth value of conjunctions and disjunctions. Construct truth tables to determine the truth value of logical statements. Vocabulary compound statement conjunction disjunction truth table TEKS G.4.A Select an appropriate representation ... in order to solve problems. Also G.3.C Symbolic logic is used by computer programmers, mathematicians, and philosophers to analyze the truth value of statements, independent of their actual meaning. A compound statement is created by combining two or more statements. Suppose p and q each represent a statement. Two compound statements can be formed by combining p and q: a conjunction and a disjunction. Compound Statements TERM WORDS SYMBOLS EXAMPLE Conjunction A compound statement that uses the word and p AND q p ⋀ q Pat is a band member
AND Pat plays tennis. Disjunction A compound statement that uses the word or p OR q p ⋁ q Pat is a band member OR Pat plays tennis. A conjunction is true only when all of its parts are true. A disjunction is true if any one of its parts is true. E X A M P L E 1 Analyzing Truth Values of Conjunctions and Disjunctions Use p, q, and r to find the truth value of each compound statement. p: Washington, D.C., is the capital of the United States. q: The day after Monday is Tuesday. r: California is the largest state in the United States. A q ⋁ r B r ⋀ p Since q is true, the disjunction is true. Since r is false, the conjunction is false. Use the information given above to find the truth value of each compound statement. 1a. r ⋁ p 1b. p ⋀ q A table that lists all possible combinations of truth values for a statement is called a truth table . A truth table shows you the truth value of a compound statement, based on the possible truth values of its parts. Make sure you include all possible combinations of truth values for each piece of the compound statement 128 128 Chapter 2 Geometric Reasoning E X A M P L E 2 Constructing Truth Tables for Compound Statements Construct a truth table for the compound statement ∼u ⋀ (v ⋁ w) . Since u, v, and w can each be either true or false, the truth table will have (2) (2) (2) = 8 rows. The negation (~) of a statement has the opposite truth valueu v ⋁ w ∼u ⋀ (v ⋁ w. Construct a truth table for the compound statement ∼u ⋀ ∼v. EXTENSION Exercises Exercises Use p, q, and r to find the truth value of each compound statement. p : The day after Friday is Sunday. q: 1 _ 2 r : If -4x - 2 = 10, then x = 3. = 0.5 1. r ⋀ q 4. q ⋀ ∼q 2. r ⋁ p 5. ∼q ⋁ q 3. p ⋁ r 6. q ⋁ r Construct a truth table for each compound statement. 7. s ⋀ ∼t 8. ∼u ⋁ t 9. ∼u ⋁ (s ⋀ t) Use a truth table to show that the two statements are logically equivalent. 10. p → q; ∼q → ∼p 11. q → p; ∼p → ∼q 12. A biconditional statement can be written as (p → q) ⋀ (q → p) . Construct a truth table for this compound statement. 13. DeMorgan’s Laws state that ∼ (p ⋀ q) = ∼p ⋁ ∼q and that ∼ (p ⋁ q) = ∼p ⋀ ∼q. a. Use truth tables to show that both statements are true. b. If you think of disjunction and conjunction as inverse operations, DeMorgan’s Laws are similar to which algebraic property? 14. The Law of Disjunctive Inference states that if p ⋁ q is true and p is false, then q must be true. a. Construct a truth table for p ⋁ q. b. Use the truth table to explain why the Law of Disjunctive Inference is true. Chapter 2 Extension 129 129 For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary biconditional statement . . . . . . 96 definition . . . . . . . . . . . . . . . . . . . 97 paragraph proof . . . . . . . . . . . . 120 conclusion . . . . . . . . . . . . . . . . . . 81 flowchart proof . . . . . . . . . . . . . 118 polygon . . . . . . . . . . . . . . . . . . . . . 98 conditional statement . . . . . . . . 81 hypothesis . . . . . . . . . . . . . . . . . . 81 proof . . . . . . . . . . . . . . . . . . . . . . 104 conjecture . . . . . . . . . . . . . . . . . . 74 inductive reasoning . . . . . . . . . . 74 quadrilateral . . . . . . . . . . . . . . . . 98 contrapositive . . . . . . . . . . . . . . . 83 inverse . . . . . . . . . . . . . . . . . . . . . . 83 theorem . . . . . . . . . . . . . . . . . . . 110 converse . . . . . . . . . . . . . . . . . . . . 83 logically equivalent triangle . . . . . . . . . . . . . . . . . . . . . 98 counterexample . . . . . . . . . . . . . 75 deductive reasoning . . . . . . . . . 88 statements . . . . . . . . . . . . . . . . 83 negation . . . . . . . . . . . . . . . . . . . . 82 truth value . . . . . . . . . . . . . . . . . . 82 two-column proof . . . . . . . . . . 111 Complete the sentences below with vocabulary words from the list above. 1. A statement you can prove and then use as a reason in later proofs is a(n) ? . ̶̶̶ 2. ? is the process of using logic to draw conclusions from given facts, definitions, ̶̶̶ and properties. 3. A(n) ? is a case in which a conjecture is not true. ̶̶̶ 4. A statement you believe to be true based on inductive reasoning is called a(n) ? . ̶̶̶ 2-1 Using Inductive Reasoning to Make Conjectures (pp. 74–79) E X A M P L E S EXERCISES TEKS G.2.B, G.3.D, G.5.B ■ Find the next item in the pattern below. Make a conjecture about each pattern. Write the next two items. The red square moves in the counterclockwise direction. The next figure is . 5. ■ Complete the conjecture “The sum of two ? .” ̶̶̶ odd numbers is List some examples and look for a pattern + 11 = 18 Complete each conjecture. 8. The sum of an even number and an odd number is ? . ̶̶̶ The sum of two odd numbers is even. 9. The square of a natural number is ? . ̶̶̶ ■ Show that the conjecture “For all non-zero integers, -x < x” is false by finding a counterexample. Pick positive and negative values for x and substitute to see if the conjecture holds. Let n = 3. Since -3 < 3, the conjecture holds. Let n = -5. Since - (-5) is 5 and 5 ≮ -5, the conjecture is false. n = -5 is a counterexample. 130 130 Chapter 2 Geometric Reasoning Determine if each conjecture is true. If not, write or draw a counterexample. 10. All whole numbers are natural numbers. ̶̶ BC . 11. If C is the midpoint of ̶̶ AB , then ̶̶ AC ≅ 12. If 2x + 3 = 15, then x = 6. 13. There are 28 days in February. 14. Draw a triangle. Construct the bisectors of each angle of the triangle. Make a conjecture about where the three angle bisectors intersect. �� 2-2 Conditional Statements (pp. 81–87) TEKS G.3.A, G.3.C E X A M P L E S EXERCISES ■ Write a conditional statement from the sentence “A rectangle has congruent diagonals.” If a figure is a rectangle, then it has congruent diagonals. ■ Write the inverse, converse, and contrapositive of the conditional statement “If m∠1 = 35°, then ∠1 is acute.” Find the truth value of each. Converse: If ∠1 is acute, then m∠1 = 35°. Not all acute angles measure 35°, so this is false. Inverse: If m∠1 ≠ 35°, then ∠1 is not acute. You can draw an acute angle that does not measure 35°, so this is false. Contrapositive: If ∠1 is not acute, then m∠1 ≠ 35°. An angle that measures 35° must be acute. So this statement is true. Write a conditional statement from each Venn diagram. 15. 16. Determine if each conditional is true. If false, give a counterexample. 17. If two angles are adjacent, then they have a common ray. 18. If you multiply two irrational numbers, the product is irrational. Write the converse, inverse, and contrapositive of each conditional statement. Find the truth value of each. 19. If ∠X is a right angle, then m∠X = 90°. 20. If x is a whole number, then x = 2. 2-3 Using Deductive Reasoning to Verify Conjectures (pp. 88–93) TEKS G.2.B, E X A M P L E S EXERCISES G.3.B, G.3.C, G.3.E ■ Determine if the conjecture is valid by the Law of Detachment or the Law of Syllogism. Given: If 5c = 8y, then 2w = -15. If 5c = 8y, then x = 17. Conjecture: If 2w = -15, then x = 17. Let p be 5c = 8y, q be 2w = -15, and r be x = 17. Using symbols, the given information is written as p → q and p → r. Neither the Law of Detachment nor the Law of Syllogism can be applied. The conjecture is not valid. Use the true statements below to determine whether each conclusion is true or false. Sue is a member of the swim team. When the team practices, Sue swims. The team begins practice when the pool opens. The pool opens at 8 A.M. on weekdays and at 12 noon on Saturday. 21. The swim team practices on weekdays only. 22. Sue swims on Saturdays. 23. Swim team practice starts at the same time every day. ■ Draw a conclusion from the given information. Given: If two points are distinct, then there is one line through them. A and B are distinct points. Let p be the hypothesis: two points are distinct. Let q be the conclusion: there is one line through the points. The statement “A and B are distinct points” matches the hypothesis, so you can conclude that there is one line through A and B. Use the following information for Exercises 24–26. The expression 2.15 + 0.07x gives the cost of a long-distance phone call, where x is the number of minutes after the first minute. If possible, draw a conclusion from the given information. If not possible, explain why. 24. The cost of Sara’s long-distance call is $2.57. 25. Paulo makes a long-distance call that lasts ten minutes. 26. Asa’s long-distance phone bill for the month is $19.05. Study Guide: Review 131 131 �� 2-4 Biconditional Statements and Definitions (pp. 96–101) TEKS G.3.A, G.3.B E X A M P L E S EXERCISES ■ For the conditional “If a number is divisible by 10, then it ends in 0”, write the converse and a biconditional statement. Converse: If a number ends in 0, then it is divisible by 10. Biconditional: A number is divisible by 10 if and only if it ends in 0. ■ Determine if the biconditional “The sides of a triangle measure 3, 7, and 15 if and only if the perimeter is 25” is true. If false, give a counterexample. Conditional: If the sides of a triangle measure 3, 7, and 15, then the perimeter is 25. True. Converse: If the perimeter of a triangle is 25, then its sides measure 3, 7, and 15. False; a triangle with side lengths of 6, 10, and 9 also has a perimeter of 25. Therefore the biconditional is false. Determine if a true biconditional can be written from each conditional statement. If not, give a counterexample. 27. If 3 - 2x_ 5 = 2, then x = 5_ . 2 28. If x < 0, then the value of x 4 is positive. 29. If a segment has endpoints at (1, 5) and (-3, 1) , then its midpoint is (-1, 3) . 30. If the measure of one angle of a triangle is 90°, then the triangle is a right triangle. Complete each statement to form a true biconditional. 31. Two angles are ? if and only if the sum of ̶̶̶ their measures is 90°. 32. x 3 >0 if and only if x is ? . ̶̶̶ 33. Trey can travel 100 miles in less than 2 hours if and only if his average speed is ? . ̶̶̶ 34. The area of a squ
are is equal to s 2 if and only if the perimeter of the square is ? . ̶̶̶ 2-5 Algebraic Proof (pp. 104–109) TEKS G.3.B, G.3.C, G.3.E E X A M P L E S EXERCISES ■ Solve the equation 5x - 3 = -18. Write a justification for each step. 5x - 3 = -18 + 3 + 3 ̶̶̶ ̶̶̶̶̶ 5x = -15 -15_ 5x_ 5 5 x = -3 = Given Add. Prop. of = Simplify. Div. Prop. of = Simplify. ■ Write a justification for each step. RS = ST Given 5x - 18 = 4x x - 18 = 0 x = 18 Subst. Prop. of = Subtr. Prop. of = Add. Prop. of = Identify the property that justifies each statement. ■ ∠X ≅ ∠2, so ∠2 ≅ ∠X. Symmetric Property of Congruence ■ If m∠2 = 180° and m∠3 = 180°, then m∠2 = m∠3. Transitive Property of Equality 132 132 Chapter 2 Geometric Reasoning Solve each equation. Write a justification for each step. 35. m_ -5 36. -47 = 3x - 59 + 3 = -4.5 Identify the property that justifies each statement. 37. a + b = a + b 38. If ∠RST ≅ ∠ABC, then ∠ABC ≅ ∠RST. 39. 2x = 9, and y = 9. So 2x = y. Use the indicated property to complete each statement. 40. Reflex. Prop. of ≅: figure ABCD ≅ ? ̶̶̶ 41. Sym. Prop. of =: If m∠2 = m∠5, then ̶̶ CD and ̶̶ AB ≅ ? . ̶̶̶ ̶̶ EF , 42. Trans. Prop. of ≅: If ? . ̶̶̶ then ̶̶ AB ≅ 43. Kim borrowed money at an annual simple interest rate of 6% to buy a car. How much did she borrow if she paid $4200 in interest over the life of the 4-year loan? Solve the equation I = Prt for P and justify each step. �� 2-6 Geometric Proof (pp. 110–116) TEKS G.1.A, G.3.B, G.3.C, G.3.E E X A M P L E S EXERCISES ■ Write a justification for each step, given that m∠2 = 2m∠1. 1. ∠1 and ∠2 supp. 2. m∠1 + m∠2 = 180° 3. m∠2 = 2m∠1 4. m∠1 + 2m∠1 = 180° 5. 3m∠1 = 180° 6. m∠1 = 60° Lin. Pair Thm. Def. of supp.  Given Subst. Steps 2, 3 Simplify Div. Prop. of = ■ Use the given plan to write a two-column proof. Given: ̶̶ AD bisects ∠BAC. ∠1 ≅ ∠3 Prove: ∠2 ≅ ∠3 Plan: Use the definition of angle bisector to show that ∠1 ≅ ∠2. Use the Transitive Property to conclude that ∠2 ≅ ∠3. Two-column proof: Statements Reasons ̶̶̶ AD bisects ∠BAC. 1. 1. Given 2. ∠1 ≅ ∠2 3. ∠1 ≅ ∠3 4. ∠2 ≅ ∠3 2. Def. of ∠ bisector 3. Given 4. Trans. Prop. of ≅ 44. Write a justification for each step, given that ∠1 and ∠2 are complementary, and ∠1 ≅ ∠3. 1. ∠1 and ∠2 comp. 2. m∠1 + m∠2 = 90° 3. ∠1 ≅ ∠3 4. m∠1 = m∠3 5. m∠3 + m∠2 = 90° 6. ∠3 and ∠2 comp. 45. Fill in the blanks to complete the two-column proof. ̶̶ TU ≅ Given: Prove: SU + TU = SV Two-column proof: ̶̶ UV Statements Reasons ̶̶ TU ≅ ̶̶ UV 1. 2. b. 3. c. ? ̶̶̶̶ ? ̶̶̶̶ 4. SU + TU = SV 1. a. ? ̶̶̶̶ 2. Def. of ≅ segs. 3. Seg. Add. Post. 4. d. ? ̶̶̶̶ Find the value of each variable. 46. 47. 2-7 Flowchart and Paragraph Proofs (pp. 118–125) TEKS G.1.A, G.2.B, G.3.C, G.3.E E X A M P L E S EXERCISES Use the two-column proof in the example for Lesson 2-6 above to write each of the following. ■ a flowchart proof ■ a paragraph proof ̶̶ AD bisects ∠BAC, ∠1 ≅ ∠2 by the Since definition of angle bisector. It is given that ∠1 ≅ ∠3. Therefore, ∠2 ≅ ∠3 by the Transitive Property of Congruence. Use the given plan to write each of the following. Given: ∠ADE and ∠DAE are complementary. ∠ADE and ∠BAC are complementary. Prove: ∠DAC ≅ ∠BAE Plan: Use the Congruent Complements Theorem to show that ∠DAE ≅ ∠BAC. Since ∠CAE ≅ ∠CAE, ∠DAC ≅ ∠BAE by the Common Angles Theorem. 48. a flowchart proof 49. a paragraph proof Find the value of each variable and name the theorem that justifies your answer. 50. 51. Study Guide: Review 133 133 �� Find the next item in each pattern. 1. 2. 405, 135, 45, 15, … 3. Complete the conjecture “The sum of two even numbers is ? . ” ̶̶̶ 4. Show that the conjecture “All complementary angles are adjacent” is false by finding a counterexample. 5. Identify the hypothesis and conclusion of the conditional statement “The show is cancelled if it rains.” 6. Write a conditional statement from the sentence “Parallel lines do not intersect.” Determine if each conditional is true. If false, give a counterexample. 7. If two lines intersect, then they form four right angles. 8. If a number is divisible by 10, then it is divisible by 5. Use the conditional “If you live in the United States, then you live in Kentucky” for Items 9–11. Write the indicated type of statement and determine its truth value. 9. converse 10. inverse 11. contrapositive 12. Determine if the following conjecture is valid by the Law of Detachment. Given: If it is colder than 50°F, Tom wears a sweater. It is 46°F today. Conjecture: Tom is wearing a sweater. 13. Use the Law of Syllogism to draw a conclusion from the given information. Given: If a figure is a square, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon. Figure ABCD is a square. 14. Write the conditional statement and converse within the biconditional “Chad will work on Saturday if and only if he gets paid overtime.” 15. Determine if the biconditional “B is the midpoint of ̶̶ AC iff AB = BC” is true. If false, give a counterexample. Solve each equation. Write a justification for each step. 16. 8 - 5s = 1 17. 0.4t + 3 = 1.6 18. 38 = -3w + 2 Identify the property that justifies each statement. 19. If 2x = y and y = 7, then 2x = 7. 21. ∠X ≅ ∠P, and ∠P ≅ ∠D. So ∠X ≅ ∠D. Use the given plan to write a proof in each format. 20. m∠DEF = m∠DEF ̶̶ XY ≅ ̶̶ XY , then ̶̶ ST ≅ 22. If ̶̶ ST .  FB bisects ∠AFC. Given: ∠AFB ≅ ∠EFD Prove: Plan: Since vertical angles are congruent, ∠EFD ≅ ∠BFC. Use the Transitive Property to conclude that ∠AFB ≅ ∠BFC. Thus  FB bisects ∠AFC by the definition of angle bisector. 23. two-column proof 24. paragraph proof 25. flowchart proof 134 134 Chapter 2 Geometric Reasoning �� FOCUS ON SAT MATHEMATICS SUBJECT TESTS Some colleges require that you take the SAT Subject Tests. There are two math subject tests—Level 1 and Level 2. Take the Mathematics Subject Test Level 1 when you have completed three years of college-prep mathematics courses. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. On SAT Mathematics Subject Test questions, you receive one point for each correct answer, but you lose a fraction of a point for each incorrect response. Guess only when you can eliminate at least one of the answer choices. 1. In the figure below, m∠1 = m∠2. What is the value of y? 3. What is the contrapositive of the statement “If it is raining, then the football team will win”? Note: Figure not drawn to scale. (A) 10 (C) 40 (E) 60 (B) 30 (D) 50 2. The statement “I will cancel my appointment if and only if I have a conflict” is true. Which of the following can be concluded? I. If I have a conflict, then I will cancel my appointment. II. If I do not cancel my appointment, then I do not have a conflict. III. If I cancel my appointment, then I have a conflict. (A) I only (C) III only (E) I, II, and III (B) II only (D) I and III (A) If it is not raining, then the football team will not win. (B) If it is raining, then the football team will not win. (C) If the football team wins, then it is raining. (D) If the football team does not win, then it is not raining. (E) If it is not raining, then the football team will win. 4. Given the points D (1, 5) and E (-2, 3) , which conclusion is NOT valid? ̶̶ (A) The midpoint of DE is (- 1 _ , 4) . 2 (B) D and E are collinear. (C) The distance between D and E is √  5 . (D) ̶̶ DE ≅ ̶̶ ED (E) D and E are distinct points. 5. For all integers x, what conclusion can be drawn about the value of the expression x 2 __ 2 ? (A) The value is negative. (B) The value is not negative. (C) The value is even. (D) The value is odd. (E) The value is not a whole number. College Entrance Exam Practice 135 135 �� Gridded Response: Record Your Answer When responding to a gridded-response test item, you must fill out the grid on your answer sheet correctly, or the item will be marked as incorrect. Gridded Response: Solve the equation 1258 - 2 (3x - 72) = -80. The value of x is 247. • Using a pencil, write your answer in the answer boxes at the top of the grid. • Put only one digit in each box. The decimal point has a designated column. • Do not leave a blank box in the middle of an answer. • For each digit, shade the bubble that is in the same column as the digit in the answer box. Gridded Response: The perimeter of a rectangle is 90 in. The width of the rectangle is 18 in. Find the length of the rectangle in feet. The length of the rectangle is 27 inches, but the problem asks for the measurement in feet. 27 inches = 9 _ 4 • Fractions and mixed numbers cannot be gridded, so you must grid , or 2.25, feet the answer as 2.25. • Using a pencil, write your answer in the answer boxes at the top of the grid. • Put only one digit in each box. The decimal point has a designated column. • Do not leave a blank box in the middle of an answer. • For each digit, shade the bubble that is in the same column as the digit in the answer box. 136 136 Chapter 2 Geometric Reasoning �� Sample C The length of a segment is 897 2 __ gridded this answer as shown. units. A student 5 �� You cannot grid a negative number in a griddedresponse item because the grid does not include the negative sign (-). So if you get a negative answer to a test item, rework the problem. You probably made a math error. Read each statement and answer the questions that follow. Sample A The correct answer to a test item is 1.6. A student gridded this answer as shown. 5. What answer does the grid show? 6. Explain why you cannot grid a fraction or a mixed number. 7. Write the answer 897 2__ 5 be entered in the grid correctly. in a form that could 1. What error did the student make when filling out the grid? 2. Another student got an answer of
-1.6. Explain why the student knew this answer was wrong. 8. Another student got an answer of 10,216.5 units. Explain why the student knew this answer was wrong. Sample D The measure of an angle is 48.9°. A student gridded this answer as shown. Sample B The perimeter of a triangle is 2 3 __ gridded this answer as shown. feet. A student 4 9. What answer does the grid show? 10. What error did the student make when filling out the grid? 11. Explain how to correctly grid the answer. 3. What error did the student make when filling out the grid? 4. Explain how to correctly grid the answer. TAKS Tackler 137 137 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–2 Multiple Choice Use the figure below for Items 1 and 2. In the figure,   DB bisects ∠ADC. 5. A diagonal of a polygon connects nonconsecutive vertices. The table shows the number of diagonals in a polygon with n sides. Number of Sides Number of Diagonals 4 5 6 7 2 5 9 14 If the pattern continues, how many diagonals does a polygon with 8 sides have? 17 19 20 21 6. Which type of transformation maps figure LMNP onto figure L’M’N’P’? 1. Which best describes the intersection of ∠ADB and ∠BDC ? Exactly one ray Exactly one point Exactly one angle Exactly one segment 2. Which expression is equal to the measure of ∠ADC ? 2 (m∠ADB) 90° - m∠BDC 180° - 2 (m∠ADC) m∠BDC - m∠ADB 3. What is the inverse of the statement, “If a polygon has 8 sides, then it is an octagon”? Reflection Rotation Translation None of these If a polygon is an octagon, then it has 8 sides. If a polygon is not an octagon, then it does not have 8 sides. If an octagon has 8 sides, then it is a polygon. If a polygon does not have 8 sides, then it is not an octagon. 4. Lily conjectures that if a number is divisible by 15, then it is also divisible by 9. Which of the following is a counterexample? 45 50 60 72 7. Miyoko went jogging on July 25, July 28, July 31, and August 3. If this pattern continues, when will Miyoko go jogging next? August 5 August 6 August 7 August 8 8. Congruent segments have equal measures. A segment bisector divides a segment into ̶̶ two congruent segments. DE at X  XY intersects ̶̶ DE . Which conjecture is valid? and bisects m∠YXD = m∠YXE Y is between D and E. DX = XE DE = YE 138 138 Chapter 2 Geometric Reasoning �� 9. Which statement is true by the Symmetric Property of Congruence? ̶̶ ST ̶̶ ST ≅ 15 + MN = MN + 15 If ∠P ≅ ∠Q, then ∠Q ≅ ∠P. If ∠D ≅ ∠E and ∠E ≅ ∠F, then ∠D ≅ ∠F. �� To find a counterexample for a biconditional statement, write the conditional statement and converse it contains. Then try to find a counterexample for one of these statements. 10. Which is a counterexample for the following biconditional statement? A pair of angles is supplementary if and only if the angles form a linear pair. The measures of supplementary angles add to 180°. A linear pair of angles is supplementary. Complementary angles do not form a linear pair. Two supplementary angles are not adjacent. 11. K is between J and L. The distance between J and K is 3.5 times the distance between K and L. If JK = 14, what is JL? 10.5 18 24.5 49 STANDARDIZED TEST PREP Short Response 16. Solve the equation 2 (AB) + 16 = 24 to find the length of segment AB. Write a justification for each step. 17. Use the given two-column proof to write a flowchart proof. ̶̶ Given: DE ≅ Prove: DE = FG + GH ̶̶ FH Two-column proof: Statements ̶̶ FH ̶̶ DE ≅ 1. Reasons 1. Given 2. DE = FH 2. Def. of ≅ segs. 3. FG + GH = FH 3. Seg. Add. Post. 4. DE = FG + GH 4. Subst. 18. Consider the following conditional statement. If two angles are complementary, then the angles are acute. a. Determine if the conditional is true or false. If false, give a counterexample. b. Write the converse of the conditional statement. 12. What is the length of the segment connecting the points (-7, -5) and (5, -2) ? c. Determine whether the converse is true or false. If false, give a counterexample. √  13 √  53 3 √  17 √  193 Gridded Response 13. A segment has an endpoint at (5, -2) . The midpoint of the segment is (2, 2) . What is the length of the segment? 14. ∠P measures 30° more than the measure of its supplement. What is the measure of ∠P in degrees? 15. The perimeter of a square field is 1.6 kilometers. What is the area of the field in square kilometers? Extended Response 19. The figure below shows the intersection of two lines. a. Name the linear pairs of angles in the figure. What conclusion can you make about each pair? Explain your reasoning. b. Name the pairs of vertical angles in the figure. What conclusion can you make about each pair? Explain your reasoning. c. Suppose m∠EBD = 90°. What are the measures of the other angles in the figure? Write a two-column proof to support your answer. Cumulative Assessment, Chapters 1–2 139 139 �� T E X A S TAKS Grades 9–11 Obj. 10 �� The Freescale Marathon Every February, runners take to the streets of Austin to participate in a 26-mile marathon. The course travels mostly downhill, with a net drop in elevation of more than 400 feet from beginning to end. But don’t be fooled; many former participants strongly recommend hill training for this course! Choose one or more strategies to solve each problem. 1. During the marathon, a runner maintains a steady pace and completes the first 2.6 miles in 20 minutes. After 1 hour 20 minutes, she has completed 10.4 miles. Make a conjecture about the runner’s average speed in miles per hour. How long do you expect her to take to complete the marathon? 2. The course features fully equipped aid stations with medical support for the complete eight-hour duration of the race. From mile 2 to mile 6, these stations are located every other mile. After that, they are located every mile to the finish. Portable toilets are available at each mile marker and at the end of the course. At how many points are there both an aid station and portable toilets? �� For 3, use the map. 3. The course includes a straight section along Forty-fifth Street from Shoal Creek Boulevard to Duval Street. The distance from Guadalupe Street to Duval is twice the distance from Burnet Road to Lamar Boulevard. The distance from Lamar to Guadalupe is 210 feet greater than the distance from Shoal Creek to Burnet. What is the distance from Guadalupe to Duval? 140 140 Chapter 2 Geometric Reasoning � � � � � � � � � � � � � � � � � � � � �� Show Caves The region between San Antonio and Austin, known as the Texas Hill Country, is home to six of the seven show caves in the state. A show cave is a cave developed for public use, typically with amenities such as lighting and groomed trails. The seventh show cave in Texas, the Caverns of Sonora, is internationally recognized as one of the most beautiful in the world. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Choose one or more strategies to solve each problem. 1. Jared took a tour at least 5 __ 8 mi long and saw a cave that is less than 10,000 ft in length. Which caves might Jared have visited? 2. A travel brochure includes the following statements about Texas show caves. Determine whether each statement is true or false. If false, explain why. a. If you tour a cave that is more Texas Show Caves Cave Length (ft) Approximate Depth (ft) Tour Length (mi) Cascade Caverns Cave Without a Name Caverns of Sonora Inner Space Cavern Longhorn Cavern Natural Bridge Caverns Wonder Cave 1,700 14,211 20,000 15,000 9,850 8,600 1,296 132 89 150 80 23 250 91 0.25 0.25 1.5 1.2 0.625 0.75 0.08 than 100 ft deep, then you’ll see a cave that is more than 8000 ft in length. b. If you haven’t been to the Caverns of Sonora, then you haven’t seen a cave that is at least 15,000 ft long. c. If you don’t want to walk more than a mile, but you want to see a cave with a depth of at least 150 ft, then you should visit Natural Bridge Caverns. 3. Inner Space Cavern has a second tour, which Ingrid completes in 18 min and 45 s. If Ingrid walks 12,672 ft in 1 h, what is the length of the second tour? Problem Solving on Location 141141 Parallel and Perpendicular Lines 3A Lines with Transversals 3-1 Lines and Angles Lab Explore Parallel Lines and Transversals 3-2 Angles Formed by Parallel Lines and Transversals 3-3 Proving Lines Parallel Lab Construct Parallel Lines 3-4 Perpendicular Lines Lab Construct Perpendicular Lines 3B Coordinate Geometry 3-5 Slopes of Lines Lab Explore Parallel and Perpendicular Lines 3-6 Lines in the Coordinate Plane KEYWORD: MG7 ChProj Sailboats at the Corpus Christi Municipal Marina 142 142 Chapter 3 Vocabulary Match each term on the left with a definition on the right. 1. acute angle A. segments that have the same length 2. congruent angles B. an angle that measures greater than 90° and less than 180° 3. obtuse angle 4. collinear C. points that lie in the same plane D. angles that have the same measure 5. congruent segments E. points that lie on the same line F. an angle that measures greater than 0° and less than 90° Conditional Statements Identify the hypothesis and conclusion of each conditional. 6. If E is on AC , then E lies in plane P. 7. If A is not in plane Q, then A is not on BD . 8. If plane P and plane Q intersect, then they intersect in a line. Name and Classify Angles Name and classify each angle. 9. 1
0. 11. 12. Angle Relationships Give an example of each angle pair. 13. vertical angles 14. adjacent angles 15. complementary angles 16. supplementary angles Evaluate Expressions Evaluate each expression for the given value of the variable. 17. 4x + 9 for x = 31 18. 6x - 16 for x = 43 19. 97 - 3x for x = 20 20. 5x + 3x + 12 for x = 17 Solve Multi-Step Equations Solve each equation for x. 21. 4x + 8 = 24 23. 4x + 3x + 6 = 90 22. 2 = 2x - 8 24. 21x + 13 + 14x - 8 = 180 Parallel and Perpendicular Lines 143 143 �� Key Vocabulary/Vocabulario alternate exterior angles alternate interior angles corresponding angles parallel lines ángulos alternos externos ángulos alternos internos ángulos correspondientes líneas paralelas perpendicular bisector mediatriz perpendicular lines líneas perpendiculares same-side interior angles ángulos internos del mismo lado slope transversal pendiente transversal Geometry TEKS G.1.A Geometric Structure* develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, answer the following questions. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The root trans- means “across.” What do you think a transversal of two lines does? 2. The slope of a mountain trail describes the steepness of the climb. What might the slope of a line describe? 3. What does the word corresponding mean? What do you think the term corresponding angles means? 4. What does the word interior mean? What might the phrase “interior of a pair of lines” describe? The word alternate means “to change from one to another.” If two lines are crossed by a third line, where do you think a pair of alternate interior angles might be? 3-2 Tech. Lab Les. 3-1 Les. 3-2 Les. 3-3 ★ 3-3 Geo. Lab Les. 3-4 ★ 3-4 Geo. Lab 3-6 Tech. Lab Les. 3-5 Les. 3-6 G.2.A Geometric Structure* use construction to ★ ★ ★ explore attributes of geometric figures and to make conjectures about geometric relationships G.3.C Geometric Structure* use logical reasoning to ★ ★ ★ ★ prove statements are true ... G.7.B Dimensionality and the geometry of location* use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines ... G.7.C Dimensionality and the geometry of location* develop and use formulas involving ... slope ... ★ ★ ★ ★ G.9.A Congruence and the geometry of ★ ★ ★ ★ ★ ★ size* formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations * Knowledge and skills are written out completely on pages TX28–TX35. 144 144 Chapter 3 Study Strategy: Take Effective Notes Taking effective notes is an important study strategy. The Cornell system of note taking is a good way to organize and review main ideas. In the Cornell system, the paper is divided into three main sections. The note-taking column is where you take notes during lecture. The cue column is where you write questions and key phrases as you review your notes. The summary area is where you write a brief summary of the lecture. Step 2: Cues After class, write down key phrases or questions in the left column. Step 3: Summary Use your cues to restate the main points in your own words. 9/4/05 Chapter 2 Lesson 6 page 1 What can you use to just ify steps in a proof? Geometric proof: Start with hypothes is , and then use defs . , and thms . to reach posts . conc lus ion . Just ify each step . , Step 1: Notes Draw a vertical line about 2.5 inches from the left side of your paper. During class, write your notes about the main points of the lecture in the right column. What k ind of ang les form a l inear pa ir? Linear Pa ir Theorem If 2 ∠ s form a l in . pa ir, then they are supp . Congruent Supp lements Theorem If 2 ∠ s are supp . to the same ∠ (or to 2 ≅ ∠ s) , then the 2 ∠ s are ≅ . What is true about two supp lements of the same ang le? S u m m a ry : def i n i t i on s , p o stu l ate s , a n d th e o re m s to sh ow th at a c on c l u s i on i s tr u e . Th e Li n e a r Pa i r Th e o re m s ay s th at two a n g le s th at fo re s u p p le m ent a ry. Th e C on g r u ent S u p p le m ent s Th e o re m s ay s th at two s u p p le m ent s to th e s a m e a n g le a re c on g r u ent . Try This 1. Research and write a paragraph describing the Cornell system of note taking. Describe how you can benefit from using this type of system. 2. In your next class, use the Cornell system of note taking. Compare these notes to your notes from a previous lecture. Parallel and Perpendicular Lines 145 145 3-1 Lines and Angles Objectives Identify parallel, perpendicular, and skew lines. Identify the angles formed by two lines and a transversal. Vocabulary parallel lines perpendicular lines skew lines parallel planes transversal corresponding angles alternate interior angles alternate exterior angles same-side interior angles Who uses this? Card architects use playing cards to build structures that contain parallel and perpendicular planes. Bryan Berg uses cards to build structures like the one at right. In 1992, he broke the Guinness World Record for card structures by building a tower 14 feet 6 inches tall. Since then, he has built structures more than 25 feet tall. Parallel, Perpendicular, and Skew Lines Parallel lines (ǁ) are coplanar and do not intersect. In the figure,   AB ǁ   EF , and   EG ǁ   FH . Perpendicular lines (⊥) intersect at 90° angles. In the figure,   AB ⊥   AE , and   EG ⊥   GH . Skew lines are not coplanar. Skew lines are not parallel and do not intersect. In the figure,   AB and   EG are skew. Parallel planes are planes that do not intersect. In the figure, plane ABE ǁ plane CDG. E X A M P L E 1 Identifying Types of Lines and Planes Identify each of the following. Arrows are used to show that   AB ǁ   EF and   EG ǁ   FH . Segments or rays are parallel, perpendicular, or skew if the lines that contain them are parallel, perpendicular, or skew. A a pair of parallel segments ̶̶ KN ǁ ̶̶ PS B a pair of skew segments ̶̶ RS are skew. ̶̶̶ LM and C a pair of perpendicular segments ̶̶̶ MR ⊥ ̶̶ RS D a pair of parallel planes plane KPS ǁ plane LQR Identify each of the following. 1a. a pair of parallel segments 1b. a pair of skew segments 1c. a pair of perpendicular segments 1d. a pair of parallel planes 146 146 Chapter 3 Parallel and Perpendicular Lines �� Angle Pairs Formed by a Transversal TERM EXAMPLE A transversal is a line that intersects two coplanar lines at two different points. The transversal t and the other two lines r and s form eight angles. Corresponding angles lie on the same side of the transversal t, on the same sides of lines r and s. Alternate interior angles are nonadjacent angles that lie on opposite sides of the transversal t, between lines r and s. Alternate exterior angles lie on opposite sides of the transversal t, outside lines r and s. Same-side interior angles or consecutive interior angles lie on the same side of the transversal t, between lines r and s. ∠1 and ∠5 ∠3 and ∠6 ∠1 and ∠8 ∠3 and ∠5 E X A M P L E 2 Classifying Pairs of Angles Give an example of each angle pair. A corresponding angles B alternate interior angles ∠4 and ∠8 ∠4 and ∠6 C alternate exterior angles D same-side interior angles ∠2 and ∠8 ∠4 and ∠5 Give an example of each angle pair. 2a. corresponding angles 2b. alternate interior angles 2c. alternate exterior angles 2d. same-side interior angles E X A M P L E 3 Identifying Angle Pairs and Transversals Identify the transversal and classify each angle pair. To determine which line is the transversal for a given angle pair, locate the line that connects the vertices. A ∠1 and ∠5 transversal: n; alternate interior angles B ∠3 and ∠6 transversal: m; corresponding angles C ∠1 and ∠4 transversal: ℓ; alternate exterior angles 3. Identify the transversal and classify the angle pair ∠2 and ∠5 in the diagram above. 3- 1 Lines and Angles 147 147 �� THINK AND DISCUSS 1. Compare perpendicular and intersecting lines. 2. Describe the positions of two alternate exterior angles formed by lines m and n with transversal p. 3. GET ORGANIZED Copy the diagram and graphic organizer. In each box, list all the angle pairs of each type in the diagram. 3-1 Exercises Exercises KEYWORD: MG7 3-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary ? are located on opposite sides of a transversal, between the two ̶̶̶̶ lines that intersect the transversal. (corresponding angles, alternate interior angles, alternate exterior angles, or same-side interior angles Identify each of the following. p. 146 2. one pair of perpendicular segments 3. one pair of skew segments 4. one pair of parallel segments 5. one pair of parallel planes Give an example of each angle pair. p. 147 6. alternate interior angles 7. alternate exterior angles 8. corresponding angles 9. same-side interior angles Identify the transversal and classify each angle pair. p. 147 10. ∠1 and ∠2 11. ∠2 and ∠3 12. ∠2 and ∠4 13. ∠4 and ∠5 148 148 Chapter 3 Parallel and Perpendicular Lines �� Independent Practice For See Exercises Example 14–17 18–21 22–25 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 PRACTICE AND PROBLEM SOLVING Identify each of the following. 14. one pair of parallel segments 15. one pair of skew segments 16. one pair of perpendicular segments 17. 17. one pair of parallel planes Give an example of each angle pair. 18. same-side interior angles 19. alternate exterior angles 20. corresponding angles 21. alternate interior angles Identify the transversal and classify each angle pair. 22. ∠2 and ∠3 23. ∠4 and ∠5 24. ∠2 and ∠4 25. ∠1 and ∠2 26. Sports A football player ru
ns across the 30-yard line at an angle. He continues in a straight line and crosses the goal line at the same angle. Describe two parallel lines and a transversal in the diagram. Name the type of angle pair shown in each letter. 27. F 28. Z 29. C Entertainment Entertainment In an Ames room, two people of the same height that are standing in different parts of the room appear to be different sizes. Entertainment Entertainment Use the following information for Exercises 30–32. In an Ames room, the floor is tilted and the back wall is closer to the front wall on one side. 30. 30. Name a pair of parallel segments in the diagram. 31. Name a pair of skew segments in the diagram. 32. Name a pair of perpendicular segments in the diagram. 3- 1 Lines and Angles 149 149 �� 33. This problem will prepare you for the Multi-Step TAKS Prep on p 180. Buildings that are tilted like the one shown are sometimes called mystery spots. a. Name a plane parallel to plane KLP, a plane parallel to plane KNP, and a plane parallel to KLM. b. In the diagram, ̶̶ QR is a transversal to ̶̶ PQ and ̶̶ RS . What type of angle pair is ∠PQR and ∠QRS? 34. Critical Thinking Line ℓ is contained in plane P and line m is contained in plane Q. If P and Q are parallel, what are the possible classifications of ℓ and m? Include diagrams to support your answer. Use the diagram for Exercises 35–40. 35. Name a pair of alternate interior angles with transversal n. 36. Name a pair of same-side interior angles with transversal ℓ. 37. Name a pair of corresponding angles with transversal m. 38. Identify the transversal and classify the angle pair for ∠3 and ∠7. 39. Identify the transversal and classify the angle pair for ∠5 and ∠8. 40. Identify the transversal and classify the angle pair for ∠1 and ∠6. 41. Aviation Describe the type of lines formed by two planes when flight 1449 is flying from San Francisco to Atlanta at 32,000 feet and flight 2390 is flying from Dallas to Chicago at 28,000 feet. 42. Multi-Step Draw line p, then draw two lines m and n that are both perpendicular to p. Make a conjecture about the relationship between lines m and n. 43. Write About It Discuss a real-world example of skew lines. Include a sketch. 44. Which pair of angles in the diagram are alternate interior angles? ∠1 and ∠5 ∠2 and ∠6 ∠7 and ∠5 ∠2 and ∠3 45. How many pairs of corresponding angles are in the diagram? 2 4 8 16 150 150 Chapter 3 Parallel and Perpendicular Lines ��San FranciscoAtlantaChicagoDallasHolt, Rinehart & WinstonHigh School Mathge07sec03101007a Locator map showing cities3rd proof�� 46. Which type of lines are NOT represented in the diagram? Parallel lines Skew lines Intersecting lines Perpendicular lines 47. For two lines and a transversal, ∠1 and ∠8 are alternate exterior angles, and ∠1 and ∠5 are corresponding angles. Classify the angle pair ∠5 and ∠8. Vertical angles Alternate interior angles Adjacent angles Same-side interior angles 48. Which angles in the diagram are NOT corresponding angles? ∠1 and ∠5 ∠2 and ∠6 ∠4 and ∠8 ∠2 and ∠7 � � � � � � � � CHALLENGE AND EXTEND Name all the angle pairs of each type in the diagram. Identify the transversal for each pair. 49. corresponding 50. alternate interior 51. alternate exterior 52. same-side interior 53. Multi-Step Draw two lines and a transversal such that ∠1 and ∠3 are corresponding angles, ∠1 and ∠2 are alternate interior angles, and ∠3 and ∠4 are alternate exterior angles. What type of angle pair is ∠2 and ∠4? 54. If the figure shown is folded to form a cube, which faces of the cube will be parallel? � � � � � � � � � � � �� SPIRAL REVIEW Evaluate each function for x = -1, 0, 1, 2, and 3. (Previous course) 55. y = 4 x 2 - 7 56. y = -2 x 2 + 5 57. y = (x + 3) (x - 3) Find the circumference and area of each circle. Use the π key on your calculator and round to the nearest tenth. (Lesson 1-5) 58. �� 59. �� Write a justification for each statement, given that ∠1 and ∠3 are right angles. (Lesson 2-6) 60. ∠1 ≅ ∠3 61. m∠1 + m∠2 = 180° 62. ∠2 ≅ ∠4 � � � � 3- 1 Lines and Angles 151 151 Systems of Equations Algebra Sometimes angle measures are given as algebraic expressions. When you know the relationship between two angles, you can write and solve a system of equations to find angle measures. See Skills Bank page S67 Solving Systems of Equations by Using Elimination Step 1 Write the system so that like terms are under one another. Step 2 Eliminate one of the variables. Step 3 Substitute that value into one of the original equations and solve. Step 4 Write the answers as an ordered pair, (x, y). Step 5 Check your solution. Example 1 Solve for x and y. Since the lines are perpendicular, all of the angles are right angles. To write two equations, you can set each expression equal to 90°. (3x + 2y)° = 90°, (6x - 2y)° = 90° Step 1 Step 2 3x + 2y = 90 6x - 2y = 90 ̶̶̶̶̶̶̶̶ 9x + 0 = 180 Write the system so that like terms are under one another. Add like terms on each side of the equations. The y-term has been eliminated. x = 20 Divide both sides by 9 to solve for x. Step 3 3x + 2y = 90 Write one of the original equations. 3 (20) + 2y = 90 Substitute 20 for x. 60 + 2y = 90 Simplify. 2y = 30 y = 15 Subtract 60 from both sides. Divide by 2 on both sides. Step 4 (20, 15) Write the solution as an ordered pair. Step 5 Check the solution by substituting 20 for x and 15 for y in the original equations. 3x + 2y = 90 6x - 2y = 90 3(20) + 2(15) 90 6(20) - 2(15) 90 60 + 30 90 120 - 30 90 90 90 ✓ 90 90 ✓ In some cases, before you can do Step 1 you will need to multiply one or both of the equations by a number so that you can eliminate a variable. 152 152 Chapter 3 Parallel and Perpendicular Lines �� Example 2 Solve for x and y. (2x + 4y)° = 72° (5x + 2y)° = 108° Vertical Angles Theorem Linear Pair Theorem The equations cannot be added or subtracted to eliminate a variable. Multiply the second equation by -2 to get opposite y-coefficients. 5x + 2y = 108 → -2 (5x + 2y) = -2 (108) → -10x - 4y = -216 Step 1 Step 2 2x + 4y = 72 -10x - 4y = -216 ̶̶̶̶̶̶̶̶̶̶̶̶ = -144 -8x Write the system so that like terms are under one another. Add like terms on both sides of the equations. The y-term has been eliminated. x = 18 Divide both sides by -8 to solve for x. Step 3 2x + 4y = 72 Write one of the original equations. 2(18) + 4y = 72 Substitute 18 for x. 36 + 4y = 72 Simplify. 4y = 36 y = 9 Subtract 36 from both sides. Divide by 4 on both sides. Step 4 (18, 9) Write the solution as an ordered pair. Step 5 Check the solution by substituting 18 for x and 9 for y in the original equations. 2x + 4y = 72 3(18) + 4(9) 36 + 36 72 72 5x + 2y = 108 5 (18) + 2 (9) 108 90 + 18 108 72 72 ✓ 108 108 ✓ Try This TAKS Grades 9–11 Obj. 2, 4, 6 Solve for x and y. 1. 3. 2. 4. On Track for TAKS 153 153 �� 3-2 Use with Lesson 3-2 Activity Explore Parallel Lines and Transversals Geometry software can help you explore angles that are formed when a transversal intersects a pair of parallel lines. TEKS G.9.A Congruence and the geometry of size: formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations .... KEYWORD: MG7 Lab3 1 Construct a line and label two points on the line A and B. 2 Create point C not on AB . Construct a line parallel to   AB through point C. Create another point on this line and label it D. 3 Create two points outside the two parallel lines and label them E and F. Construct transversal   EF . Label the points of intersection G and H. 4 Measure the angles formed by the parallel lines and the transversal. Write the angle measures in a chart like the one below. Drag point E or F and chart with the new angle measures. What relationships do you notice about the angle measures? What conjectures can you make? ∠AGE ∠BGE ∠AGH ∠BGH ∠CHG ∠DHG ∠CHF ∠DHF Angle Measure Measure Try This 1. Identify the pairs of corresponding angles in the diagram. What conjecture can you make about their angle measures? Drag a point in the figure to confirm your conjecture. 2. Repeat steps in the previous problem for alternate interior angles, alternate exterior angles, and same-side interior angles. 3. Try dragging point C to change the distance between the parallel lines. What happens to the angle measures in the figure? Why do you think this happens? 154 154 Chapter 3 Parallel and Perpendicular Lines 3-2 Angles Formed by Parallel Lines and Transversals TEKS G.3.C Geometric structure: use logical reasoning to prove statements are true .... Also G.3.E, G.9.A Objective Prove and use theorems about the angles formed by parallel lines and a transversal. Who uses this? Piano makers use parallel strings for the higher notes. The longer strings used to produce the lower notes can be viewed as transversals. (See Example 3.) When parallel lines are cut by a transversal, the angle pairs formed are either congruent or supplementary. Postulate 3-2-1 Corresponding Angles Postulate THEOREM HYPOTHESIS CONCLUSION If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. ∠1 ≅ ∠3 ∠2 ≅ ∠4 ∠5 ≅ ∠7 ∠6 ≅ ∠8 E X A M P L E 1 Using the Corresponding Angles Postulate Find each angle measure. A m∠ABC x = 80 m∠ABC = 80° B m∠DEF Corr.  Post. (2x - 45) ° = (x + 30) ° Corr.  Post. x - 45 = 30 x = 75 m∠DEF = x + 30 Subtract x from both sides. Add 45 to both sides. = 75 + 30 = 105° Substitute 75 for x. 1. Find m∠QRS. Remember that postulates are statements that are accepted without proof. Since the Corresponding Angles Postulate is given as a postulate, it can be used to prove the next three theorems. 3- 2 Angles Formed by Parallel Lines and Transversals 155 155 �� Theorems Parallel Lines and Angle Pairs TH
EOREM HYPOTHESIS CONCLUSION If a transversal is perpendicular to two parallel lines, all eight angles are congruent. 3-2-2 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3-2-3 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent. 3-2-4 Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. ∠1 ≅ ∠3 ∠2 ≅ ∠4 ∠5 ≅ ∠7 ∠6 ≅ ∠8 m∠1 + m∠4 = 180° m∠2 + m∠3 = 180° You will prove Theorems 3-2-3 and 3-2-4 in Exercises 25 and 26. PROOF PROOF Alternate Interior Angles Theorem Given: ℓ ǁ m Prove: ∠ 2 ≅ ∠3 Proof: E X A M P L E 2 Finding Angle Measures Find each angle measure. A m∠EDF x = 125 m∠EDF = 125° Alt. Ext.  Thm. B m∠TUS 13x° + 23x° = 180° Same-Side Int.  Thm. 36x = 180 x = 5 m∠TUS = 23 (5) = 115° Combine like terms. Divide both sides by 36. Substitute 5 for x. 2. Find m∠ABD. 156 156 Chapter 3 Parallel and Perpendicular Lines �� Parallel Lines and Transversals When I solve problems with parallel lines and transversals, I remind myself that every pair of angles is either congruent or supplementary. If r ǁ s, all the acute angles are congruent and all the obtuse angles are congruent. The acute angles are supplementary to the obtuse angles. Nancy Martin East Branch High School E X A M P L E 3 Music Application The treble strings of a grand piano are parallel. Viewed from above, the bass strings form transversals to the treble strings. Find x and y in the diagram. By the Alternate Exterior Angles Theorem, (25x + 5y) ° = 125°. By the Corresponding Angles Postulate, (25x + 4y) ° = 120°. 25x + 5y = 125 - (25x + 4y = 120) ̶̶̶̶̶̶̶̶̶̶̶̶ y = 5 25x + 5 (5) = 125 x = 4, y = 5 Subtract the second equation from the first equation. Substitute 5 for y in 25x + 5y = 125. Simplify and solve for x. 3. Find the measures of the acute angles in the diagram. THINK AND DISCUSS 1. Explain why a transversal that is perpendicular to two parallel lines forms eight congruent angles. 2. GET ORGANIZED Copy the diagram and graphic organizer. Complete the graphic organizer by explaining why each of the three theorems is true. 3- 2 Angles Formed by Parallel Lines and Transversals 157 157 �� 3-2 Exercises Exercises GUIDED PRACTICE Find each angle measure. p. 155 1. m∠JKL 2. m∠BEF KEYWORD: MG7 3-2 KEYWORD: MG7 Parent . m∠1 4. m∠CBY p. 156 . Safety The railing of p. 157 a wheelchair ramp is parallel to the ramp. Find x and y in the diagram. Independent Practice For See Exercises Example 6–7 8–11 12 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 PRACTICE AND PROBLEM SOLVING Find each angle measure. 6. m∠KLM 7. m∠VYX 8. m∠ ABC 9. m∠EFG 10. m∠PQR 11. m∠STU 158 158 Chapter 3 Parallel and Perpendicular Lines �� 12. Parking In the parking lot shown, the lines that mark the width of each space are parallel. m∠1 = (2x - 3y) ° m∠2 = (x + 3y) ° Find x and y. Find each angle measure. Justify each answer with a postulate or theorem. 13. m∠1 14. m∠2 15. m∠3 16. m∠4 17. m∠5 18. m∠6 19. m∠7 Algebra State the theorem or postulate that is related to the measures of the angles in each pair. Then find the angle measures. 20. m∠1 = (7x + 15) °, m∠2 = (10x - 9) ° 21. m∠3 = (23x + 11) °, m∠4 = (14x + 21) ° 22. m∠4 = (37x - 15) °, m∠5 = (44x - 29) ° 23. m∠1 = (6x + 24) °, m∠4 = (17x - 9) ° 24. Architecture The Luxor Hotel in Las Vegas, Nevada, is a 30-story pyramid. The hotel uses an elevator called an inclinator to take people up the side of the pyramid. The inclinator travels at a 39° angle. Which theorem or postulate best illustrates the angles formed by the path of the inclinator and each parallel floor? (Hint: Draw a picture.) 25. Complete the two-column proof of the Alternate Exterior Angles Theorem. Given: ℓ ǁ m Prove: ∠1 ≅ ∠2 Proof: Statements Reasons Architecture Architecture The Luxor hotel is 600 feet wide, 600 feet long, and 350 feet high. The atrium in the hotel measures 29 million cubic feet. 1. ℓ ǁ m 2. a. ? ̶̶̶̶̶ 3. ∠3 ≅ ∠2 4. c. ? ̶̶̶̶̶ 1. Given 2. Vert.  Thm. 3. b. 4. d. ? ̶̶̶̶̶ ? ̶̶̶̶̶ 26. Write a paragraph proof of the Same-Side Interior Angles Theorem. Given: r ǁ s Prove: m∠1 + m∠2 = 180° Draw the given situation or tell why it is impossible. 27. Two parallel lines are intersected by a transversal so that the corresponding angles are supplementary. 28. Two parallel lines are intersected by a transversal so that the same-side interior angles are complementary. 3- 2 Angles Formed by Parallel Lines and Transversals 159 159 �� 29. This problem will prepare you for the Multi-Step TAKS Prep on page 180. In the diagram, which represents the side view of a mystery spot, m∠SRT = 25°.   RT is a transversal to   PS and   QR . a. What type of angle pair is ∠QRT and ∠STR? b. Find m∠STR. Use a theorem or postulate to justify your answer. 30. Land Development A piece of property lies between two parallel streets as shown. m∠1 = (2x + 6) °, and m∠2 = (3x + 9) °. What is the relationship between the angles? What are their measures? 31. /////ERROR ANALYSIS///// In the figure, m∠ABC = (15x + 5) °, and m∠BCD = (10x + 25) °. Which value of m∠BCD is incorrect? Explain. 32. Critical Thinking In the diagram, ℓ ǁ m. Explain why x _ y = 1. 33. Write About It Suppose that lines ℓ and m are intersected by transversal p. One of the angles formed by ℓ and p is congruent to every angle formed by m and p. Draw a diagram showing lines ℓ, m, and p, mark any congruent angles that are formed, and explain what you know is true. 34. m∠RST = (x + 50) °, and m∠STU = (3x + 20) °. Find m∠RVT. 15° 27.5° 65° 77.5° 160 160 Chapter 3 Parallel and Perpendicular Lines �� 35. For two parallel lines and a transversal, m∠1 = 83°. For which pair of angle measures is the sum the least? ∠1 and a corresponding angle ∠1 and a same-side interior angle ∠1 and its supplement ∠1 and its complement 36. Short Response Given a ǁ b with transversal t, explain why ∠1 and ∠3 are supplementary. � � � � � � CHALLENGE AND EXTEND Multi-Step Find m∠1 in each diagram. (Hint: Draw a line parallel to the given parallel lines.) 37. �� 38. � �� 39. Find x and y in the diagram. Justify your answer. � 40. Two lines are parallel. The measures of two corresponding angles are a° and 2b°, and the measures of two same-side interior angles are a° and b°. Find the value of a. � �� SPIRAL REVIEW If the first quantity increases, tell whether the second quantity is likely to increase, decrease, or stay the same. (Previous course) 41. time in years and average cost of a new car 42. age of a student and length of time needed to read 500 words Use the Law of Syllogism to draw a conclusion from the given information. (Lesson 2-3) 43. If two angles form a linear pair, then they are supplementary. If two angles are supplementary, then their measures add to 180°. ∠1 and ∠2 form a linear pair. 44. If a figure is a square, then it is a rectangle. If a figure is a rectangle, then its sides are perpendicular. Figure ABCD is a square. Give an example of each angle pair. (Lesson 3-1) 45. alternate interior angles 46. alternate exterior angles 47. same-side interior angles � � � � � � � � 3- 2 Angles Formed by Parallel Lines and Transversals 161 161 3-3 Proving Lines Parallel TEKS G.3.C Geometric structure: use logical reasoning to prove statements are true .... Also G.1.A, G.3.E, G.9.A Objective Use the angles formed by a transversal to prove two lines are parallel. Who uses this? Rowers have to keep the oars on each side parallel in order to travel in a straight line. (See Example 4.) Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem. Postulate 3-3-1 Converse of the Corresponding Angles Postulate THEOREM HYPOTHESIS CONCLUSION If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. ∠1 ≅ ∠ Using the Converse of the Corresponding Angles Postulate Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. A ∠1 ≅ ∠5 ∠1 ≅ ∠5 ℓ ǁ m ∠1 and ∠5 are corresponding angles. Conv. of Corr. ∠s Post. B m∠4 = (2x + 10) °, m∠8 = (3x - 55) °, x = 65 m∠4 = 2 (65) + 10 = 140 m∠8 = 3 (65) - 55 = 140 m∠4 = m∠8 ∠4 ≅ ∠8 ℓ ǁ m Substitute 65 for x. Substitute 65 for x. Trans. Prop. of Equality Def. of ≅  Conv. of Corr.  Post. Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. 1a. m∠1 = m∠3 1b. m∠7 = (4x + 25) °, m∠5 = (5x + 12) °, x = 13 162 162 Chapter 3 Parallel and Perpendicular Lines �� Postulate 3-3-2 Parallel Postulate Through a point P not on line ℓ, there is exactly one line parallel to ℓ. The Converse of the Corresponding Ang
les Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ. Construction Parallel Lines  Draw a line ℓ and a point P that is not on ℓ.  Draw a line m through P that intersects ℓ. Label the angle 1.  Construct an angle congruent to ∠1 at P. By the converse of the Corresponding Angles Postulate, ℓ ǁ n. Theorems Proving Lines Parallel THEOREM HYPOTHESIS CONCLUSION 3-3-3 Converse of the Alternate ∠1 ≅ ∠2 Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. 3-3-4 Converse of the Alternate ∠3 ≅ ∠4 Exterior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel. 3-3-5 Converse of the Same-Side m∠5 + m∠6 = 180° Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel You will prove Theorems 3-3-3 and 3-3-5 in Exercises 38–39. 3- 3 Proving Lines Parallel 163 163 �� PROOF PROOF Converse of the Alternate Exterior Angles Theorem Given: ∠1 ≅ ∠2 Prove: ℓ ǁ m Proof: It is given that ∠1 ≅ ∠2. Vertical angles are congruent, so ∠1 ≅ ∠3. By the Transitive Property of Congruence, ∠2 ≅ ∠3. So ℓ ǁ m by the Converse of the Corresponding Angles Postulate. E X A M P L E 2 Determining Whether Lines are Parallel Use the given information and the theorems you have learned to show that r ǁ s. A ∠2 ≅ ∠6 ∠2 ≅ ∠6 r ǁ s ∠2 and ∠6 are alternate interior angles. Conv. of Alt. Int.  Thm. B m∠6 = (6x + 18) °, m∠7 = (9x + 12) °, x = 10 m∠6 = 6x + 18 = 6 (10) + 18 = 78° Substitute 10 for x. m∠7 = 9x + 12 = 9 (10) + 12 = 102° m∠6 + m∠7 = 78° + 102° Substitute 10 for x. = 180° ∠6 and ∠7 are same-side interior angles. r ǁ s Conv. of Same-Side Int.  Thm. Refer to the diagram above. Use the given information and the theorems you have learned to show that r ǁ s. 2a. m∠4 = m∠8 2b. m∠3 = 2x°, m∠7 = (x + 50) °, x = 50 E X A M P L E 3 Proving Lines Parallel Given: ℓ ǁ m, ∠1 ≅ ∠3 Prove: r ǁ p Proof: Statements Reasons 1. ℓ ǁ m 2. ∠1 ≅ ∠2 3. ∠1 ≅ ∠3 4. ∠2 ≅ ∠3 5. r ǁ p 1. Given 2. Corr.  Post. 3. Given 4. Trans. Prop. of ≅ 5. Conv. of Alt. Ext.  Thm. 3. Given: ∠1 ≅ ∠4, ∠3 and ∠4 are supplementary. Prove: ℓ ǁ m 164 164 Chapter 3 Parallel and Perpendicular Lines �� E X A M P L E 4 Sports Application During a race, all members of a rowing team should keep the oars parallel on each side. If m∠1 = (3x + 13) °, m∠2 = (5x - 5) °, and x = 9, show that the oars are parallel. A line through the center of the boat forms a transversal to the two oars on each side of the boat. ∠1 and ∠2 are corresponding angles. If ∠1 ≅ ∠2, then the oars are parallel. Substitute 9 for x in each expression: m∠1 = 3x + 13 = 3 (9) + 13 = 40° Substitute 9 for x in each expression. m∠2 = 5x - 5 = 5 (9) - 5 = 40° m∠1 = m∠2, so ∠1 ≅ ∠2. The corresponding angles are congruent, so the oars are parallel by the Converse of the Corresponding Angles Postulate. 4. What if…? Suppose the corresponding angles on the opposite side of the boat measure (4y - 2) ° and (3y + 6) °, where y = 8. Show that the oars are parallel. THINK AND DISCUSS 1. Explain three ways of proving that two lines are parallel. 2. If you know m∠1, how could you use the measures of ∠5, ∠6, ∠7, or ∠8 to prove m ǁ n? 3. GET ORGANIZED Copy and complete the graphic organizer. Use it to compare the Corresponding Angles Postulate with the Converse of the Corresponding Angles Postulate. 3- 3 Proving Lines Parallel 165 165 21�� KEYWORD: MG7 3-3 KEYWORD: MG7 Parent 3-3 Exercises Exercises GUIDED PRACTICE . 162 Use the Converse of the Corresponding Angles Postulate and the given information to show that p ǁ q. 1. ∠4 ≅ ∠5 2. m∠1 = (4x + 16) °, m∠8 = (5x - 12) °, x = 28 3. m∠4 = (6x - 19) °, m∠5 = (3x + 14) °, x = 11 Use the theorems and given information to show that r ǁ s. p. 164 4. ∠1 ≅ ∠5 5. m∠3 + m∠4= 180° 6. ∠3 ≅ ∠7 7. m∠4 = (13x - 4) °, m∠8 = (9x + 16) °, x = 5 8. m∠8 = (17x + 37) °, m∠7 = (9x - 13) °, x = 6 9. m∠2 = (25x + 7) °, m∠6 = (24x + 12) °, 10. Complete the following two-column proof. p. 164 Given: ∠1 ≅ ∠2, ∠3 ≅ ∠1 Prove: XY ǁ WV Proof: Statements Reasons 1. ∠1 ≅ ∠2, ∠3 ≅ ∠1 1. Given 2. ∠2 ≅ ∠3 3. b. ? ̶̶̶̶̶ 2. a. 3. c. ? ̶̶̶̶̶ ? ̶̶̶̶̶ 11. Architecture In the fire escape, p. 165 m∠1 = (17x + 9) °, m∠2 = (14x + 18) °, and x = 3. Show that the two landings are parallel. PRACTICE AND PROBLEM SOLVING Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ ǁ m. 12. ∠3 ≅ 7 13. m∠4 = 54°, m∠8 = (7x + 5) °, x = 7 14. m∠2 = (8x + 4) °, m∠6 = (11x - 41) °, x = 15 15. m∠1 = (3x + 19) °, m∠5 = (4x + 7) °, x = 12 166 166 Chapter 3 Parallel and Perpendicular Lines �� Independent Practice Use the theorems and given information to show that n ǁ p. For See Exercises Example 12–15 16–21 22 23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 16. ∠3 ≅ ∠6 17. ∠2 ≅ ∠7 18. m∠4 + m∠6 = 180° 19. m∠1 = (8x - 7) °, m∠8 = (6x + 21) °, x = 14 20. m∠4 = (4x + 3) °, m∠5 = (5x -22) °, x = 25 21. m∠3 = (2x + 15) °, m∠5 = (3x + 15) °, x = 30 22. Complete the following two-column proof. Given: Prove: ̶̶ AB ǁ ̶̶ BC ǁ ̶̶ CD , ∠1 ≅ ∠2, ∠3 ≅ ∠4 ̶̶ DE Proof: Statements Reasons ̶̶ AB ǁ ̶̶ CD 1. 2. ∠1 ≅ ∠3 3. ∠1 ≅ ∠2, ∠3 ≅ ∠4 4. ∠2 ≅ ∠4 5. d. ? ̶̶̶̶̶̶ 1. Given 2. a. 3. b. 4. c. 5. e. ? ̶̶̶̶̶̶ ? ̶̶̶̶̶̶ ? ̶̶̶̶̶̶ ? ̶̶̶̶̶̶ 23. Art Edmund Dulac used perspective when drawing the floor titles in this illustration for The Wind’s Tale by Hans Christian Andersen. Show that DJ ǁ EK if m∠1 = (3x + 2) °, m∠2 = (5x - 10) °, and x = 6. � � � � � � � � � � � � Name the postulate or theorem that proves that ℓ ǁ m. 24. ∠8 ≅ ∠6 25. ∠8 ≅ ∠4 26. ∠2 ≅ ∠6 27. ∠7 ≅ ∠5 28. ∠3 ≅ ∠7 29. m∠2 + m∠3 = 180° For the given information, tell which pair of lines must be parallel. Name the postulate or theorem that supports your answer. 30. m∠2 = m∠10 31. m∠8 + m∠9 = 180° 32. ∠1 ≅ ∠7 33. m∠10 = m∠6 34. ∠11 ≅ ∠5 35. m∠2 + m∠5 = 180° 36. Multi-Step Two lines are intersected by a transversal so that ∠1 and ∠2 are corresponding angles, ∠1 and ∠3 are alternate exterior angles, and ∠3 and ∠4 are corresponding angles. If ∠2 ≅ ∠4, what theorem or postulate can be used to prove the lines parallel? 3- 3 Proving Lines Parallel 167 167 �� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 180. In the diagram, which represents the side view of a mystery spot, m∠SRT = 25°, and m∠SUR = 65°. a. Name a same-side interior angle of ∠SUR ̶̶ for lines   SU and   RT with transversal RU . What is its measure? Explain your reasoning. b. Prove that   SU and   RT are parallel. 38. Complete the flowchart proof of the Converse of the Alternate Interior Angles Theorem. Given: ∠2 ≅ ∠3 Prove: ℓ ǁ m Proof: 39. Use the diagram to write a paragraph proof of the Converse of the Same-Side Interior Angles Theorem. Given: ∠1 and ∠2 are supplementary. Prove: ℓ ǁ m 40. Carpentry A plumb bob is a weight hung at the end of a string, called a plumb line. The weight pulls the string down so that the plumb line is perfectly vertical. Suppose that the angle formed by the wall and the roof is 123° and the angle formed by the plumb line and the roof is 123°. How does this show that the wall is perfectly vertical? 41. Critical Thinking Are the Reflexive, Symmetric, and Transitive Properties true for parallel lines? Explain why or why not. Reflexive: ℓ ǁ ℓ Symmetric: If ℓ ǁ m, then m ǁ ℓ. Transitive: If ℓ ǁ m and m ǁ n, then ℓ ǁ n. 42. Write About It Does the information given in the diagram allow you to conclude that a ǁ b? Explain. 43. Which postulate or theorem can be used to prove ℓ ǁ m? Converse of the Corresponding Angles Postulate Converse of the Alternate Interior Angles Theorem Converse of the Alternate Exterior Angles Theorem Converse of the Same-Side Interior Angles Theorem 168 168 Chapter 3 Parallel and Perpendicular Lines ��PlumblineWallRoof123˚123˚�� 44. Two coplanar lines are cut by a transversal. Which condition does NOT guarantee that the two lines are parallel? A pair of alternate interior angles are congruent. A pair of same-side interior angles are supplementary. A pair of corresponding angles are congruent. A pair of alternate exterior angles are complementary. 45. Gridded Response Find the value of x so that ℓ ǁ m. �� CHALLENGE AND EXTEND Determine which lines, if any, can be proven parallel using the given information. Justify your answers. 46. ∠1 ≅ ∠15 48. ∠3 ≅ ∠7 50. ∠6 ≅ ∠8 47. ∠8 ≅ ∠14 49. ∠8 ≅ ∠10 51. ∠13 ≅ ∠11 52. m∠12 + m∠15 = 180° 53. m∠5 + m∠8 = 180° 54. Write a paragraph ̶̶ BD . proof that ̶̶ AE ǁ � � � �� Use the diagram for Exercises 55 and 56. 55. Given: m∠2 + m∠3 = 180° Prove: ℓ ǁ m 56. Given: m∠2 + m∠5 = 180° Prove: ℓ ǁ n � � � � � � � � � � � � � � � � �� SPIRAL REVIEW Solve each equation for the indicated variable. (Previous course) 57. a - b = -c, for a 58. y = 1 _ 2 x - 10, for x 59. 4y + 6x = 12, for y Write the converse, inverse, and contrapositive of each conditional statement. Find the truth value of each. (Lesson 2-2) 60. If an animal is a bat, then it has wings. 61. If a polygon is a triangle, then it has exactly three sides. 62. If the digit in the ones place of a whole number is 2, then the number is even. Identify each of the following. (Lesson 3-1) � 63. one pair of parallel segments 64. one pair of skew segments 65. one pair of perpendicular segments � � � � 3- 3 Proving Lines Parallel 169 1
69 3-3 Construct Parallel Lines In Lesson 3-3, you learned one method of constructing parallel lines using a compass and straightedge. Another method, called the rhombus method, uses a property of a figure called a rhombus, which you will study in Chapter 6. The rhombus method is shown below. Use with Lesson 3-3 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.9.A Activity 1 1 Draw a line ℓ and a point P not on the line. 2 Choose a point Q on the line. Place your compass point at Q and draw an arc through P that intersects ℓ. Label the intersection R. 3 Using the same compass setting as the 4 Draw PS ǁ ℓ. first arc, draw two more arcs: one from P, the other from R. Label the intersection of the two arcs S. Try This 1. Repeat Activity 1 using a different point not on the line. Are your results the same? 2. Using the lines you constructed in Problem 1, draw transversal PQ . Verify that the lines are parallel by using a protractor to measure alternate interior angles. 3. What postulate ensures that this construction is always possible? 4. A rhombus is a quadrilateral with four congruent sides. Explain why this method is called the rhombus method. 170 170 Chapter 3 Parallel and Perpendicular Lines �� Activity 2 1 Draw a line ℓ and point P on a piece of 2 Fold the paper through P so that both sides patty paper. of line ℓ match up 3 Crease the paper to form line m. P should 4 Fold the paper again through P so that be on line m. both sides of line m match up. 5 Crease the paper to form line n. Line n is parallel to line ℓ through P. Try This 5. Repeat Activity 2 using a point in a different place not on the line. Are your results the same? 6. Use a protractor to measure corresponding angles. How can you tell that the lines are parallel? 7. Draw a triangle and construct a line parallel to one side through the vertex that is not on that side. 8. Line m is perpendicular to both ℓ and n. Use this statement to complete the following conjecture: If two lines in a plane are perpendicular to the same line, then ? . ̶̶̶̶̶̶̶̶̶̶ 3- 3 Geometry Lab 171 171 3-4 Perpendicular Lines TEKS G.1.A Geometric structure: ... connecting definitions ... logical reasoning, and theorems. Also G.2.A, G.3.C, G.3.E, G.9.A Objective Prove and apply theorems about perpendicular lines. Vocabulary perpendicular bisector distance from a point to a line Why learn this? Rip currents are strong currents that flow away from the shoreline and are perpendicular to it. A swimmer who gets caught in a rip current can get swept far out to sea. (See Example 3.) The perpendicular bisector of a segment is a line perpendicular to a segment at the segment’s midpoint. A construction of a perpendicular bisector is shown below. Construction Perpendicular Bisector of a Segment  ̶̶ AB . Open the compass Draw wider than half of AB and draw an arc centered at A.  Using the same compass setting, draw an arc centered at B that intersects the first arc at C and D.  Draw   CD .   CD is the perpendicular bisector of ̶̶ AB . The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the distance from a point to a line as the length of the perpendicular segment from the point to the line. E X A M P L E 1 Distance From a Point to a Line A Name the shortest segment from P to    AC . The shortest distance from a point to a line is the length of the perpendicular segment, ̶̶ PB is the shortest segment from P to   AC . so B Write and solve an inequality for x. ̶̶ PB is the shortest segment. Substitute x + 3 for PA and 5 for PB. Subtract 3 from both sides of the inequality. PA > PB x + 3 > 5 - 3 - 3 ̶̶̶ ̶̶̶̶ x > 2 1a. Name the shortest segment from A to   BC . 1b. Write and solve an inequality for x. 172 172 Chapter 3 Parallel and Perpendicular Lines �� Theorems THEOREM DIAGRAM EXAMPLE 3-4-1 If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. (2 intersecting lines form lin. pair of ≅  → lines ⊥.) 3-4-2 Perpendicular Transversal Theorem In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. (2 lines ⊥ to same line → 2 lines ǁ.) 3-4- You will prove Theorems 3-4-1 and 3-4-3 in Exercises 37 and 38. PROOF PROOF Perpendicular Transversal Theorem Given: BC ǁ DE , AB ⊥ BC Prove: AB ⊥ DE Proof: It is given that BC ǁ DE , so ∠ABC ≅ ∠BDE by the Corresponding Angles Postulate. It is also given that   AB ⊥   BC , so m∠ABC = 90°. By the definition of congruent angles, m∠ABC = m∠BDE, so m∠BDE = 90° by the Transitive Property of Equality. By the definition of perpendicular lines,   AB ⊥   DE . E X A M P L E 2 Proving Properties of Lines Write a two-column proof. Given:   AD ǁ   BC ,   AD ⊥   AB ,   BC ⊥   DC Prove:   AB ǁ   DC Proof: Statements Reasons 1.   AD ǁ   BC ,   BC ⊥   DC 1. Given 2.   AD ⊥   DC 3.   AD ⊥   AB 4.   AB ǁ   DC 2. ⊥ Transv. Thm. 3. Given 4. 2 lines ⊥ to same line → 2 lines ǁ. 2. Write a two-column proof. Given: ∠EHF ≅ ∠HFG,   FG ⊥   GH Prove:   EH ⊥   GH 3- 4 Perpendicular Lines 173 173 �� E X A M P L E 3 Oceanography Application Oceanography The National Weather Service in Brownsville provides a rip current forecast for South Padre Island and other Texas beaches. Park officials also post flags on the beach to warn of rip current danger. Source: www.ripcurrents. noaa.gov Rip currents may be caused by a sandbar parallel to the shoreline. Waves cause a buildup of water between the sandbar and the shoreline. When this water breaks through the sandbar, it flows out in a direction perpendicular to the sandbar. Why must the rip current be perpendicular to the shoreline? The rip current forms a transversal to the shoreline and the sandbar. The shoreline and the sandbar are parallel, and the rip current is perpendicular to the sandbar. So by the Perpendicular Transversal Theorem, the rip current is perpendicular to the shoreline. 3. A swimmer who gets caught in a rip current should swim in a direction perpendicular to the current. Why should the path of the swimmer be parallel to the shoreline? THINK AND DISCUSS 1. Describe what happens if two intersecting lines form a linear pair of congruent angles. 2. Explain why a transversal that is perpendicular to two parallel lines forms eight congruent angles. 3. GET ORGANIZED Copy and complete the graphic organizer. Use the diagram and the theorems from this lesson to complete the table. 174 174 Chapter 3 Parallel and Perpendicular Lines Rip currentSandbarSandbarShoreline�� 3-4 Exercises Exercises KEYWORD: MG7 3-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary   CD is the perpendicular bisector of ̶̶ AB .   CD intersects What can you say about ̶̶ AB and   CD ? What can you say about ̶̶ AC and ̶̶ AB at C. ̶̶ BC ? . Name the shortest segment from p. 172 point E to   AD . 3. Write and solve an inequality for x. Complete the two-column proof. p. 173 Given: ∠ABC ≅ ∠CBE,   DE ⊥   AF Prove:   CB ǁ   DE Proof: Statements Reasons 1. ∠ABC ≅ ∠CBE 1. Given 2.   CB ⊥   AF 3. b. ? ̶̶̶̶̶̶ 4.   CB ǁ   DE 2. a. ? ̶̶̶̶̶̶ 3. Given 4. c. ? ̶̶̶̶̶ . 174 5. Sports The center line in a tennis court is perpendicular to both service lines. Explain why the service lines must be parallel to each other. Independent Practice For See Exercises Example 6–7 8 9 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S8 Application Practice p. S30 PRACTICE AND PROBLEM SOLVING 6. Name the shortest segment from point W to ̶̶ XZ . 7. Write and solve an inequality for x. 8. Complete the two-column proof below. Given:   AB ⊥   BC , m∠1 + m∠2 = 180° Prove:   BC ⊥   CD Proof: Statements Reasons 1.   AB ⊥   BC 2. m∠1 + m∠2 = 180° 1. Given 2. a. ? ̶̶̶̶̶ 3. ∠1 and ∠2 are supplementary. 3. Def. of supplementary 4. b. ? ̶̶̶̶̶ 5.   BC ⊥   CD 4. Converse of the Same-Side Interior Angles Theorem 5. c. ? ̶̶̶̶̶ 3- 4 Perpendicular Lines 175 175 ��ServicelineCenterlineServicelinege07se_c03l04003ad2ndpass3/18/05NPatel�� 9. Music The frets on a guitar are all perpendicular to one of the strings. Explain why the frets must be parallel to each other. For each diagram, write and solve an inequality for x. 10. 11. Multi-Step Solve to find x and y in each diagram. 12. 14. 13. �� 15. Determine if there is enough information given in the diagram to prove each statement. 16. ∠1 ≅ ∠2 17. ∠1 ≅ ∠3 18. ∠2 ≅ ∠3 19. ∠2 ≅ ∠4 20. ∠3 ≅ ∠4 21. ∠3 ≅ ∠5 22. Critical Thinking Are the Reflexive, Symmetric, and Transitive Properties true for perpendicular lines? Explain why or why not. Reflexive: ℓ ⊥ ℓ Symmetric: If ℓ ⊥ m, then m ⊥ ℓ. Transitive: If ℓ ⊥ m and m ⊥ n, then ℓ ⊥ n. 23. This problem will prepare you for the Multi-Step TAKS Prep on page 180. a mystery spot, In the diagram, which represents the side view of ̶̶ PQ , ̶̶ PS ⊥ ̶̶ PQ ǁ ̶̶ RS . ̶̶ RS , and a. Prove ̶̶ PS ǁ ̶̶ QR . ̶̶ QR ⊥ ̶̶ RS and ̶̶ PS . ̶̶ QR ⊥ ̶̶ PQ ⊥ b. Prove 176 176 Chapter 3 Parallel and Perpendicular Lines �� 24. Geography Felton Avenue, Arlee Avenue, and Viehl Avenue are all parallel. Broadway Street is perpendicular to Felton Avenue. Use the satellite photo and the given information to determine the values of x and y. 25. Estimation Copy the diagram onto a grid with 1 cm by 1 cm squares. Estimate the distance from point P to line ℓ. ��
�� 26. Critical Thinking Draw a figure to show that Theorem 3-4-3 is not true if the lines are not in the same plane. 27. Draw a figure in which perpendicular bisector of ̶̶ AB . ̶̶ AB is a perpendicular bisector of ̶̶ XY but ̶̶ XY is not a 28. Write About It A ladder is formed by rungs that are perpendicular to the sides of the ladder. Explain why the rungs of the ladder are parallel. �� Construction Construct a segment congruent to each given segment and then �� construct its perpendicular bisector. 29. 30. 31. Which inequality is correct for the given diagram? 2x + 5 < 3x x > 1 2x + 5 > 3x x > 5 32. In the diagram, ℓ ⊥ m. Find x and y. x = 5, y = 7 x = 7, y = 5 x = 90, y = 90 x = 10, y = 5 33. If ℓ ⊥ m, which statement is NOT correct? m∠2 = 90° m∠1 + m∠2 = 180° ∠1 ≅ ∠2 ∠1 ⊥ ∠2 3- 4 Perpendicular Lines 177 177 �� 34. In a plane, both lines m and n are perpendicular to both lines p and q. Which conclusion CANNOT be made All angles formed by lines m, n, p, and q are congruent. 35. Extended Response Lines m and n are parallel. Line p intersects line m at A and line n at B, and is perpendicular to line m. a. What is the relationship between line n and line p? Draw a diagram to support your answer. b. What is the distance from point A to line n? What is the distance from point B to line m? Explain. c. How would you define the distance between two parallel lines in a plane? CHALLENGE AND EXTEND 36. Multi-Step Find m∠1 in the diagram. (Hint: Draw a line parallel to the given parallel lines.) 37. Prove Theorem 3-4-1: If two intersecting lines form a linear pair of congruent angles, then the two lines are perpendicular. 38. Prove Theorem 3-4-3: If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. � � � � � � SPIRAL REVIEW 39. A soccer league has 6 teams. During one season, each team plays each of the other teams 2 times. What is the total number of games played in the league during one season? (Previous course) Find the measure of each angle. (Lesson 1-4) 40. the supplement of ∠DJE 41. the complement of ∠FJG 42. the supplement of ∠GJH For the given information, name the postulate or theorem that proves ℓ ǁ m. (Lesson 3-3) 43. ∠2 ≅ ∠7 44. ∠3 ≅ ∠6 45. m∠4 + m∠6 = 180° � � �� 178 178 Chapter 3 Parallel and Perpendicular Lines 3-4 Use with Lesson 3-4 Activity Construct Perpendicular Lines In Lesson 3-4, you learned to construct the perpendicular bisector of a segment. This is the basis of the construction of a line perpendicular to a given line through a given point. The steps in the construction are the same whether the point is on or off the line. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.9.A Copy the given line ℓ and point P. 1 Place the compass point on P and draw an arc that intersects ℓ at two points. Label the points A and B. 2 Construct the perpendicular bisector of ̶̶ AB . Try This Copy each diagram and construct a line perpendicular to line ℓ through point P. Use a protractor to verify that the lines are perpendicular. 1. 2. 3. Follow the steps below to construct two parallel lines. Explain why ℓ ǁ n. Step 1 Given a line ℓ, draw a point P not on ℓ. Step 2 Construct line m perpendicular to ℓ through P. Step 3 Construct line n perpendicular to m through P. 3- 4 Geometry Lab 179 179 �� SECTION 3A Parallel and Perpendicular Lines and Transversals On the Spot Inside a mystery spot building, objects can appear to roll uphill, and people can look as if they are standing at impossible angles. This is because there is no view of the outside, so the room appears to be normal. Suppose that the ground is perfectly level and the floor of the building forms a 25° angle with the ground. The floor and ceiling are parallel, and the walls are perpendicular to the floor. 1. A table is placed in the room. The legs of the table are perpendicular to the floor, and the top is perpendicular to the legs. Draw a diagram and describe the relationship of the tabletop to the floor, walls, and ceiling of the room. 2. Find the angle of the table top relative to the ground. Suppose a ball is placed on the table. Describe what would happen and how it would appear to a person in the room. 3. Two people of the same height are standing on opposite ends of a board that makes a 25° angle with the floor, as shown. Explain how you know that the board is parallel to the ground. What would appear to be happening from the point of view of a person inside the room? 4. In the room, a lamp hangs from the ceiling along a line perpendicular to the ground. Find the angle the line makes with the walls. Describe how it would appear to a person standing in the room. 180 180 Chapter 3 Parallel and Perpendicular Lines �� Quiz for Lessons 3-1 Through 3-4 SECTION 3A 3-1 Lines and Angles Identify each of the following. 1. a pair of perpendicular segments 2. a pair of skew segments 3. a pair of parallel segments 4. a pair of parallel planes Give an example of each angle pair. 5. alternate interior angles 6. alternate exterior angles 7. corresponding angles 8. same-side interior angles 3-2 Angles Formed by Parallel Lines and Transversals Find each angle measure. 9. 10. 11. 3-3 Proving Lines Parallel Use the given information and the theorems and postulates you have learned to show that a ǁ b. 12. m∠8 = (13x + 20) °, m∠6 = (7x + 38) °, x = 3 13. ∠1 ≅ ∠5 14. m∠8 + m∠7 = 180° 15. m∠8 = m∠4 16. The tower shown is supported by guy wires such that m∠1 = (3x + 12) °, m∠2 = (4x - 2) °, and x = 14. Show that the guy wires are parallel. 3-4 Perpendicular Lines 17. Write a two-column proof. Given: ∠1 ≅ ∠2, ℓ ⊥ n Prove: ℓ ⊥ p Ready to Go On? 181 181 �� 3-5 Slopes of Lines TEKS G.7.B Dimensionality and the geometry of location: use slopes ... to investigate geometric relationships, including parallel lines, perpendicular lines, .... Also G.7.A, G.7.C Objectives Find the slope of a line. Use slopes to identify parallel and perpendicular lines. Vocabulary rise run slope Why learn this? You can use the graph of a line to describe your rate of change, or speed, when traveling. (See Example 2.) The slope of a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope. Slope of a Line DEFINITION EXAMPLE The rise is the difference in the y-values of two points on a line. The run is the difference in the x-values of two points on a line. The slope of a line is the ratio of rise to run. If ( x 1 , y 1 ) and ( x 2 , y 2 ) are any two points on line, the slope of the line is Finding the Slope of a Line Use the slope formula to determine the slope of each line. A    AB B    CD A fraction with zero in the denominator is undefined because it is impossible to divide by zero. Substitute (2, 3) for ( x 1 , y 1 ) and (7, 5) for ( x 2 , y 2 ) in the slope formula and then simplify Substitute (4, -3) for ( x 1 , y 1 ) and (4, 5) for ( x 2 , y 2 ) in the slope formula and then simplify. 5 - (-3 The slope is undefined. = 8 _ 0 m = 182 182 Chapter 3 Parallel and Perpendicular Lines �� Use the slope formula to determine the slope of each line. C   EF D  GH Substitute (3, 4) for ( x 1 , y 1 ) and (6, 4) for ( x 2 , y 2 ) in the slope formula and then simplify0 Substitute (6, 2) for ( x 1 , y 1 ) and (2, 6) for ( x 2 , y 2 ) in the slope formula and then simplify4 = -1 1. Use the slope formula to determine the slope of   JK through J (3, 1) and K (2, -1) . Positive Slope Negative Slope Zero Slope Undefined Slope Summary: Slope of a Line One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour. E X A M P L E 2 Transportation Application Transportation A trip from Dallas to Atlanta, Georgia covers 781 miles and crosses three states but is still shorter than a trip across Texas. The distance from Orange, Texas to El Paso via Interstate 10 is 859 miles. Tony is driving from Dallas, Texas, to Atlanta, Georgia. At 3:00 P.M., he is 180 miles from Dallas. At 5:30 P.M., he is 330 miles from Dallas. Graph the line that represents Tony’s distance from Dallas at a given time. Find and interpret the slope of the line. Use the points (3, 180) and (5.5, 330) to graph the line and find the slope. m = 330 - 180 _ 5.5 - 3 = 150 _ 2.5 = 60 The slope is 60, which means he is traveling at an average speed of 60 miles per hour. 2. What if…? Use the graph above to estimate how far Tony will have traveled by 6:30 P.M. if his average speed stays the same. 3- 5 Slopes of Lines 183 183 �� Slopes of Parallel and Perpendicular Lines 3-5-1 Parallel Lines Theorem In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. 3-5-2 Perpendicular Lines Theorem In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. If a line has a slope of a_ b and - b _ a are called opposite reciprocals. The ratios a _ b , then the slope of a perpendicular line is - b_ a. E X A M P L E 3 Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. A B C Four given points do not always determine two lines. Graph t
he lines to make sure the points are not collinear.    AB and    CD for A (2, 1) , B (1, 5) , C (4, 2) , and D (5, -2) slope of   AB = 5 - 1 _ = -4 1 - 2 = 4 _ -1 = -4 _ 1 slope of   CD = -2 - 2 _ 5 - 4 The lines have the same slope, so they are parallel. = -4    ST and    UV for S (-2, 2) , T (5, -1) , U (3, 4) , and V (-1, -4) slope of   ST = -1 - 2 _ 5 - (-2) slope of   UV = -4 - 4 _ -1 - 3 = -8 _ -4 = 2 The slopes are not the same, so the lines are not parallel. The product of the slopes is not -1, so the lines are not perpendicular.    FG and    HJ for F (1, 1) , G (2, 2) , H (2, 1) , and J (1, 2) slope of   FG = 2 - 1 _ = 1 slope of   HJ = 2 - 1 _ = -1 = 1 _ 1 = 1 _ -1 2 - 1 1 - 2 The product of the slopes is 1 (-1) = -1, so the lines are perpendicular. Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 3a.   WX and   YZ for W (3, 1) , X (3, -2) , Y (-2, 3) , and Z (4, 3) 3b.   KL and   MN for K (-4, 4) , L (-2, -3) , M (3, 1) , and N (-5, -1) 3c.   BC and   DE for B (1, 1) , C (3, 5) , D (-2, -6) , and E (3, 4) 184 184 Chapter 3 Parallel and Perpendicular Lines �� THINK AND DISCUSS 1. Explain how to find the slope of a line when given two points. 2. Compare the slopes of horizontal and vertical lines. 3. GET ORGANIZED Copy and complete the graphic organizer. 3-5 Exercises Exercises KEYWORD: MG7 3-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary The slope of a line is the ratio of its ? to its ̶̶̶ ? . (rise or run) ̶̶̶ Use the slope formula to determine the slope of each line. p. 182 2.  MN 3.   CD 4.  AB 5.   ST . 183 6. Biology A migrating bird flying at a constant speed travels 80 miles by 8:00 A.M. and 200 miles by 11:00 A.M. Graph the line that represents the bird’s distance traveled. Find and interpret the slope of the line. 184 Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 7.   HJ and   KM for H (3, 2) , J (4, 1) , K (-2, -4) , and M (-1, -5) 8.   LM and   NP for L (-2, 2) , M (2, 5) , N (0, 2) , and P (3, -2) 9.   QR and   ST for Q (6, 1) , R (-2, 4) , S (5, 3) , and T (-3, -1) 3- 5 Slopes of Lines 185 185 �� Independent Practice Use the slope formula to determine the slope of each line. PRACTICE AND PROBLEM SOLVING For See Exercises Example 10.  AB 11.   CD 10–13 14 15–17 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S9 Application Practice p. S30 12.  EF 13.   GH 14. Aviation A pilot traveling at a constant speed flies 100 miles by 2:30 P.M. and 475 miles by 5:00 P.M. Graph the line that represents the pilot’s distance flown. Find and interpret the slope of the line. Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 15.  AB and   CD for A (2, -1) , B (7, 2) , C (2, -3) , and D (-3, -6) 16.  XY and   ZW for X (-2, 5) , Y (6, -2) , Z (-3, 6) , and W (4, 0) 17.  JK and   JL for J (-4, -2) , K (4, -2) , and L (-4, 6) 18. Geography The Rio Grande has an elevation of about 1150 meters above sea level in El Paso. The length of the river from that point to Brownsville where it enters the sea is about 2400 km. Find and interpret the slope of the river. For F (7, 6) , G (-3, 5) , H (-2, -3) , J (4, -2) , and K (6, 1) , find each slope. 19.   FG 20.   GJ 21.  HK 22.   GK 23. Critical Thinking The slope of AB is greater than 0 and less than 1. Write an inequality for the slope of a line perpendicular to   AB . 24. Write About It Two cars are driving at the same speed. What is true about the lines that represent the distance traveled by each car at a given time? 25. This problem will prepare you for the Multi-Step TAKS Prep on page 200. A traffic engineer calculates the speed of vehicles as they pass a traffic light. While the light is green, a taxi passes at a constant speed. After 2 s the taxi is 132 ft past the light. After 5 s it is 330 ft past the light. a. Find the speed of the taxi in feet per second. b. Use the fact that 22 ft/s = 15 mi/h to find the taxi’s speed in miles per hour. 186 186 Chapter 3 Parallel and Perpendicular Lines �� 26.   AB ⊥   CD for A (1, 3) , B (4, -2) , C (6, 1) , and D (x, y) . Which are possible values of x and y? x = 1, y = -2 x = 3, y = 6 x = 3, y = -4 x = -2, y = -4 27. Classify   MN and   PQ for M (-3, 1) , N (1, 3) , P (8, 4) , and Q (2, 1) . Parallel Perpendicular Vertical Skew 28. In the formula d = rt, d represents distance, and r represents the rate of change, or slope. Which ray on the graph represents a slope of 45 miles per hour? A B C D CHALLENGE AND EXTEND Use the given information to classify   29. a = c JK for J (a, b) and K (c, d) . 30. b = d 31. The vertices of square ABCD are A (0, -2) , B (6, 4) , C (0, 10) , D (-6, 4) . a. Show that the opposite sides are parallel. b. Show that the consecutive sides are perpendicular. c. Show that all sides are congruent. 32.   ST ǁ   VW for S (-3, 5) , T (1, -1) , V (x, -3) , and W (1, y) . Find a set of possible values for x and y. 33.  MN ⊥   PQ for M (2, 1) , N (-3, 0) , P (x, 4) , and Q (3, y) . Find a set of possible values for x and y. SPIRAL REVIEW Find the x- and y-intercepts of the line that contains each pair of points. (Previous course) 34. (-5, 0) and (0, -5) 35. (0, 1) and (2, -7) 36. (1, -3) and (3, 3) Use the given paragraph proof to write a two-column proof. (Lesson 2-7) 37. Given: ∠1 is supplementary to ∠3. Prove: ∠2 ≅ ∠3 Proof: It is given that ∠1 is supplementary to ∠3. ∠1 and ∠2 are a linear pair by the definition of a linear pair. By the Linear Pair Theorem, ∠1 and ∠2 are supplementary. Thus ∠2 ≅ ∠3 by the Congruent Supplements Theorem. Given that m∠2 = 75°, tell whether each statement is true or false. Justify your answer with a postulate or theorem. (Lesson 3-2) 38. ∠1 ≅ ∠8 39. ∠2 ≅ ∠6 40. ∠3 ≅ ∠5 3- 5 Slopes of Lines 187 187 �� 3-6 Use with Lesson 3-6 Explore Parallel and Perpendicular Lines A graphing calculator can help you explore graphs of parallel and perpendicular lines. To graph a line on a calculator, you can enter the equation of the line in slope-intercept form. The slope-intercept form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. For example, the line y = 2x + 3 has a slope of 2 and crosses the y-axis at (0, 3). TEKS G.7.B Dimensionality and the geometry of location: use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, .... KEYWORD: MG7 Lab3 Activity 1 1 On a graphing calculator, graph the lines y = 3x – 4, y = –3x – 4, and y = 3x + 1. Which lines appear to be parallel? What do you notice about the slopes of the parallel lines? 2 Graph y = 2x. Experiment with other equations to find a line that appears parallel to y = 2x. If necessary, graph y = 2x on graph paper and construct a parallel line. What is the slope of this new line? 3 Graph y = - 1__ 2x + 3. Try to graph a line that appears parallel to y = - 1 __ 2 x + 3. What is the slope of this new line? Try This 1. Create two new equations of lines that you think will be parallel. Graph these to confirm your conjecture. 2. Graph two lines that you think are parallel. Change the window settings on the calculator. Do the lines still appear parallel? Describe your results. 3. Try changing the y-intercepts of one of the parallel lines. Does this change whether the lines appear to be parallel? 188 188 Chapter 3 Parallel and Perpendicular Lines On a graphing calculator, perpendicular lines may not appear to be perpendicular on the screen. This is because the unit distances on the x-axis and y-axis can have different lengths. To make sure that the lines appear perpendicular on the screen, use a square window, which shows the x-axis and y-axis as having equal unit distances. One way to get a square window is to use the Zoom feature. On the Zoom menu, the ZDecimal and ZSquare commands change the window to a square window. The ZStandard command does not produce a square window. Activity 2 1 Graph the lines y = x and y = -x in a square window. Do the lines appear to be perpendicular? 2 Graph y = 3x - 2 in a square window. Experiment with other equations to find a line that appears perpendicular to y = 3x - 2. If necessary, graph y = 3x - 2 on graph paper and construct a perpendicular line. What is the slope of this new line? 3 Graph y = 2 __ 3 x in a square window. Try to graph a line that appears perpendicular to y = 2 __ 3 x. What is the slope of this new line? Try This 4. Create two new equations of lines that you think will be perpendicular. Graph these in a square window to confirm your conjecture. 5. Graph two lines that you think are perpendicular. Change the window settings on the calculator. Do the lines still appear perpendicular? Describe your results. 6. Try changing the y-intercepts of one of the perpendicular lines. Does this change whether the lines appear to be perpendicular? 3- 6 Technology Lab 189 189 3-6 Lines in the Coordinate Plane TEKS G.7.B Dimensionality and the geometry of location: use... equations of lines to investigate... parallel lines, perpendicular lines, .... Also G.3.C, G.3.E, G.7.A, G.7.C Objectives Graph lines and write their equations in slope-intercept and point-slope form. Classify lines as parallel, intersecting, or coinciding. Vocabulary point-slope form slope-intercept form Why learn this? The cost of some health club plans includes a one-time enrollment fee and a monthly fee. You can use the equations of lines to determine which plan is best for you. (See Example 4.) The equation of a line can be written in many differe
nt forms. The point-slope and slope-intercept forms of a line are equivalent. Because the slope of a vertical line is undefined, these forms cannot be used to write the equation of a vertical line © Forms of the Equation of a Line FORM EXAMPLE The point-slope form of a line is y - y 1 = m (x - x 1 ) , where m is the slope and ( x 1 , y 1 ) is a given point on the line. y - 3 = 2 (x - 4 ) m = 23, 4 ) The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. y = 3x + 6 m = 3, b = 6 The equation of a vertical line is x = a, where a is the x-intercept. The equation of a horizontal line is y = b, where b is the y-intercept. x = 5 y = 2 You will prove the slope-intercept form of a line in Exercise 48. PROOF PROOF Point-Slope Form of a Line Given: The slope of a line through points ( x 1 , y 1 ) and ( x 2 , y 2 ) is m = Prove: The equation of the line through ( x 1 , y 1 ) with slope m is x - x 1 ) . Proof: Let (x, y) be any point on the linex - x 1 ) m = (x - x 1 ) m (x - x 1 ) = ( Slope formula Substitute (x, y) for ( x 2 , y 2 ) . Multiply both sides by (x - x 1 ) . Simplify. y - y 1 = m (x - x 1 ) Sym. Prop. of = 190 190 Chapter 3 Parallel and Perpendicular Lines E X A M P L E 1 Writing Equations of Lines Write the equation of each line in the given form. A the line with slope 3 through (2, 1) in point-slope form y - y 1 = m (x - 2) Point-slope form Substitute 3 for m, 2 for x 1 , and 1 for y 1 . B the line through (0, 4) and (-1, 2) in slope-intercept form = -2 _ -1 - 0 y = mx + b 4 = 2 (0) + b 4 = b y = 2x + 4 Find the slope. Slope-intercept form Substitute 2 for m, 0 for x, and 4 for y to find b. Simplify. Write in slope-intercept form using m = 2 and b = 4. A line with y-intercept b contains the point (0, b) . A line with x-intercept a contains the point (a, 0) . C the line with x-intercept 2 and y-intercept 3 in point-slope form x - 2) 2 y = - 3 _ (x - 2) 2 Use the points (2, 0) and (0, 3) to find the slope. Point-slope form 3 _ for m, 2 for x 1 , and 0 for y 1 . Substitute - 2 Simplify. Write the equation of each line in the given form. 1a. the line with slope 0 through (4, 6) in slope-intercept form 1b. the line through (-3, 2) and (1, 2) in point-slope form E X A M P L E 2 Graphing Lines Graph each line The equation is given in slope-intercept form, with a slope of 3 __ 2 and a y-intercept of 3. Plot the point (0, 3) and then rise 3 and run 2 to find another point. Draw the line containing the two points. B y + 3 = -2 (x - 1) The equation is given in point-slope form, with a slope of -2 = -2 ___ 1 through the point (1, -3) . Plot the point (1, -3) and then rise -2 and run 1 to find another point. Draw the line containing the two points. 3- 6 Lines in the Coordinate Plane 191 191 �� Graph the line. C x = 3 The equation is given in the form for a vertical line with an x-intercept of 3. The equation tells you that the x-coordinate of every point on the line is 3. Draw the vertical line through (3, 0) . Graph each line. 2a. y = 2x - 3 2b. y - 1 = - 2 _ (x + 2) 3 2c. y = -4 A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms. Pairs of Lines Parallel Lines Intersecting Lines Coinciding Lines y = 5x + 8 y = 5x - 4 Same slope different y-intercept y = 2x - 5 y = 4x + 3 Different slopes y = 2x - 4 y = 2x - 4 Same slope same y-intercept E X A M P L E 3 Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. A y = 2x + 3, y = 2x - 1 Both lines have a slope of 2, and the y-intercepts are different. So the lines are parallel. B y = 3x - 5, 6x - 2y = 10 Solve the second equation for y to find the slope-intercept form. 6x - 2y = 10 -2y = -6x + 10 y = 3x - 5 Both lines have a slope of 3 and a y-intercept of -5, so they coincide. C 3x + 2y = 7, 3y = 4x + 7 Solve both equations for y to find the slope-intercept form. 3x + 2y = 7 3y = 4x + The slope is 4 _ . 3 2y = -3x + . The slope is - 2 The lines have different slopes, so they intersect. 3. Determine whether the lines 3x + 5y = 2 and 3x + 6 = -5y are parallel, intersect, or coincide. 192 192 Chapter 3 Parallel and Perpendicular Lines �� E X A M P L E 4 Problem-Solving Application Audrey is trying to decide between two health club plans. After how many months would both plans’ total costs be the same? Plan A Plan B Enrollment Fee $140 $60 Monthly Fee $35 $55 Understand the Problem The answer is the number of months after which the costs of the two plans would be the same. Plan A costs $140 for enrollment and $35 per month. Plan B costs $60 for enrollment and $55 per month. Make a Plan Write an equation for each plan, and then graph the equations. The solution is the intersection of the two lines. Find the intersection by solving the system of equations. Solve Plan A: y = 35x + 140 Plan B: y = 55x + 60 0 = -20x + 80 Subtract the second x = 4 y = 35 (4) + 140 = 280 equation from the first. Solve for x. Substitute 4 for x in the first equation. The lines cross at (4, 280) . Both plans cost $280 after 4 months. Look Back Check your answer for each plan in the original problem. For 4 months, plan A costs $140 plus $35 (4) = $140 + $140 = $280. Plan B costs $60 + $55 (4) = $60 + $220 = $280, so the plans cost the same. Use the information above to answer the following. 4. What if…? Suppose the rate for Plan B was also $35 per month. What would be true about the lines that represent the cost of each plan? THINK AND DISCUSS 1. Explain how to use the slopes and y-intercepts to determine if two lines are parallel. 2. Describe the relationship between the slopes of perpendicular lines. 3. GET ORGANIZED Copy and complete the graphic organizer. 3- 6 Lines in the Coordinate Plane 193 193 1234�� 3-6 Exercises Exercises KEYWORD: MG7 3-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary How can you recognize the slope-intercept form of an equation Write the equation of each line in the given form. p. 191 2. the line through (4, 7) and (-2, 1) in slope-intercept form 3. the line through (-4, 2) with slope 3 _ 4 4. the line with x-intercept 4 and y-intercept -2 in slope-intercept form in point-slope form Graph each line. p. 191 . 192 5. y = -3x + 4 6. y + 4 = 2 _ (x - 6) 3 Determine whether the lines are parallel, intersect, or coincide. 7. x = 5 8. y = -3x + 4, y = -3x + 1 x + 2 _ 10. y = 1 _ 3 3 , 3y = x + 2 9. 6x - 12y = -24, 3y = 2x + 18 11. 4x + 2y = 10, y = -2x + 15 . 193 12. Transportation A speeding ticket in Conroe costs $115 for the first 10 mi/h over the speed limit and $1 for each additional mi/h. In Lakeville, a ticket costs $50 for the first 10 mi/h over the speed limit and $10 for each additional mi/h. If the speed limit is 55 mi/h, at what speed will the tickets cost approximately the same? Homework Help See For Exercises Example 13–15 16–18 19–22 23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S9 Application Practice p. S30 PRACTICE AND PROBLEM SOLVING Write the equation of each line in the given form. 13. the line through (0, -2) and (4, 6) in point-slope form 14. the line through (5, 2) and (-2, 2) in slope-intercept form 15. the line through (6, -4) with slope 2 _ 3 in point-slope form Graph each line. 16. y - 7 = x + 4 17. y = 1 _ 2 x - 2 18. y = 2 Determine whether the lines are parallel, intersect, or coincide. 19. y = x - 7, y = -x + 3 21. x + 2y = 6 20. y = 5 _ 2 x + 4, 2y = 5x - 4 22. 7x + 2y = 10, 3y = 4x - 5 23. Business Chris is comparing two sales positions that he has been offered. The first pays a weekly salary of $375 plus a 20% commission. The second pays a weekly salary of $325 plus a 25% commission. How much must he make in sales per week for the two jobs to pay the same? Write the equation of each line in slope-intercept form. Then graph the line. 24. through (-6, 2) and (3, 6) 26. through (5, -2) with slope 2 _ 3 27. x-intercept 4, y-intercept -3 25. horizontal line through (2, 3) Write the equation of each line in point-slope form. Then graph the line. 28. slope - 1 _ 2 30. through (5, -1) with slope -1 29. slope 3 _ 4 31. through (4, 6) and (-2, -5) , x-intercept -2 , y-intercept 2 194 194 Chapter 3 Parallel and Perpendicular Lines 32. ////ERROR ANALYSIS///// Write the equation of the line with slope -2 through the point (-4, 3) in slope-intercept form. Which equation is incorrect? Explain. Determine whether the lines are perpendicular. 33. y = 3x - 5, y = -3x + 1 x + 5, y = 3 _ 35, 34. y = -x + 1, y = x + 2 36. y = -2x + 4 Multi-Step Given the equation of the line and point P not on the line, find the equation of a line parallel to the given line and a line perpendicular to the given line through the given point. 37. y = 3x + 7, P (2, 3) 38. y = -2x - 5, P (-1, 4) 39. 4x + 3y = 8, P (4, -2) 40. 2x - 5y = 7, P (-2, 4) Food Multi-Step Use slope to determine if each triangle is a right triangle. If so, which angle is the right angle? 41. A (-5, 3) , B (0, -2) , C (5, 3) 42. D (1, 0) , E (2, 7) , F (5, 1) 43. G (3, 4) , H (-3, 4) , J (1, -2) 44. K (-2, 4) , L (2, 1) , M (1, 8) In 2004, the world’s largest pizza was baked in Italy. The diameter of the pizza was 5.19 m (about 17 ft) and it weighed 124 kg (about 273 lb). 45. Food A restaurant charges $8 for a large cheese pizza plus $1.50 per topping. Another restaurant charges $11 for a large cheese pizza plus $0.75 per topping. How many toppings does a pizza have that costs the same at both restaurants? 46. Estimation Estimate the solution of the system of equations represented by the lines in the graph. Write the equation of the perpendicular bisector of the segment with the given endpoints. 47. (2, 5) and (4, 9) 48. (1, 1) and (3, 1) 49. (1, 3) and (-1, 4)
50. (-3, 2) and (-3, -10) 51. Line ℓ has equation y = - 1 __ 2 x + 4, and point P has coordinates (3, 5) . a. Find the equation of line m that passes through P and is perpendicular to ℓ. b. Find the coordinates of the intersection of ℓ and m. c. What is the distance from P to ℓ? 52. Line p has equation y = x + 3, and line q has equation y = x - 1. a. Find the equation of a line r that is perpendicular to p and q. b. Find the coordinates of the intersection of p and r and the coordinates of the intersection of q and r. c. Find the distance between lines p and q. 3- 6 Lines in the Coordinate Plane 195 195 �� 53. This problem will prepare you for the Multi-Step TAKS Prep on page 200. For a car moving at 60 mi/h, the equation d = 88t gives the distance in feet d that the car travels in t seconds. a. Graph the line d = 88t. b. On the same graph you made for part a, graph the line d = 300. What does the intersection of the two lines represent? c. Use the graph to estimate the number of seconds it takes the car to travel 300 ft. 54. Prove the slope-intercept form of a line, given the point-slope form. Given: The equation of the line through ( x 1 , y 1 ) with slope m is y - y 1 = m (x - x 1 ) . Prove: The equation of the line through (0, b) with slope m is y = mx + b. Plan: Substitute (0, b) for ( x 1 , y 1 ) in the equation y - y 1 = m (x - x 1 ) and simplify. 55. Data Collection Use a graphing calculator and a motion detector to do the following: Walk in front of the motion detector at a constant speed, and write the equation of the resulting graph. 56. Critical Thinking A line contains the points (-4, 6) and (2, 2) . Write a convincing argument that the line crosses the x-axis at (5, 0) . Include a graph to verify your argument. 57. Write About It Determine whether the lines are parallel. Use slope to explain your answer. 58. Which graph best represents a solution to this system of equations? ⎧ -3x + y = 7 ⎨ 2x + y = -3 ⎩ 196 196 Chapter 3 Parallel and Perpendicular Lines �� 59. Which line is parallel to the line with the equation y = -2x + 5?   AB through A (2, 3) and B (1, 1 4x + 2y = 10 x + 1 _ 2 y = 1 60. Which equation best describes the graph shown = 3x - � � � � � � � � 61. Which line includes the points (-4, 2) and (6, -3) ? y = 2x - 4 y = 2x CHALLENGE AND EXTEND 62. A right triangle is formed by the x-axis, the y-axis, and the line y = -2x + 5. Find the length of the hypotenuse. 63. If the length of the hypotenuse of a right triangle is 17 units and the legs lie along the x-axis and y-axis, find a possible equation that describes the line that contains the hypotenuse. 64. Find the equations of three lines that form a triangle with a hypotenuse of 13 units. 65. Multi-Step Are the points (-2, -4) , (5, -2) and (2, -3) collinear? Explain the method you used to determine your answer. 66. For the line y = x + 1 and the point P (3, 2) , let d represent the distance from P to a point (x, y) on the line. a. Write an expression for d 2 in terms of x and y. Substitute the expression x + 1 for y and simplify. b. How could you use this expression to find the shortest distance from P to the line? Compare your result to the distance along a perpendicular line. SPIRAL REVIEW 67. The cost of renting DVDs from an online company is $5.00 per month plus $2.50 for each DVD rented. Write an equation for the total cost c of renting d DVDs from the company in one month. Graph the equation. How many DVDs did Sean rent from the company if his total bill for one month was $20.00? (Previous course) Use the coordinate plane for Exercises 68–70. Find the coordinates of the midpoint of each segment. (Lesson 1-6) ̶̶ AB ̶̶ BC ̶̶ AC 70. 68. 69. Use the slope formula to find the slope of each segment. (Lesson 3-5) ̶̶ AB ̶̶ BC ̶̶ AC 73. 71. 72. � � � � �� 3- 6 Lines in the Coordinate Plane 197 197 Scatter Plots and Lines of Best Fit Data Analysis Recall that a line has an infinite number of points on it. You can compute the slope of a line if you can identify two distinct points on the line. See Skills Bank page S79 Example 1 The table shows several possible measures of an angle and its supplement. Graph the points in the table. Then draw the line that best represents the data and write the equation of the line. Step 1 Use the table to write ordered pairs (x, 180 - x) and then plot the points. (30, 150) , (60, 120) , (90, 90) , (120, 60) , (150, 30) Step 2 Draw a line that passes through all the points. x 30 60 90 120 150 y = 180 - x 150 120 90 60 30 Step 3 Choose two points from the line, such as (30, 150) and (120, 60) . m = Use them to find the slope = 60 - 150 _ 120 - 30 = -90 _ 90 = -1 Slope formula Substitute (30, 150) for ( x 1 , y 1 ) and (120, 60) for ( x 2 , y 2 ) . Simplify. Step 4 Use the point-slope form to find the equation of the line and then simplify. y - y 1 = m (x - x 1 ) Point-slope form y - 150 = -1 (x - 30) Substitute (30, 150) for ( x 1 , y 1 ) and -1 for m. y = -x + 180 Simplify. 198 198 Chapter 3 Parallel and Perpendicular Lines �� If you can draw a line through all the points in a set of data, the relationship is linear. If the points are close to a line, you can approximate the relationship with a line of best fit. Example 2 A physical therapist evaluates a client’s progress by measuring the angle of motion of an injured joint. The table shows the angle of motion of a client’s wrist over six weeks. Estimate the equation of the line of best fit. Step 1 Use the table to write ordered pairs and then plot the points. (1, 30) , (2, 36) , (3, 46) , (4, 48) , (5, 54) , (6, 62) Step 2 Use a ruler to estimate a line of best fit. Try to get the edge of the ruler closest to all the points on the line. Week Angle Measure 1 2 3 4 5 6 30 36 46 48 54 62 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Step 3 A line passing through (2, 36) and (6, 62) seems to be closest to all the points. Draw this line. Use the points (2, 36) and (6, 62) to find the slope of the line = 62 - 36 _ _ 6 - 2 = 6.5 Substitute (2, 36) for ( x 1 , y 1 ) and (6, 62) for ( x 2 , y 2 ) . Step 4 Use the point-slope form to find the equation of the line and then simplify. y - y 1 = m (x - x 1 ) y - 36 = 6.5 (x - 2) Point-slope form Substitute (2, 36) for ( x 1 , y 1 ) and 6.5 for m. y = 6.5x + 23 Simplify. Try This TAKS Grades 9–11 Obj. 2, 3, 9 Estimate the equation of the line of best fit for each relationship. 1. 2. the relationship between an angle and its complement 3. Data Collection Use a graphing calculator and a motion detector to do the following: Set the equipment so that the graph shows distance on the y-axis and time on the x-axis. Walk in front of the motion dector while varying your speed slightly and use the resulting graph. On Track for TAKS 199 199 �� SECTION 3B Coordinate Geometry Red Light, Green Light When a driver approaches an intersection and sees a yellow traffic light, she must decide if she can make it through the intersection before the light turns red. Traffic engineers use graphs and equations to study this situation. 1. Traffic engineers can set the duration of the yellow lights on Lincoln Road for any length of time t up to 10 seconds. For each value of t, there is a critical distance d. If a car moving at the speed limit is more than d feet from the light when it turns yellow, the driver will have to stop. If the car is less than d feet from the light, the driver can continue through the intersection. The graph shows the relationship between t and d. Find the speed limit on Lincoln Road in miles per hour. (Hint : 22 ft/s = 15 mi/h) 2. Traffic engineers use the equation d = 22 __ 15 st to determine the critical distance for various durations of a yellow light. In the equation, s is the speed limit. The speed limit on Porter Street is 45 mi/h. Write the equation of the critical distance for a yellow light on Porter Street and then graph the line. Does this line intersect the line for Lincoln Road? If so, where? Is the line for Porter Street steeper or flatter than the line for Lincoln Road? Explain how you know. 200 200 Chapter 3 Parallel and Perpendicular Lines �� SECTION 3B Quiz for Lesson 3-5 Through 3-6 3-5 Slopes of Lines Use the slope formula to determine the slope of each line. 1.  AC 2.   CD 3.  AB 4.   BD Find the slope of the line through the given points. 6. F (-1, 4) and G (5, -1) 8. K (4, 2) and L (-3, 2) 5. M (2, 3) and N (0, 7) 7. P (4, 0) and Q (1, -3) 9. Sonia is walking 4 miles home from school. She leaves at 4:00 P.M., and gets home at 4:45 P.M. Graph the line that represents Sonia’s distance from school at a given time. Find and interpret the slope of the line. Graph each pair of lines and use their slopes to determine if they are parallel, perpendicular, or neither. 10.  EF and   GH for E (-2, 3) , F (6, 1) , G (6, 4) , and H (2, 5) JK and   LM for J (4, 3) , K (5, -1) , L (-2, 4) , and M (3, -5) 11.  12.  NP and   QR for N (5, -3) , P (0, 4) , Q (-3, -2) , and R (4, 3) 13.  ST and   VW for S (0, 3) , T (0, 7) , V (2, 3) , and W (5, 3) 3-6 Lines in the Coordinate Plane Write the equation of each line in the given form. 14. the line through (3, 8) and (-3, 4) in slope-intercept form 15. the line through (-5, 4) with slope 2 _ 3 16. the line with y-intercept 2 through the poi
nt (4, 1) in slope-intercept form in point-slope form Graph each line. 17. y = -2x + 5 18. y + 3 = 1 _ (x - 4) 4 19. x = 3 Write the equation of each line. 20. 21. 22. Determine whether the lines are parallel, intersect, or coincide. 23. y = -2x + 5 y = -2x - 5 24. 3x + 2y = 25. y = 4x -5 3x + 4y = 7 Ready to Go On? 201 201 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary alternate exterior angles . . . . . 147 parallel planes . . . . . . . . . . . . . . 146 same-side interior angles . . . . 147 alternate interior angles . . . . . 147 perpendicular bisector . . . . . . 172 skew lines . . . . . . . . . . . . . . . . . . 146 corresponding angles . . . . . . . 147 perpendicular lines . . . . . . . . . 146 slope . . . . . . . . . . . . . . . . . . . . . . . . .182 distance from a point-slope form . . . . . . . . . . . 190 slope-intercept form . . . . . . . . 190 point to a line . . . . . . . . . . . . 172 parallel lines . . . . . . . . . . . . . . . 146 rise . . . . . . . . . . . . . . . . . . . . . . . . 182 transversal . . . . . . . . . . . . . . . . . 147 run . . . . . . . . . . . . . . . . . . . . . . . . 182 Complete the sentences below with vocabulary words from the list above. 1. Angles on opposite sides of a transversal and between the lines it intersects are ? . ̶̶̶̶ 2. Lines that are in different planes are ? . ̶̶̶̶ 3. A(n) 4. The ? is a line that intersects two coplanar lines at two points. ̶̶̶̶ ? is used to write the equation of a line with a given slope that passes ̶̶̶̶ through a given point. 5. The slope of a line is the ratio of the ? to the ̶̶̶̶ ? . ̶̶̶̶ 3-1 Lines and Angles (pp. 146–151) E X A M P L E S EXERCISES Identify each of the following. Identify each of the following. ■ a pair of parallel segments ̶̶ AB ǁ ̶̶ CD ■ a pair of parallel planes plane ABC ǁ plane EFG ■ a pair of perpendicular segments ̶̶ AB ⊥ ̶̶ AE ■ a pair of skew segments ̶̶ AB and ̶̶ FG are skew. 6. a pair of skew segments 7. a pair of parallel segments 8. a pair of perpendicular segments 9. a pair of parallel planes 202 202 Chapter 3 Parallel and Perpendicular Lines �� Identify the transversal and classify each angle pair. ■ ∠4 and ∠6 p, corresponding angles ■ ∠1 and ∠2 q, alternate interior angles ■ ∠3 and ∠4 p, alternate exterior angles ■ ∠6 and ∠7 r, same-side interior angles Identify the transversal and classify each angle pair. 10. ∠5 and ∠2 11. ∠6 and ∠3 12. ∠2 and ∠4 13. ∠1 and ∠2 3-2 Angles Formed by Parallel Lines and Transversals (pp. 155–161) TEKS G.2.A, E X A M P L E S Find each angle measure. ■ m∠TUV EXERCISES Find each angle measure. 14. m∠WYZ G.3.C, G.3.E, G.9.A By the Same-Side Interior Angles Theorem, (6x + 10) + (4x + 20) = 180. 15. m∠KLM x = 15 Solve for x. Substitute the value for x into the expression for m∠TUV. m∠TUV = 4 (15) + 20 = 80° ■ m∠ABC 16. m∠DEF 17. m∠QRS By the Corresponding Angles Postulate, 8x + 28 = 10x + 4. x = 12 Solve for x. Substitute the value for x into the expression for one of the obtuse angles. 10 (12) + 4 = 124° ∠ABC is supplementary to the 124° angle, so m∠ABC = 180 - 124 = 56°. Study Guide Review 203 203 �� 3-3 Proving Lines Parallel (pp. 162–169) TEKS G.1.A, G.3.C, G.3.E, G.9.A EXERCISES Use the given information and theorems and postulates you have learned to show that c ǁ d. 18. m∠4 = 58°, m∠6 = 58° 19. m∠1 = (23x + 38) °, m∠5 = (17x + 56) °, x = 3 20. m∠6 = (12x + 6) °, m∠3 = (21x + 9) °, x = 5 21. m∠1 = 99°, m∠7 = (13x + 8) °, Use the given information and theorems and postulates you have learned to show that p ǁ q. ■ m∠2 + m∠3 = 180° ∠2 and ∠3 are supplementary, so p ǁ q by the Converse of the Same-Side Interior Angles Theorem. ■ ∠8 ≅ ∠6 ∠8 ≅ ∠6, so p ǁ q by the Converse of the Corresponding Angles Postulate. ■ m∠1 = (7x - 3) °, m∠5 = 5x + 15, x = 9 m∠1 = 60°, and m∠5 = 60°. So ∠1 ≅ ∠5. p ǁ q by the Converse of the Alternate Exterior Angles Theorem. 3-4 Perpendicular Lines (pp. 172–178) TEKS G.1.A, G.2.A, G.3.C, G.3.E, G.9.A EXERCISES 22. Name the shortest segment from point K to ̶̶ LN . 23. Write and solve an inequality for x. 24. Given: Prove: ̶̶ AD ǁ ̶̶ AB ǁ ̶̶ BC , ̶̶ CD ̶̶ AD ⊥ ̶̶ AB , ̶̶ DC ⊥ ̶̶ BC E X A M P L E S ■ Name the shortest segment from ̶̶ WY . point X to ̶̶ XZ ■ Write and solve an inequality for x. x + 3 > 3 x > 0 Subtract 3 from both sides. ■ Given: m ⊥ p, ∠1 and ∠2 are complementary. Prove: p ǁ q Proof: It is given that m ⊥ p. ∠1 and ∠2 are complementary, so m∠1 + m∠2 = 90°. Thus m ⊥ q. Two lines perpendicular to the same line are parallel, so p ǁ q. 204 204 Chapter 3 Parallel and Perpendicular Lines �� 3-5 Slopes of Lines (pp. 182–187) TEKS G.7.A, G.7.B, G.7.C E X A M P L E S ■ Use the slope formula to determine the slope of the line. EXERCISES Use the slope formula to determine the slope of each line. 25. 26. slope of   WX = - (-3) _ 2 - (-4) = 6 _ 6 = 1 slope of   AB = ■ Use slopes to determine whether    AB and    CD are parallel, perpendicular, or neither for A (-1, 5) , B (-3, 4) , C (3, -1) , and D (4, -33 - (-1) -3 - (-1) = -2 _ _ 4 - 3 1 The slopes are opposite reciprocals, so the slope of   CD = = -2 Use slopes to determine if the lines are parallel, perpendicular, or neither. 27. EF and GH for E (8, 2) , F (-3, 4) , G (6, 1) , and H (-4, 3) 28. JK and LM for J (4, 3) , K (-4, -2) , L (5, 6) , and M (-3, 1) 29. ST and UV for S (-4, 5) , T (2, 3) , U (3, 1) , and V (4, 4) lines are perpendicular. 3-6 Lines in the Coordinate Plane (pp. 190–197) TEKS G.3.C, G.3.E, G.7.A, G.7.B, G.7.C E X A M P L E S EXERCISES ■ Write the equation of the line through (5, -2) with slope 3 __ in slope-intercept form. 5 y - (-2) = 3 _ (x - 5 Solve for y. Simplify. x - 3 x - 5 Point-slope form ■ Determine whether the lines y = 4x + 6 and 8x - 2y = 4 are parallel, intersect, or coincide. Solve the second equation for y to find the slope-intercept form. 8x - 2y = 4 y = 4x - 2 Both the lines have a slope of 4 and have different y-intercepts, so they are parallel. Write the equation of each line in the given form. 30. the line through (6, 1) and (-3, 5) in slope-intercept form 31. the line through (-3, -4) with slope 2 _ 3 in slope-intercept form 32. the line with x-intercept 1 and y-intercept -2 in point-slope form Determine whether the lines are parallel, intersect, or coincide. 33. -3x + 2y = 5, 6x - 4y = 8 34. y = 4x - 3, 5x + 2y = 1 35. y = 2x + 1, 2x - y = -1 Study Guide Review 205 205 �� Identify each of the following. 1. a pair of parallel planes 2. a pair of parallel segments 3. a pair of skew segments Find each angle measure. � � � � � � � � �� 4. 5. 6. Use the given information and the theorems and postulates you have learned to show f ǁ g. 7. m∠4 = (16x + 20) °, m∠5 = (12x + 32) °, x = 3 8. m∠3 = (18x + 6) °, m∠5 = (21x + 18) °, x = 4 Write a two-column proof. 9. Given: ∠1 ≅ ∠2, n ⊥ ℓ Prove: n ⊥ m Use the slope formula to determine the slope of each line. 10. 11. 12. 13. Greg is on a 32-mile bicycle trail from Elroy, Wisconsin, to Sparta, Wisconsin. He leaves Elroy at 9:30 A.M. and arrives in Sparta at 2:00 P.M. Graph the line that represents Greg’s distance from Elroy at a given time. Find and interpret the slope of the line. 14. Graph   QR and   ST for Q (3, 3) , R (6, -5) , S (-4, 6) , and T (-1, -2) . Use slopes to determine whether the lines are parallel, perpendicular, or neither. 15. Write the equation of the line through (-2, -5) with slope - 3 _ 4 16. Determine whether the lines 6x + y = 3 and 2x + 3y = 1 are parallel, intersect, in point-slope form. or coincide. 206 206 Chapter 3 Parallel and Perpendicular Lines �� FOCUS ON ACT When you take the ACT Mathematics Test, you receive a separate subscore for each of the following areas: • Pre-Algebra/Elementary Algebra, • Intermediate Algebra/Coordinate Geometry, and • Plane Geometry/Trigonometry. You may want to time yourself as you take this practice test. It should take you about 5 minutes to complete. 1. Which of the following is an equation of the line that passes through the point (2, -3) and is parallel to the line 4x - 5y = 1? (A) -4x + 5y = -23 (B) -5x - 4y = 2 (C) -2x - 5y = 11 (D) -4x - 5y = 7 (E) -5x + 4y = -22 2. In the figure below, line t crosses parallel lines ℓ and m. Which of the following statements are true? I. ∠1 and ∠6 are alternate interior angles. II. ∠2 ≅ ∠4 III. ∠2 ≅ ∠8 (F) I only (G) II only (H) III only (J) I and II only (K) II and III only Find out what percent of questions are from each area and concentrate on content that represents the greatest percent of questions. 3. In the standard (x, y) coordinate plane, the line that passes through (1, -7) and (-8, 5) is perpendicular to the line that passes through (3, 6) and (-1, b) . What is the value of b? (A) 2 (B) 3 (C) 7 (D) 9 (E) 10 4. Lines m and n are cut by a transversal so that ∠2 and ∠5 are corresponding angles. If m∠2 = (x + 18) ° and m∠5 = (2x - 28) °, which value of x makes lines m and n parallel? (F) 3 1 _ 3 (G) 33 1 _ 3 (H) 46 (J) 63 1 _ 3 (K) 72 5. What is the distance between point G (4, 2) and the line through the points E (1, -2) and F (7, -2) ? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 College Entrance Exam Practice 207 207 �� Any Question Type: Interpret Coordinate Graphs When test items refer to a coordinate plane, it is important to interpret the coordinate graphs correctly. It is also important to understand the relationship between an equation and its graph. Multiple Choice Which statement best describes the graph of the following equations? y = 2x + 6 2y = 4x + 6 The lines are parallel. The lines are pe
rpendicular. The lines coincide. The lines have the same y-intercept. It may help to graph the lines or visualize the graph in order to answer the question. The lines appear to be parallel. Write the equations of both lines in slope-intercept form. y = 2x + 6 y = 2x + 3 The slope is 2 and the y-intercept is 6. The slope is 2 and the y-intercept is 3. The lines have the same slope and different y-intercepts, so they are parallel. The answer is A. Gridded Response What is the rate of change of the following graph? Remember that the rate of change of the graph of a line is its slope. Choose two points on the line and use their coordinates to calculate the slope of the line. Use the points (5, 0) and (0, -2) . m = - The slope is 2 _ 5 = 0.4. Enter 0.4 in the answer grid. 208 208 Chapter 3 Parallel and Perpendicular Lines �� A quick look at the graph of a line can tell you whether the slope is positive or negative. This may help you eliminate answer choices. Item C Multiple Choice Which equation describes the line through the point (4, 2) that is perpendicular to the line 3x - y = 7? Read each test item and answer the questions that follow. Item A Multiple Choice The line segment on the graph shows the altitude of a hot air balloon during a landing. Which statement best describes the slope of the line segment? The balloon descends about 5 feet per 8 seconds. The balloon descends about 8 feet per 5 seconds. The balloon descends about 1 foot per 2 seconds. The balloon descends about 2 feet per second. 1. What are the coordinates of two points on the graph? 2. How does the scale of the graph affect the appearance of the slope? 3. How is the slope of the line related to the rate of descent? Item B Gridded Response What is the y-intercept of the line through (5, 2) that is parallel to the line x - 4y = 8? y = 3x - 10 y = -3x + 14 . Graph the line represented by 3x - y = 7. What is its slope? 8. Is the slope of a line perpendicular to the line represented by 3x - y = 7 positive or negative? Based on your answer, can you eliminate any answer choices? 9. What is the slope of a line perpendicular to the line represented by 3x - y = 7? Item D Multiple Choice Which graph best represents a solution to the following system of equations? -2x + y = 6 3x + y = -2 10. What is the slope of the line represented by -2x + y = 6? Based on your answer, can you eliminate any answer choices? 4. Graph the line x - 4y = 8. What is its slope? 11. What is the slope of the line represented 5. Write an equation in point-slope form for the line through (5, 2) that is parallel to x - 4y = 8. 6. How would you use the equation in point- slope form to find the y-intercept of the line? by 3x + y = -2? Based on your answer, can you eliminate any answer choices? TAKS Tackler 209 209 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–3 Multiple Choice Use the diagram below for Items 1 and 2. 1. What type of angle pair are ∠JKM and ∠KMN? Corresponding angles Alternate exterior angles Same-side interior angles Alternate interior angles 2. What is m∠KML? 57° 80° 102° 125° 3. What is a possible value of x in the diagram? 2 3 4 5 4. A graphic artist used a computer illustration program to draw a line connecting points with coordinates (3, -1) and (4, 6) . She needs to draw a second line parallel to the first line. What slope should the second line have. Which term describes a pair of vertical angles that are also supplementary? Acute Obtuse Right Straight 6. What is the equation of the line that passes through the points (-1, 8) and (4, -2) ? y = -2x + = 2x + 10 7. Given the points R (-5, 3) , S (-5, 4) , T (-3, 4) , and U (-3, 1) , which line is perpendicular to   TU ?   RS   RT   ST   SU 8. Which of following is true if   XY and   UV are skew?   XY and   UV are coplanar. X, Y, and U are noncollinear.   XY ǁ   UV   XY ⊥   UV �� Make sure that you answer the question that is asked. Some problems require more than one step. You must perform all of the steps to get the correct answer. 9. Point C is the midpoint of B (7, 2) . What is the length of nearest tenth. ̶̶ AB for A (1, -2) and ̶̶ AC ? Round to the 3.0 3.6 5.0 7.2 Use the diagram below for Items 10 and 11. 10.  AD bisects ∠CAE, and  AE bisects ∠CAF. If m∠DAF = 120°, what is m∠DAE? 40° 60° 80° 100° 210 210 Chapter 3 Parallel and Perpendicular Lines A F 11. What is the intersection of  AD ?  AF and ̶̶ FD ∠DAF �� 12. Which statement is true by the Transitive Property of Equality? If x + 3 = y, then y = x + 3. If k = 6, then 2k = 12. If a = b and b = 8, then a = 8. If m = n, then m + 7 = n + 7. 13. Which condition guarantees that r \|\| s? STANDARDIZED TEST PREP Short Response 21. Given ℓ ǁ m with transversal t, explain why ∠1 and ∠8 are supplementary. ∠1 ≅ ∠2 ∠2 ≅ ∠7 ∠2 ≅ ∠3 ∠1 ≅ ∠4 22. Read the following conditional statement. If two angles are vertical angles, then they are congruent. a. Write the converse of this conditional statement. b. Give a counterexample to show that the 14. What is the converse of the following statement? converse is false. If x = 2, then x + 3 = 5. If x ≠ 2, then x + 3 = 5. If x = 2, then x + 3 ≠ 5. If x + 3 ≠ 5, then x ≠ 2. If x + 3 = 5, then x = 2. Gridded Response 15. Two lines a and b are cut by a transversal so that ∠1 and ∠2 are same-side interior angles. If m∠1 = (2x + 30) ° and m∠2 = (4x - 75) °, what value of x proves that a ǁ b? 23. Assume that the following statements are true when the bases are loaded in a baseball game. If a batter hits the ball over the fence, then the batter hits a home run. A batter hits a home run if and only if the result is four runs scored. a. If a batter hits the ball over the fence when the bases are loaded, can you conclude that four runs were scored? Explain your answer. b. If a batter hits a home run when the bases are loaded, can you conclude that the batter hit the ball over the fence? Explain your answer. 16. What is the slope of the line that passes through (3, 7) and (-5, 1)? Extended Response 17. ∠1 and ∠2 form a linear pair. m∠1 = (4x + 18)° and m∠2 = (3x - 6)°. What is the value of x? 18. Points A, B, and C are collinear, and B is between A and C. AB = 16 and AC = 27. What is the distance BC? 19. Ms. Nelson wants to put a chain-link fence around 3 sides of a square-shaped lawn. Chain-link fencing is sold in sections that are each 6 feet wide. If Ms. Nelson’s lawn has an area of 3600 square feet, how many sections of fencing will she need? 20. What is the next number in this pattern? 67, 76, 83, 88,… 24. A car passes through a tollbooth at 8:00 A.M. and begins traveling east at an average speed of 45 miles per hour. A second car passes through the same tollbooth an hour later and begins traveling east at an average speed of 60 miles per hour. a. Write an equation for each car that relates the number of hours x since 8:00 A.M. to the distance in miles y the car has traveled. Explain what the slope of each equation represents. b. Graph the system of equations on the coordinate plane. c. If neither car stops, at what time will the second car catch up to the first car? Explain how you determined your answer. Cumulative Assessment, Chapters 1–3 211 211 �� Triangle Congruence 4A Triangles and Congruence 4-1 Classifying Triangles Lab Develop the Triangle Sum Theorem 4-2 Angle Relationships in Triangles 4-3 Congruent Triangles 4B Proving Triangle Congruence Lab Explore SSS and SAS Triangle Congruence 4-4 Triangle Congruence: SSS and SAS Lab Predict Other Triangle Congruence Relationships 4-5 Triangle Congruence: ASA, AAS, and HL 4-6 Triangle Congruence: CPCTC 4-7 Introduction to Coordinate Proof 4-8 Isosceles and Equilateral Triangles Ext Proving Constructions Valid KEYWORD: MG7 ChProj The Bob Bullock Texas State History Museum opened in Austin in April 2001. 212 212 Chapter 4 Vocabulary Match each term on the left with a definition on the right. 1. acute angle A. a statement that is accepted as true without proof 2. congruent segments B. an angle that measures greater than 90° and less than 180° 3. obtuse angle C. a statement that you can prove 4. postulate 5. triangle D. segments that have the same length E. a three-sided polygon F. an angle that measures greater than 0° and less than 90° Measure Angles Use a protractor to measure each angle. 6. 7. Use a protractor to draw an angle with each of the following measures. 8. 20° 10. 105° 9. 63° 11. 158° Solve Equations with Fractions x + 7 = 25 Solve. 12. 9_ 2 14. x - 1_ 5 = 12_ 5 13. 3x - 2_ 3 = 4_ 3 15. 2y = 5y - 21_ 2 Connect Words and Algebra Write an equation for each statement. 16. Tanya’s age t is three times Martin’s age m. 17. Twice the length of a segment x is 9 ft. 18. The sum of 53° and twice an angle measure y is 90°. 19. The price of a radio r is $25 less than the price of a CD player p. 20. Half the amount of liquid j in a jar is 5 oz more than the amount of liquid b in a bowl. Triangle Congruence 213 213 Key Vocabulary/Vocabulario acute triangle triángulo acutángulo congruent polygons polígonos congruentes corollary corolario equilateral triangle triángulo equilátero exterior angle ángulo externo interior angle ángulo interno isosceles triangle triángulo isósceles obtuse triangle triángulo obtusángulo right triangle triángulo rectángulo scalene triangle triángulo escaleno Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The Latin word acutus means “pointed” or “sharp.” Draw a triangle that looks pointed or sharp. Do you think this is an acute triangle ? 2. Consider the everyday meaning of the word exterior. Where do you think an exterior angle of a triangle is located? 3. You already
know the definition of an obtuse angle. Use this meaning to make a conjecture about an obtuse triangle . Geometry TEKS G.1.A Geometric structure* develop an awareness of the structure of a mathematical system … G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures … G.2.B Geometric structure* make conjectures about angles, lines, polygons … and determine validity of the conjectures, choosing from a variety of approaches … 4-2 Geo. Lab Les. 4-1 ★ Les. 4-2 Les. 4-3 ★ 4-4 Geo. Lab 4-5 Tech. Lab Les. 4-4 Les. 4-5 Les. 4-6 Les. 4-7 Les. 4-8 Ext.5.A Geometric patterns* use … geometric patterns ★ to develop algebraic expressions representing geometric properties G.7.A Dimensionality and the geometry of location* use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures ★ G.9.B Congruence and the geometry of ★ ★ ★ ★ ★ size* formulate and test conjectures about the properties and attributes of polygons … based on explorations … G.10.B Congruence and the geometry of size* justify ★ ★ ★ ★ ★ ★ ★ ★ and apply triangle congruence relationships * Knowledge and skills are written out completely on pages TX28–TX35. 214 214 Chapter 4 Reading Strategy: Read Geometry Symbols In Geometry we often use symbols to communicate information. When studying each lesson, read both the symbols and the words slowly and carefully. Reading aloud can sometimes help you translate symbols into words. � �� Throughout this course, you will use these symbols and combinations of these symbols to represent various geometric statements. Symbol Combinations Translated into Words   ST ǁ   UV ̶̶̶ ̶̶ GH BC ⊥ p → q Line ST is parallel to line UV. Segment BC is perpendicular to segment GH. If p, then q. m∠QRS = 45° The measure of angle QRS is 45 degrees. ∠CDE ≅ ∠LMN Angle CDE is congruent to angle LMN. Try This Rewrite each statement using symbols. 1. the absolute value of 2 times pi 2. The measure of angle 2 is 125 degrees. 3. Segment XY is perpendicular to line BC. 4. If not p, then not q. Translate the symbols into words. 5. m∠FGH = m∠VWX 6. ZA ǁ TU 7. ∼p → q 8. ST bisects ∠TSU. Triangle Congruence 215 215 4-1 Classifying Triangles TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning ... Objectives Classify triangles by their angle measures and side lengths. Use triangle classification to find angle measures and side lengths. Vocabulary acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle Who uses this? Manufacturers use properties of triangles to calculate the amount of material needed to make triangular objects. (See Example 4.) A triangle is a steel percussion instrument in the shape of an equilateral triangle. Different-sized triangles produce different musical notes when struck with a metal rod. Recall that a triangle (△) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths. ̶̶ BC , and ̶̶ AB , A, B, and C are the triangle’s vertices. ̶̶ AC are the sides of △ABC. Triangle Classification By Angle Measures Acute Triangle Equiangular Triangle Right Triangle Obtuse Triangle Three acute angles Three congruent acute angles One right angle One obtuse angle E X A M P L E 1 Classifying Triangles by Angle Measures Classify each triangle by its angle measures. A △EHG ∠EHG is a right angle. So △EHG is a right triangle. B △EFH ∠EFH and ∠HFG form a linear pair, so they are supplementary. Therefore m∠EFH + m∠HFG = 180°. By substitution, m∠EFH + 60° = 180°. So m∠EFH = 120°. △EFH is an obtuse triangle by definition. 1. Use the diagram to classify △FHG by its angle measures. 216 216 Chapter 4 Triangle Congruence �� Triangle Classification By Side Lengths Equilateral Triangle Isosceles Triangle Scalene Triangle Three congruent sides At least two congruent sides No congruent sides E X A M P L E 2 Classifying Triangles by Side Lengths Classify each triangle by its side lengths. When you look at a figure, you cannot assume segments are congruent based on their appearance. They must be marked as congruent. A △ABC From the figure, and △ABC is equilateral. ̶̶ AB ≅ ̶̶ AC . So AC = 15, B △ABD By the Segment Addition Postulate, BD = BC + CD = 15 + 5 = 20. Since no sides are congruent, △ABD is scalene. 2. Use the diagram to classify △ACD by its side lengths. E X A M P L E 3 Using Triangle Classification Find the side lengths of the triangle. Step 1 Find the value of x. ̶̶ ̶̶ KL JK ≅ JK = KL (4x - 1.3) = (x + 3.2) 3x = 4.5 x = 1.5 Given Def. of ≅ segs. Substitute (4x - 13) for JK and (x + 3.2) for KL. Add 1.3 and subtract x from both sides. Divide both sides by 3. Step 2 Substitute 1.5 into the expressions to find the side lengths. JK = 4x - 1.3 = 4 (1.5) - 1.3 = 4.7 KL = x + 3.2 = (1.5) + 3.2 = 4.7 JL = 5x - 0.2 = 5 (1.5) - 0.2 = 7.3 3. Find the side lengths of equilateral △FGH. 4-1 Classifying Triangles 217 217 �� E X A M P L E 4 Music Application �� A manufacturer produces musical triangles by bending pieces of steel into the shape of an equilateral triangle. The triangles are available in side lengths of 4 inches, 7 inches, and 10 inches. How many 4-inch triangles can the manufacturer produce from a 100 inch piece of steel? The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle. P = 3 (4) = 12 in. To find the number of triangles that can be made from 100 inches. of steel, divide 100 by the amount of steel needed for one triangle. 100 ÷ 12 = 8 1 _ 3 triangles There is not enough steel to complete a ninth triangle. So the manufacturer can make 8 triangles from a 100 in. piece of steel. Each measure is the side length of an equilateral triangle. Determine how many triangles can be formed from a 100 in. piece of steel. 4a. 7 in. 4b. 10 in. THINK AND DISCUSS 1. For △DEF, name the three pairs of consecutive sides and the vertex formed by each. 2. Sketch an example of an obtuse isosceles triangle, or explain why it is not possible to do so. 3. Is every acute triangle equiangular? Explain and support your answer with a sketch. 4. Use the Pythagorean Theorem to explain why you cannot draw an equilateral right triangle. 5. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe each type of triangle. 218 218 Chapter 4 Triangle Congruence �� 4-1 Exercises Exercises KEYWORD: MG7 4-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In △JKL, JK, KL, and JL are equal. How does this help you classify △JKL by its side lengths? 2. △XYZ is an obtuse triangle. What can you say about the types of angles in △XYZ Classify each triangle by its angle measures. p. 216 3. △DBC 4. △ABD 5. △ADC Classify each triangle by its side lengths. p. 217 6. △EGH 7. △EFH 8. △HFG Multi-Step Find the side lengths of each triangle. p. 217 9. 10. 218 11. Crafts A jeweler creates triangular earrings by bending pieces of silver wire. Each earring is an isosceles triangle with the dimensions shown. How many earrings can be made from a piece of wire that is 50 cm long? Independent Practice For See Exercises Example 12–14 15–17 18–20 21–22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S10 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Classify each triangle by its angle measures. 12. △BEA 13. △DBC 14. △ABC Classify each triangle by its side lengths. 15. △PST 16. △RSP 17. △RPT Multi-Step Find the side lengths of each triangle. 18. 19. �� 20. Draw a triangle large enough to measure. Label the vertices X, Y, and Z. a. Name the three sides and three angles of the triangle. b. Use a ruler and protractor to classify the triangle by its side lengths and angle measures. 4-1 Classifying Triangles 219 219 �� Carpentry Use the following information for Exercises 21 and 22. A manufacturer makes trusses, or triangular supports, for the roofs of houses. Each truss is the shape of an isosceles triangle in which base ̶̶ ̶̶ PQ ≅ PR . The length of the ̶̶ QR is 4 __ 3 the length of each of the congruent sides. 21. The perimeter of each truss is 60 ft. Find each side length. 22. How many trusses can the manufacturer make from 150 feet of lumber? Draw an example of each type of triangle or explain why it is not possible. 23. isosceles right 24. equiangular obtuse 25. scalene right 26. equilateral acute 27. scalene equiangular 28. isosceles acute 29. An equilateral triangle has a perimeter of 105 in. What is the length of each side of the triangle? Architecture Classify each triangle by its angles and sides. 30. △ABC 31. △ACD Daniel Burnham designed and built the 22-story Flatiron Building in New York City in 1902. Source: www.greatbuildings.com 32. An isosceles triangle has a perimeter of 34 cm. The congruent sides measure (4x - 1) cm. The length of the third side is x cm. What is the value of x? 33. Architecture The base of the Flatiron Building is a triangle bordered by three streets: Broadway, Fifth Avenue, and East Twenty-second Street. The Fifth Avenue side is 1 ft shorter than twice the East Twenty-second Street side. The East Twenty-second Street side is 8 ft shorter than half the Broadway side. The Broadway side is 190 ft. a. Find the two unknown side lengths. b. Classify the triangle by its side lengths. 34. Critical Thinking Is every i
sosceles triangle equilateral? Is every equilateral triangle isosceles? Explain. Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 35. An acute triangle is a scalene triangle. 36. A scalene triangle is an obtuse triangle. 37. An equiangular triangle is an isosceles triangle. 38. Write About It Write a formula for the side length s of an equilateral triangle, given the perimeter P. Explain how you derived the formula. 39. Construction Use the method for constructing congruent segments to construct an equilateral triangle. 40. This problem will prepare you for the Multi-Step TAKS Prep on page 238. Marc folded a rectangular sheet of paper, ABCD, in half ̶̶ EF . He folded the resulting square diagonally and along then unfolded the paper to create the creases shown. a. Use the Pythagorean Theorem to find DE and CE. b. What is the m∠DEC? c. Classify △DEC by its side lengths and by its angle measures. 220 220 Chapter 4 Triangle Congruence �� 41. What is the side length of an equilateral triangle with a perimeter of 36 2 __ inches? 3 36 2 _ 3 18 1 _ 3 inches inches 12 1 _ 3 12 2 _ 9 inches inches 42. The vertices of △RST are R (3, 2) , S (-2, 3) , and T (-2, 1) . Which of these best describes △RST? Isosceles Scalene Equilateral Right 43. Which of the following is NOT a correct classification of △LMN? Acute Isosceles Equiangular Right ̶̶ 44. Gridded Response △ABC is isosceles, and AB ≅ - x) . What is the perimeter of △ABC ? BC = ( 5 __ 2 ̶̶ AC . AB = ( 1 __ x + 1 __ 4 2 ) , and CHALLENGE AND EXTEND 45. A triangle has vertices with coordinates (0, 0) , (a, 0) , and (0, a) , where a ≠ 0. Classify the triangle in two different ways. Explain your answer. 46. Write a two-column proof. Given: △ABC is equiangular. EF ǁ AC Prove: △EFB is equiangular. 47. Two sides of an equilateral triangle measure (y + 10) units and ( y 2 - 2) units. If the perimeter of the triangle is 21 units, what is the value of y? 48. Multi-Step The average length of the sides of △PQR is 24. How much longer then the average is the longest side? SPIRAL REVIEW Name the parent function of each function. (Previous course) 49. y = 5x 2 + 4 50. 2y = 3x + 4 51. y = 2 (x - 8) 2 + 6 Determine if each biconditional is true. If false, give a counterexample. (Lesson 2-4) 52. Two lines are parallel if and only if they do not intersect. 53. A triangle is equiangular if and only if it has three congruent angles. 54. A number is a multiple of 20 if and only if the number ends in a 0. Determine whether each line is parallel to, is perpendicular to, or coincides with y = 4x. (Lesson 3-6) 55. y = 4x + 2 57. 1 _ 2 y = 2x 56. 4y = -x + 8 58. -2y = 1 _ x 2 4-1 Classifying Triangles 221 221 �� 4-2 Use with Lesson 4-2 Activity Develop the Triangle Sum Theorem In this lab, you will use patty paper to discover a relationship between the measures of the interior angles of a triangle. TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about the properties and attributes of polygons … based on explorations. Also G.3.D G.4.A, G.5.A 1 Draw and label △ABC on a sheet of notebook paper. 2 On patty paper draw a line ℓ and label a point P on the line. ̶̶ AB is on line ℓ and P and B 3 Place the patty paper on top of the triangle you drew. Align the papers so that coincide. Trace ∠B. Rotate the triangle and trace ∠C adjacent to ∠B. Rotate the triangle again and trace ∠A adjacent to ∠C. The diagram shows your final step. Try This 1. What do you notice about the three angles of the triangle that you traced? 2. Repeat the activity two more times using two different triangles. Do you get the same results each time? 3. Write an equation describing the relationship among the measures of the angles of △ABC. 4. Use inductive reasoning to write a conjecture about the sum of the measures of the angles of a triangle. 222 222 Chapter 4 Triangle Congruence 4-2 Angle Relationships in Triangles TEKS G.2.B Geometric structure: make conjectures about angles, lines, polygons …. Also G.1.A Objectives Find the measures of interior and exterior angles of triangles. Apply theorems about the interior and exterior angles of triangles. Vocabulary auxiliary line corollary interior exterior interior angle exterior angle remote interior angle Who uses this? Surveyors use triangles to make measurements and create boundaries. (See Example 1.) Triangulation is a method used in surveying. Land is divided into adjacent triangles. By measuring the sides and angles of one triangle and applying properties of triangles, surveyors can gather information about adjacent triangles. This engraving shows the county surveyor and commissioners laying out the town of Baltimore in 1730. Theorem 4-2-1 Triangle Sum Theorem The sum of the angle measures of a triangle is 180°. m∠A + m∠B + m∠C = 180° The proof of the Triangle Sum Theorem uses an auxiliary line. An auxiliary line is a line that is added to a figure to aid in a proof. PROOF PROOF Triangle Sum Theorem Given: △ABC Prove: m∠1 + m∠2 + m∠3 = 180° Proof: Whenever you draw an auxiliary line, you must be able to justify its existence. Give this as the reason: Through any two points there is exactly one line. 4-2 Angle Relationships in Triangles 223 223 �� E X A M P L E 1 Surveying Application The map of France commonly used in the 1600s was significantly revised as a result of a triangulation land survey. The diagram shows part of the survey map. Use the diagram to find the indicated angle measures. A m∠NKM m∠KMN + m∠MNK + m∠NKM = 180° △ Sum Thm. 88 + 48 + m∠NKM = 180 136 + m∠NKM = 180 m∠NKM = 44° Substitute 88 for m∠KMN and 48 for m∠MNK. Simplify. Subtract 136 from both sides. B m∠JLK Step 1 Find m∠JKL. m∠NKM + m∠MKJ + m∠JKL = 180° Lin. Pair Thm. & ∠ Add. Post. 44 + 104 + m∠JKL = 180 148 + m∠JKL = 180 m∠JKL = 32° Substitute 44 for m∠NKM and 104 for m∠MKJ. Simplify. Subtract 148 from both sides. Step 2 Use substitution and then solve for m∠JLK. m∠JLK + m∠JKL + m∠KJL = 180° △ Sum Thm. m∠JLK + 32 + 70 = 180 Substitute 32 for m∠JKL and m∠JLK + 102 = 180 m∠JLK = 78° 70 for m∠KJL. Simplify. Subtract 102 from both sides. 1. Use the diagram to find m∠MJK. A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem. Corollaries COROLLARY HYPOTHESIS CONCLUSION 4-2-2 The acute angles of a right triangle are complementary. 4-2-3 The measure of each angle of an equiangular triangle is 60°. ∠D and ∠E are complementary. m∠D + m∠E = 90° m∠A = m∠B = m∠C = 60° You will prove Corollaries 4-2-2 and 4-2-3 in Exercises 24 and 25. 224 224 Chapter 4 Triangle Congruence 70°104°88°48°�� E X A M P L E 2 Finding Angle Measures in Right Triangles One of the acute angles in a right triangle measures 22.9°. What is the measure of the other acute angle? Let the acute angles be ∠M and ∠N, with m∠M = 22.9°. m∠M + m∠N = 90 22.9 + m∠N = 90 Acute  of rt. △ are comp. Substitute 22.9 for m∠M. m∠N = 67.1° Subtract 22.9 from both sides. The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? 2a. 63.7° 2b. x ° ° 2c. 48 2 _ 5 The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and the extension of an adjacent side. Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. ∠4 is an exterior angle. Its remote interior angles are ∠1 and ∠2. Theorem 4-2-4 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. m∠4 = m∠1 + m∠2 You will prove Theorem 4-2-4 in Exercise 28. E X A M P L E 3 Applying the Exterior Angle Theorem Find m∠J. m∠J + m∠H = m∠FGH 5x + 17 + 6x - 1 = 126 11x + 16 = 126 11x = 110 x = 10 Ext. ∠ Thm. Substitute 5x + 17 for m∠J, 6x - 1 for m∠H, and 126 for m∠FGH. Simplify. Subtract 16 from both sides. Divide both sides by 11. m∠J = 5x + 17 = 5 (10) + 17 = 67° 3. Find m∠ACD. 4-2 Angle Relationships in Triangles 225 225 �� Theorem 4-2-5 Third Angles Theorem THEOREM HYPOTHESIS CONCLUSION If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. ∠N ≅ ∠T You will prove Theorem 4-2-5 in Exercise 27. E X A M P L E 4 Applying the Third Angles Theorem Find m∠C and m∠F. ∠C ≅ ∠F m∠C = m∠F y 2 = 3y 2 - 72 Third  Thm. Def. of ≅ . Substitute y 2 for m∠C and 3 y 2 - 72 for m∠F. Subtract 3 y 2 from both sides. Divide both sides by -2. -2y 2 = -72 y 2 = 36 So m∠C = 36°. Since m∠F = m∠C, m∠F = 36°. 4. Find m∠P and m∠T. THINK AND DISCUSS 1. Use the Triangle Sum Theorem to explain why the supplement of one of the angles of a triangle equals in measure the sum of the other two angles of the triangle. Support your answer with a sketch. 2. Sketch a triangle and draw all of its exterior angles. How many exterior angles are there at each vertex of the triangle? How many total exterior angles does the triangle have? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write each theorem in words and then draw a diagram to represent it. 226 226 Chapter 4 Triangle Congruence �� 4-2 Exercises Exercises KEYWORD: MG7 4-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each qu
estion. 1. To remember the meaning of remote interior angle, think of a television remote control. What is another way to remember the term remote? 2. An exterior angle is drawn at vertex E of △DEF. What are its remote interior angles? 3. What do you call segments, rays, or lines that are added to a given diagram. 224 Astronomy Use the following information for Exercises 4 and 5. An asterism is a group of stars that is easier to recognize than a constellation. One popular asterism is the Summer Triangle, which is composed of the stars Deneb, Altair, and Vega. 4. What is the value of y? 5. What is the measure of each angle in the Summer Triangle. 225 The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? 6. 20.8° 7 Find each angle measure. p. 225 9. m∠M ° 8. 24 2 _ 3 10. m∠L 11. In △ABC, m∠A = 65°, and the measure of an exterior angle at C is 117°. Find m∠B and the m∠BCA 12. m∠C and m∠F 13. m∠S and m∠U p. 226 14. For △ABC and △XYZ, m∠A = m∠X and m∠B = m∠Y. Find the measures of ∠C and ∠Z if m∠C = 4x + 7 and m∠Z = 3 (x + 5) . 4-2 Angle Relationships in Triangles 227 227 �� Independent Practice For See Exercises Example 15 16–18 19–20 21–22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S10 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 15. Navigation A sailor on ship A measures the angle between ship B and the pier and finds that it is 39°. A sailor on ship B measures the angle between ship A and the pier and finds that it is 57°. What is the measure of the angle between ships A and B? The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? ° 16. 76 1 _ 4 17. 2x° 18. 56.8° Find each angle measure. 19. m∠XYZ 20. m∠C 21. m∠N and m∠P 22. m∠Q and m∠S 23. Multi-Step The measures of the angles of a triangle are in the ratio 1 : 4 : 7. What are the measures of the angles? (Hint: Let x, 4x, and 7x represent the angle measures.) 24. Complete the proof of Corollary 4-2-2. Given: △DEF with right ∠F Prove: ∠D and ∠E are complementary. Proof: Statements Reasons 1. △DEF with rt. ∠F 2. b. ? ̶̶̶̶ 3. m∠D + m∠E + m∠F = 180° 4. m∠D + m∠E + 90° = 180° 5. e. ? ̶̶̶̶ 6. ∠D and ∠E are comp. 1. a. ? ̶̶̶̶ 2. Def. of rt. ∠ 3. c. ? ̶̶̶̶ ? ̶̶̶̶ 5. Subtr. Prop. 4. d. 6. f. ? ̶̶̶̶ 25. Prove Corollary 4-2-3 using two different methods of proof. Given: △ABC is equiangular. Prove: m∠A = m∠B = m∠C = 60° 26. Multi-Step The measure of one acute angle in a right triangle is 1 1 __ 4 times the measure of the other acute angle. What is the measure of the larger acute angle? 27. Write a two-column proof of the Third Angles Theorem. 228 228 Chapter 4 Triangle Congruence Ship AShip BPier57º39º�� 28. Prove the Exterior Angle Theorem. Given: △ABC with exterior angle ∠ACD Prove: m∠ACD = m∠A + m∠B (Hint: ∠BCA and ∠DCA form a linear pair.) Find each angle measure. 29. ∠UXW 31. ∠WZX 30. ∠UWY 32. ∠XYZ 33. Critical Thinking What is the measure of any exterior angle of an equiangular triangle? What is the sum of the exterior angle measures? 34. Find m∠SRQ, given that ∠P ≅ ∠U, ∠Q ≅ ∠T, and m∠RST = 37.5°. 35. Multi-Step In a right triangle, one acute angle measure is 4 times the other acute angle measure. What is the measure of the smaller angle? 36. Aviation To study the forces of lift and drag, the Wright brothers built a glider, attached two ropes to it, and flew it like a kite. They modeled the two wind forces as the legs of a right triangle. a. What part of a right triangle is formed by each rope? b. Use the Triangle Sum Theorem to write an equation relating the angle measures in the right triangle. c. Simplify the equation from part b. What is the relationship between x and y? d. Use the Exterior Angle Theorem to write an expression for z in terms of x. e. If x = 37°, use your results from parts c and d to find y and z. 37. Estimation Draw a triangle and two exterior angles at each vertex. Estimate the measure of each angle. How are the exterior angles at each vertex related? Explain. 38. Given: Prove: ̶̶ AB ⊥ ̶̶ AD ǁ ̶̶ BD , ̶̶ CB ̶̶ BD ⊥ ̶̶ DC , ∠A ≅ ∠C 39. Write About It A triangle has angle measures of 115°, 40°, and 25°. Explain how to find the measures of the triangle’s exterior angles. Support your answer with a sketch. 40. This problem will prepare you for the Multi-Step TAKS Prep on page 238. One of the steps in making an origami crane involves folding a square sheet of paper into the shape shown. a. ∠DCE is a right angle. ̶̶ FC bisects ∠DCE, and ̶̶ BC bisects ∠FCE. Find m∠FCB. b. Use the Triangle Sum Theorem to find m∠CBE. 4-2 Angle Relationships in Triangles 229 229 ��DragLiftRopezºyºxºge07sec04ll02005aABoehm�� 41. What is the value of x? 19 52 42. Find the value of s. 23 28 57 71 34 56 43. ∠A and ∠B are the remote interior angles of ∠BCD in △ABC. Which of these equations must be true? m∠A - 180° = m∠B m∠A = 90° - m∠B m∠BCD = m∠BCA - m∠A m∠B = m∠BCD - m∠A 44. Extended Response The measures of the angles in a triangle are in the ratio 2 : 3 : 4. Describe how to use algebra to find the measures of these angles. Then find the measure of each angle and classify the triangle. CHALLENGE AND EXTEND 45. An exterior angle of a triangle measures 117°. Its remote interior angles measure ( 2y 2 + 7) ° and (61 - y 2 ) °. Find the value of y. 46. Two parallel lines are intersected by a transversal. What type of triangle is formed by the intersection of the angle bisectors of two same-side interior angles? Explain. (Hint: Use geometry software or construct a diagram of the angle bisectors of two same-side interior angles.) 47. Critical Thinking Explain why an exterior angle of a triangle cannot be congruent to a remote interior angle. 48. Probability The measure of each angle in a triangle is a multiple of 30°. What is the probability that the triangle has at least two congruent angles? 49. In △ABC, m∠B is 5° less than 1 1 __ 2 times m∠A. m∠C is 5° less than 2 1 __ 2 times m∠A. What is m∠A in degrees? SPIRAL REVIEW Make a table to show the value of each function when x is -2, 0, 1, and 4. (Previous course) 52. f (x) = (x - 3) 2 + 5 50. f (x) = 3x - 4 51. f (x) = x 2 + 1 53. Find the length of ̶̶̶ NQ . Name the theorem or postulate that justifies your answer. (Lesson 2-7) Classify each triangle by its side lengths. (Lesson 4-1) 54. △ACD 55. △BCD 56. △ABD 57. What if…? If CA = 8, What is the effect on the classification of △ACD? 230 230 Chapter 4 Triangle Congruence �� 4-3 Congruent Triangles TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.2.B Objectives Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence. Who uses this? Machinists used triangles to construct a model of the International Space Station’s support structure. Vocabulary corresponding angles corresponding sides congruent polygons Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. Two polygons are congruent polygons if and only if their corresponding angles and sides are congruent. Thus triangles that are the same size and shape are congruent. Properties of Congruent Polygons DIAGRAM CORRESPONDING ANGLES CORRESPONDING SIDES Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices. △ABC ≅ △DEF polygon PQRS ≅ polygon WXYZ ∠A ≅ ∠D ∠B ≅ ∠E ∠C ≅ ∠F ∠P ≅ ∠W ∠Q ≅ ∠X ∠R ≅ ∠Y ∠S ≅ ∠ Z ̶̶ AB ≅ ̶̶ BC ≅ ̶̶ AC ≅ ̶̶ DE ̶̶ EF ̶̶ DF ̶̶ PQ ≅ ̶̶ QR ≅ ̶̶ RS ≅ ̶̶ PS ≅ ̶̶̶ WX ̶̶ XY ̶̶ YZ ̶̶ WZ To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS. In a congruence statement, the order of the vertices indicates the corresponding parts. E X A M P L E 1 Naming Congruent Corresponding Parts △RST and △XYZ represent the triangles of the space station’s support structure. If △RST ≅ △XYZ, identify all pairs of congruent corresponding parts. Angles: ∠R ≅ ∠X, ∠S ≅ ∠Y, ∠T ≅ ∠Z ̶̶ ST ≅ Sides: ̶̶ RT ≅ ̶̶ RS ≅ ̶̶ XY , ̶̶ YZ , ̶̶ XZ 1. If polygon LMNP ≅ polygon EFGH, identify all pairs of corresponding congruent parts. 4-3 Congruent Triangles 231 231 �� E X A M P L E 2 Using Corresponding Parts of Congruent Triangles Given: △EFH ≅ △GFH When you write a statement such as △ABC ≅ △DEF, you are also stating which parts are congruent. A Find the value of x. ∠FHE and ∠FHG are rt. . Def. of ⊥ lines ∠FHE ≅ ∠FHG m∠FHE = m∠FHG (6x - 12) ° = 90° 6x = 102 x = 7 B Find m∠GFH. Rt. ∠ ≅ Thm. Def. of ≅  Substitute values for m∠FHE and m∠FHG. Add 12 to both sides. Divide both sides by 6. m∠EFH + m∠FHE + m∠E = 180° △ Sum Thm. m∠EFH + 90 + 21.6 = 180 Substitute values for m∠FHE m∠EFH + 111.6 = 180 m∠EFH = 68.4 and m∠E. Simplify. Subtract 111.6 from both sides. ∠GFH ≅ ∠EFH m∠GFH = m∠EFH m∠GFH = 68.4° Corr.  of ≅  are ≅. Def. of ≅  Trans. Prop. of = Given: △ABC ≅ △DEF 2a. Find the value of x. 2b. Find m∠F. E X A M P L E 3 Proving Triangles Congruent Given: ∠P and ∠M are right angles. R is the midpoint of ̶̶ ̶̶ NR PQ ≅ ̶̶ PM . ̶̶̶ MN , ̶̶ QR ≅ Prove: △PQR ≅ △MNR Proof: Statements Reasons 1. ∠P and ∠M are rt.  1. Given 2. ∠P ≅ ∠M 3. ∠PRQ ≅ ∠MRN 4. ∠Q ≅ ∠N ̶̶̶ PM . 6. 5. R is the mdpt. of ̶̶̶ MR ̶̶̶ MN ; ̶̶ PR ≅ ̶̶ PQ ≅ ̶̶ QR ≅ 7. ̶̶ NR 2. Rt. ∠ ≅ Thm. 3. Vert.  Thm. 4. Third  Thm. 5. Given 6. Def. of mdpt. 7. Given 8. △PQR ≅ △MNR 8. Def. of ≅  3. Given: ̶̶ AD bisects ̶̶ BE bisects ̶̶ AB ≅ Prove: △ABC ≅ △DEC ̶̶ BE . ̶̶ AD . ̶̶ DE , ∠A ≅ ∠D 232 232 Chapter 4 Triangle Congruence ��
�� Overlapping Triangles “With overlapping triangles, it helps me to redraw the triangles separately. That way I can mark what I know about one triangle without getting confused by the other one.” Cecelia Medina Lamar High School E X A M P L E 4 Engineering Application Engineering The Rattler is one of the tallest wood-tracked roller coasters in the world. The ride sits on the side of a cliff at Six Flags Fiesta Texas. It is 5080 ft long and 179 ft high. The coaster travels at a speed of 65 mi/h. The bars that give structural support to a roller coaster form triangles. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. ̶̶ ML ⊥ Given: ̶̶ JL ≅ Prove: △JKL ≅ △MLK Proof: ̶̶ KL , ∠KLJ ≅ ∠LKM, ̶̶ MK ̶̶ JK ⊥ ̶̶ JK ≅ ̶̶ KL , ̶̶ ML , ̶̶ JK ⊥ ̶̶ KL , 1. Statements ̶̶̶ ML ⊥ ̶̶ KL Reasons 1. Given 2. ∠JKL and ∠MLK are rt. . 2. Def. of ⊥ lines 3. ∠JKL ≅ ∠MLK 4. ∠KLJ ≅ ∠LKM 5. ∠KJL ≅ ∠LMK ̶̶ JK ≅ ̶̶ KL ≅ ̶̶̶ ML , ̶̶ LK 6. 7. ̶̶ JL ≅ ̶̶̶ MK 8. △JKL ≅ △MLK 3. Rt. ∠ ≅ Thm. 4. Given 5. Third  Thm. 6. Given 7. Reflex. Prop. of ≅ 8. Def. of ≅  4. Use the diagram to prove the following. ̶̶ JK ≅ ̶̶̶ MK bisects ̶̶ JL bisects ̶̶ JL . Given: Prove: △JKN ≅ △LMN ̶̶̶ MK . ̶̶̶ ML , ̶̶ JK ǁ ̶̶̶ ML THINK AND DISCUSS 1. A roof truss is a triangular structure that supports a roof. How can you be sure that two roof trusses are the same size and shape? 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, name the congruent corresponding parts. 4-3 Congruent Triangles 233 233 �� 4-3 Exercises Exercises KEYWORD: MG7 4-3 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An everyday meaning of corresponding is “matching.” How can this help you find the corresponding parts of two triangles? 2. If △ABC ≅ △RST, what angle corresponds to ∠S. 231 Given: △RST ≅ △LMN. Identify the congruent corresponding parts. ̶̶ RS ≅ ̶̶ TS ≅ 3. 6. ? ̶̶̶̶ ? ̶̶̶̶ ̶̶ LN ≅ 4. 7. ∠L ≅ ? ̶̶̶̶ ? ̶̶̶̶ 5. ∠S ≅ 8. ∠N ≅ ? ̶̶̶̶ ? ̶̶̶̶ Given: △FGH ≅ △JKL. Find each value. p. 232 9. KL 10. 232 . 233 11. Given: E is the midpoint of ̶̶ AB ≅ ̶̶ CD , ̶̶ AB ǁ ̶̶ CD ̶̶ AC and ̶̶ BD . Prove: △ABE ≅ △CDE Proof: Statements Reasons 1. a. 2. b. ? ̶̶̶̶ ? ̶̶̶̶ ? ̶̶̶̶ ? ̶̶̶̶ 5. Def. of mdpt. 4. d. 3. c. 6. f. 7. g. ? ̶̶̶̶ ? ̶̶̶̶ ̶̶ AB ǁ ̶̶ CD 1. ̶̶ AB ≅ ̶̶ CD 3. 2. ∠ABE ≅ ∠CDE, ∠BAE ≅ ∠DCE 4. E is the mdpt. of ̶̶ AC and ̶̶ BD . 5. e. ? ̶̶̶̶ 6. ∠AEB ≅ ∠CED 7. △ABE ≅ △CDE 12. Engineering The McDonald Observatory has four research telescopes and is a leading center for astronomical study. Prove that the triangles that make up the observatory dome are congruent. Given: ̶̶ ̶̶ ̶̶ SU ≅ SR , ST ≅ ∠UST ≅ ∠RST, and ∠U ≅ ∠R ̶̶ TU ≅ ̶̶ TR , Prove: △RTS ≅ △UTS 234 234 Chapter 4 Triangle Congruence �� Independent Practice Given: Polygon CDEF ≅ polygon KLMN. Identify the congruent corresponding parts. PRACTICE AND PROBLEM SOLVING 13. ̶̶ DE ≅ 15. ∠F ≅ ? ̶̶̶̶ ? ̶̶̶̶ 14. ̶̶ KN ≅ 16. ∠L ≅ ? ̶̶̶̶ ? ̶̶̶̶ For See Exercises Example 13–16 17–18 19 20 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S10 Application Practice p. S31 Given: △ABD ≅ △CBD. Find each value. 17. m∠C 18. y 19. Given: ̶̶̶ MP bisects ∠NMR. P is the midpoint of ̶̶ NR . ̶̶̶ MR , ∠N ≅ ∠R ̶̶̶ MN ≅ Prove: △MNP ≅ △MRP Proof: Statements Reasons 1. ∠N ≅ ∠R ̶̶̶ MP bisects ∠NMR. 2. 3. c. 4. d. ? ̶̶̶̶ ? ̶̶̶̶ 5. P is the mdpt. of ̶̶ NR . 6. f. ? ̶̶̶̶ ̶̶̶ MN ≅ ̶̶̶ MP ≅ ̶̶̶ MR ̶̶̶ MP 7. 8. 9. △MNP ≅ △MRP 1. a. 2. b. ? ̶̶̶̶ ? ̶̶̶̶ 3. Def. of ∠ bisector 4. Third  Thm. 5. e. ? ̶̶̶̶ 6. Def. of mdpt. 7. g. 8. h. ? ̶̶̶̶ ? ̶̶̶̶ 9. Def. of ≅  20. Hobbies In a garden, triangular flower beds are separated by straight rows of grass as shown. Given: ∠ADC and ∠BCD are right angles. ̶̶ AD ≅ ̶̶ BC ̶̶ ̶̶ AC ≅ BD , ∠DAC ≅ ∠CBD Prove: △ADC ≅ △BCD 21. For two triangles, the following corresponding parts are given: ̶̶ ̶̶ ̶̶ PH , KH , GS ≅ ∠S ≅ ∠P, ∠G ≅ ∠K, and ∠R ≅ ∠H. Write three different congruence statements. ̶̶ GR ≅ ̶̶ SR ≅ ̶̶ KP , 22. The two polygons in the diagram are congruent. Complete the following congruence statement for the polygons. polygon R ? ≅ polygon V ̶̶̶̶ ? ̶̶̶̶ Write and solve an equation for each of the following. 23. △ABC ≅ △DEF. AB = 2x - 10, and DE = x + 20. Find the value of x and AB. 24. △JKL ≅ △MNP. m∠L = ( x 2 + 10) °, and m∠P = ( 2x 2 + 1) °. What is m∠L? 25. Polygon ABCD ≅ polygon PQRS. BC = 6x + 5, and QR = 5x + 7. Find the value of x and BC. 4-3 Congruent Triangles 235 235 ��ge07se_c04l03005aAEBDC�� 26. This problem will prepare you for the Multi-Step TAKS Prep on page 238. ̶̶ JL and Many origami models begin with a square piece of paper, JKLM, that is folded along both diagonals to make the ̶̶̶ MK are perpendicular bisectors creases shown. of each other, and ∠NML ≅ ∠NKL. ̶̶ a. Explain how you know that KL and b. Prove △NML ≅ △NKL. ̶̶̶ ML are congruent. 27. Draw a diagram and then write a proof. ̶̶ BD ⊥ ̶̶ AC . D is the midpoint of ̶̶ AC . ̶̶ AB ≅ ̶̶ CB , and ̶̶ BD bisects ∠ABC. Given: Prove: △ABD ≅ △CBD 28. Critical Thinking Draw two triangles that are not congruent but have an area of 4 cm 2 each. 29. /////ERROR ANALYSIS///// Given △MPQ ≅ △EDF. Two solutions for finding m∠E are shown. Which is incorrect? Explain the error. 30. Write About It Given the diagram of the triangles, is there enough information to prove that △HKL is congruent to △YWX? Explain. 31. Which congruence statement correctly indicates that the two given triangles are congruent? △ABC ≅ △DEF △ABC ≅ △FED △ABC ≅ △EFD △ABC ≅ △FDE 32. △MNP ≅ △RST. What are the values of x and y? x = 26, y = 21 1 _ 3 x = 27, y = 20 x = 25, y = 20 2 _ 3 , y = 16 2 _ x = 30 1 _ 3 3 33. △ABC ≅ △XYZ. m∠A = 47.1°, and m∠C = 13.8°. Find m∠Y. 13.8 42.9 76.2 119.1 34. △MNR ≅ △SPQ, NL = 18, SP = 33, SR = 10, RQ = 24, and QP = 30. What is the perimeter of △MNR? 79 85 87 97 236 236 Chapter 4 Triangle Congruence �� CHALLENGE AND EXTEND 35. Multi-Step Given that the perimeter of TUVW is 149 units, find the value of x. Is △TUV ≅ △TWV? Explain. 36. Multi-Step Polygon ABCD ≅ polygon EFGH. ∠A is a right angle. m∠E = ( y 2 - 10) °, and m∠H = ( 2y 2 - 132) °. Find m∠D. 37. Given: ̶̶ RS ≅ ̶̶ RT , ∠S ≅ ∠T Prove: △RST ≅ △RTS SPIRAL REVIEW Two number cubes are rolled. Find the probability of each outcome. (Previous course) 38. Both numbers rolled are even. 39. The sum of the numbers rolled is 5. Classify each angle by its measure. (Lesson 1-3) 40. m∠DOC = 40° 41. m∠BOA = 90° 42. m∠COA = 140° Find each angle measure. (Lesson 4-2) 43. ∠Q 44. ∠P 45. ∠QRS KEYWORD: MG7 Career Q: What math classes did you take in high school? A: Algebra 1 and 2, Geometry, Precalculus Q: What kind of degree or certification will you receive? A: I will receive an associate’s degree in applied science. Then I will take an exam to be certified as an EMT or paramedic. Q: How do you use math in your hands-on training? A: I calculate dosages based on body weight and age. I also calculate drug doses in milligrams per kilogram per hour or set up an IV drip to deliver medications at the correct rate. Q: What are your future career plans? A: When I am certified, I can work for a private ambulance service or with a fire department. I could also work in a hospital, transporting critically ill patients by ambulance or helicopter. 4-3 Congruent Triangles 237 237 Jordan Carter Emergency Medical Services Program �� SECTION 4A Triangles and Congruence Origami Origami is the Japanese art of paper folding. The Japanese word origami literally means “fold paper.” This ancient art form relies on properties of geometry to produce fascinating and beautiful shapes. Each of the figures shows a step in making an origami swan from a square piece of paper. The final figure shows the creases of an origami swan that has been unfolded. Step 1 Step 2 � Step 3 � � � � � � � � � � � � Fold the paper in half diagonally and crease it. Turn it over. Fold corners A and C to the center line and crease. Turn it over. � Step 4 Step 5 � � �� Fold in half along the center crease so that and ̶̶ DF are together. ̶̶ DE Step 6 � �� Fold the narrow point upward at a 90° angle and crease. Push in the fold so that the neck is inside the body. Fold the tip downward and crease. Push in the fold so that the head is inside the neck. Fold up the flap to form the wing. � 1. Use the fact that ABCD is a square � 2. to classify △ABD by its side lengths and by its angle measures. ̶̶ ̶̶ DB bisects ∠ABC and ∠ADC. DE bisects ∠ADB. Find the measures of the angles in △EDB. Explain how you found the measures. ̶̶ 3. Given that DB bisects ∠ABC and ̶̶ ∠EDF, DE ≅ prove that △EDB ≅ △ FDB. ̶̶ BF , and ̶̶ BE ≅ ̶̶ DF , � � � � � � 238 238 Chapter 4 Triangle Congruence SECTION 4A Quiz for Lessons 4-1 Through 4-3 4-1 Classifying Triangles Classify each triangle by its angle measures. 1. △ACD 2. △ABD 3. △ADE Classify each triangle by its side lengths. 4. △PQR 5. △PRS 6. △PQS 4-2 Angle Relationships in Triangles Find each angle measure. 7. m∠M 8. m∠ABC 9. A carpenter built a triangular support structure for a roof. Two of the angles of the structure measure 37° and 55°. Find the measure of ∠RTP, the angle formed by the roof of the house and the roof of the patio. 4-3 Congruent Triangles Given: △JKL ≅ △DEF. Identify the congruent corresponding parts. 10. ̶̶ KL ≅ ? ̶̶̶̶ 11. ̶̶ DF ≅ ? ̶̶̶̶ 12. ∠K ≅ ? ̶̶̶̶ 13. ∠F ≅ ? ̶̶̶̶ Given: △PQR ≅ △STU. Find each value. 14. PQ 15. y 16. Given:   AB ǁ   CD , ̶̶ CD , ̶̶ AB ≅ ̶̶ ̶̶ DB ⊥ AC ⊥ Prove: △ACD ≅ △DBA ̶̶ CD , ̶̶ AB ̶̶ AC ≅ ̶̶ BD , Proof: Statements Reasons 1.   AB ǁ   CD 2. ∠BAD
≅ ∠CDA ̶̶ AC ⊥ ̶̶ CD , ̶̶ DB ⊥ ̶̶ AB 3. 4. ∠ACD and ∠DBA are rt.  5. e. ? ̶̶̶̶̶ ? ̶̶̶̶̶ ̶̶ CD , 6. f. ̶̶ AB ≅ 7. ̶̶ AC ≅ ̶̶ BD 8. h. ? ̶̶̶̶̶ 9. △ACD ≅ △DBA 1. a. 2. b. 3. c. 4. d. ? ̶̶̶̶̶ ? ̶̶̶̶̶ ? ̶̶̶̶̶ ? ̶̶̶̶̶ 5. Rt. ∠ ≅ Thm. 6. Third  Thm. 7. g. ? ̶̶̶̶̶ 8. Reflex Prop. of ≅ 9. i . ? ̶̶̶̶̶ Ready to Go On? 239 239 �� 4-4 Explore SSS and SAS Triangle Congruence Use with Lesson 4-4 In Lesson 4-3, you used the definition of congruent triangles to prove triangles congruent. To use the definition, you need to prove that all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. In this lab, you will discover some shortcuts for proving triangles congruent. Activity 1 TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about the properties and attributes of polygons … based on explorations. Also G.4.A G.10.B 1 Measure and cut six pieces from the straws: two that are 2 inches long, two that are 4 inches long, and two that are 5 inches long. 2 Cut two pieces of string that are each about 20 inches long. 3 Thread one piece of each size of straw onto a piece of string. Tie the ends of the string together so that the pieces of straw form a triangle. 4 Using the remaining pieces, try to make another triangle with the same side lengths that is not congruent to the first triangle. Try This 1. Repeat Activity 1 using side lengths of your choice. Are your results the same? 2. Do you think it is possible to make two triangles that have the same side lengths but that are not congruent? Why or why not? 3. How does your answer to Problem 2 provide a shortcut for proving triangles congruent? 4. Complete the following conjecture based on your results. Two triangles are congruent if ? . ̶̶̶̶̶̶̶̶̶̶̶̶̶ 240 240 Chapter 4 Triangle Congruence Activity 2 1 Measure and cut two pieces from the straws: one that is 4 inches long and one that is 5 inches long. 2 Use a protractor to help you bend a paper clip to form a 30° angle. 3 Place the pieces of straw on the sides of the 30° angle. The straws will form two sides of your triangle. 4 Without changing the angle formed by the paper clip, use a piece of straw to make a third side for your triangle, cutting it to fit as necessary. Use additional paper clips or string to hold the straws together in a triangle. Try This 5. Repeat Activity 2 using side lengths and an angle measure of your choice. Are your results the same? 6. Suppose you know two side lengths of a triangle and the measure of the angle between these sides. Can the length of the third side be any measure? Explain. 7. How does your answer to Problem 6 provide a shortcut for proving triangles congruent? 8. Use the two given sides and the given angle from Activity 2 to form a triangle that is not congruent to the triangle you formed. (Hint: One of the given sides does not have to be adjacent to the given angle.) 9. Complete the following conjecture based on your results. Two triangles are congruent if ? . ̶̶̶̶̶̶̶̶̶̶̶̶̶ 4- 4 Geometry Lab 241 241 4-4 Triangle Congruence: SSS and SAS TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.2.A, G.3.B, G.3.E Objectives Apply SSS and SAS to construct triangles and to solve problems. Prove triangles congruent by using SSS and SAS. Vocabulary triangle rigidity included angle Who uses this? Engineers used the property of triangle rigidity to design the internal support for the Statue of Liberty and to build bridges, towers, and other structures. (See Example 2.) In Lesson 4-3, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate. Postulate 4-4-1 Side-Side-Side (SSS) Congruence POSTULATE HYPOTHESIS CONCLUSION If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. △ABC ≅ △FDE E X A M P L E 1 Using SSS to Prove Triangle Congruence Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts. Use SSS to explain why △PQR ≅ △PSR. ̶̶ ̶̶ SR . By QR ≅ It is given that ̶̶ PR ≅ the Reflexive Property of Congruence, Therefore △PQR ≅ △PSR by SSS. ̶̶ PS and that ̶̶ PQ ≅ ̶̶ PR . 1. Use SSS to explain why △ABC ≅ △CDA. An included angle is an angle formed by two adjacent sides of a polygon. ∠B is the included angle between sides ̶̶ AB and ̶̶ BC . 242 242 Chapter 4 Triangle Congruence �� It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent. Postulate 4-4-2 Side-Angle-Side (SAS) Congruence POSTULATE HYPOTHESIS CONCLUSION If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. △ABC ≅ △EFD E X A M P L E 2 Engineering Application The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides. The figure shows part of the support structure of the Statue of Liberty. Use SAS to explain why △KPN ≅ △LPM. It is given that ̶̶ NP ≅ and that By the Vertical Angles Theorem, ∠KPN ≅ ∠LPM. Therefore △KPN ≅ △LPM by SAS. ̶̶ KP ≅ ̶̶̶ MP . ̶̶ LP 2. Use SAS to explain why △ABC ≅ △DBC. The SAS Postulate guarantees that if you are given the lengths of two sides and the measure of the included angle, you can construct one and only one triangle. Construction Congruent Triangles Using SAS Use a straightedge to draw two segments and one angle, or copy the given segments and angle.    ̶̶ AB congruent to one Construct of the segments. Construct ∠A congruent to the given angle. ̶̶ Construct AC congruent to the other segment. Draw to complete △ABC. ̶̶ CB 4-4 Triangle Congruence: SSS and SAS 243 243 KPMNLge07se_c04l04009a3rd pass4/25/5cmurphy�� E X A M P L E 3 Verifying Triangle Congruence Show that the triangles are congruent for the given value of the variable. A △UVW ≅ △YXW, x = 3 ZY = x - 1 = 3 - 1 = 2 XZ = x = 3 XY = 3x - 5 = 3 (3) - 5 = 4 ̶̶ ̶̶̶ ̶̶ UV ≅ XZ , and VW ≅ So △UVW ≅ △YXZ by SSS. ̶̶ YX . ̶̶̶ UW ≅ ̶̶ YZ . B △DEF ≅ △JGH, y = 7 JG = 2y + 1 = 2 (7) + 1 = 15 GH = y 2 - 4y + 3 = (7) 2 - 4 (7) + 3 = 24 m∠G = 12y + 42 = 12 (7) + 42 = 126° ̶̶ ̶̶ ̶̶ DE ≅ EF ≅ JG . So △DEF ≅ △JGH by SAS. ̶̶̶ GH , and ∠E ≅ ∠G. 3. △ADB ≅ △CDB Proving Triangles Congruent ̶̶ EG ≅ ̶̶ Given: ℓ ǁ m, HF Prove: △EGF ≅ △HFG Proof: Statements Reasons ̶̶ EG ≅ ̶̶ HF 1. 2. ℓ ǁ m 3. ∠EGF ≅ ∠HFG ̶̶ FG ≅ ̶̶ FG 4. 1. Given 2. Given 3. Alt. Int.  Thm. 4. Reflex Prop. of ≅ 5. △EGF ≅ △HFG 5. SAS Steps 1, 3, 4 4. Given: Prove: △RQP ≅ △SQP  QP bisects ∠RQS. ̶̶ QR ≅ ̶̶ QS 244 244 Chapter 4 Triangle Congruence �� THINK AND DISCUSS 1. Describe three ways you could prove that △ABC ≅ △DEF. 2. Explain why the SSS and SAS Postulates are shortcuts for proving triangles congruent. 3. GET ORGANIZED Copy and complete the graphic organizer. Use it to compare the SSS and SAS postulates. 4-4 Exercises Exercises KEYWORD: MG7 4-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In △RST which angle is the included angle of sides ̶̶ ST and ̶̶ TR ? Use SSS to explain why the triangles in each pair are congruent. p. 242 2. △ABD ≅ △CDB 3. △MNP ≅ △MQP . Design This Texas flag consists of a blue, p. 243 perpendicular stripe with a white star in the center. The star consists of five triangles. GJ = LG = 20 in., and GK = GH = 13 in. Use SAS to explain why △JGK ≅ △LGH Show that the triangles are congruent for the given value of the variable. p. 244 5. △GHJ ≅ △IHJ, x = 4 6. △RST ≅ △TUR, x = 18 4-4 Triangle Congruence: SSS and SAS 245 245 �� . Given: ̶̶ JK ≅ ̶̶̶ ML , ∠JKL ≅ ∠MLK p. 244 Prove: △JKL ≅ △MLK Proof: Statements Reasons ̶̶ JK ≅ ̶̶̶ ML 1. 2. b. ̶̶ KL ≅ ? ̶̶̶̶ ̶̶ LK 3. 4. △JKL ≅ △MLK 1. a. ? ̶̶̶̶ 2. Given 3. c. 4. d. ? ̶̶̶̶ ? ̶̶̶̶ Independent Practice Use SSS to explain why the triangles in each pair are congruent. PRACTICE AND PROBLEM SOLVING For See Exercises Example 8–9 10 11–12 13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 8. △BCD ≅ △EDC 9. △GJK ≅ △GJL 10. Theater The lights shining on a stage appear to form two congruent right triangles. Given △ECB ≅ △DBC. ̶̶ DB , use SAS to explain why ̶̶ EC ≅ Show that the triangles are congruent for the given value of the variable. 11. △MNP ≅ △QNP, y = 3 12. △XYZ ≅ △STU, t = 5 13. Given: B is the midpoint of ̶̶ DC . ̶̶ AB ⊥ ̶̶ DC Prove: △ABD ≅ △ABC Proof: Statements ̶̶ DC . 1. B is the mdpt. of 2. b. 3. c. ? ̶̶̶̶ ? ̶̶̶̶ 4. ∠ABD and ∠ABC are rt. . 5. ∠ABD ≅ ∠ABC 6. f. ? ̶̶̶̶ 7. △ABD ≅ △ABC 246 246 Chapter 4 Triangle Congruence Reasons 1. a. ? ̶̶̶̶ 2. Def. of mdpt. 3. Given 4. d. 5. e. ? ̶̶̶̶ ? ̶̶̶̶ 6. Reflex. Prop. of ≅ 7. g. ? ̶̶̶̶ �� Which postulate, if any, can be used to prove the triangles congruent? 14. 16. 15. 17. 18. Explain what additional information, if any, you would need to prove △ABC ≅ △DEC by each postulate. b. SAS a. SSS Multi-Step Graph each triangle. Then use the Distance Formula and the SSS Postulate to determine whether the triangles are congruent. 19. △QRS and △TUV 20. △ABC and △DEF Q (-2, 0) , R (1, -2) , S (-3, -2) T (5, 1) , U (3, -2) , V (3, 2) A
(2, 3) , B (3, -1) , C (7, 2) D (-3, 1) , E (1, 2) , F (-3, 5) 21. Given: ∠ZVY ≅ ∠WYV, ∠ZVW ≅ ∠WYZ, ̶̶ ̶̶̶ YZ VW ≅ Prove: △ZVY ≅ △WYV Proof: Statements Reasons 1. ∠ZVY ≅ ∠WYV, ∠ZVW ≅ WYZ 2. m∠ZVY = m∠WYV, m∠ZVW = m∠WYZ 3. m∠ZVY + m∠ZVW = m∠WYV + m∠WYZ 4. c. ? ̶̶̶̶ 5. ∠WVY ≅ ∠ZYV ̶̶̶ VW ≅ ̶̶ YZ 6. 1. a. 2. b. ? ̶̶̶̶ ? ̶̶̶̶ 3. Add. Prop. of = 4. ∠ Add. Post. 5. d. 6. e. ? ̶̶̶̶ ? ̶̶̶̶ 7. f. ? ̶̶̶̶ 8. △ZVY ≅ △WYV 7. Reflex. Prop. of ≅ 8. g. ? ̶̶̶̶ 22. This problem will prepare you for the Multi-Step TAKS Prep on page 280. The diagram shows two triangular trusses that were built for the roof of a doghouse. a. You can use a protractor to check that ∠A and ∠D are right angles. Explain how you could make just two additional measurements on each truss to ensure that the trusses are congruent. b. You verify that the trusses are congruent and find ̶̶ EF to the that AB = AC = 2.5 ft. Find the length of nearest tenth. Explain. 4-4 Triangle Congruence: SSS and SAS 247 247 �� 23. Critical Thinking Draw two isosceles triangles that are not congruent but that have a perimeter of 15 cm each. Ecology 24. △ABC ≅ △ADC for what value of x? Explain why the SSS Postulate can be used to prove the two triangles congruent. 25. Ecology A wing deflector is a triangular structure made of logs that is filled with large rocks and placed in a stream to guide the current or prevent erosion. Wing deflectors are often used in pairs. Suppose an engineer wants to build two wing deflectors. The logs that form the sides of each wing deflector are perpendicular. How can the engineer make sure that the two wing deflectors are congruent? Wing deflectors are designed to reduce the width-to-depth ratio of a stream. Reducing the width increases the velocity of the stream. 26. Write About It If you use the same two sides and included angle to repeat the construction of a triangle, are your two constructed triangles congruent? Explain. 27. Construction Use three segments (SSS) to construct a scalene triangle. Suppose you then use the same segments in a different order to construct a second triangle. Will the result be the same? Explain. 28. Which of the three triangles below can be proven congruent by SSS or SAS? I and II II and III I and III I, II, and III 29. What is the perimeter of polygon ABCD? 29.9 cm 39.8 cm 49.8 cm 59.8 cm 30. Jacob wants to prove that △FGH ≅ △JKL using SAS. ̶̶ ̶̶ JL . What additional JK and ̶̶ FH ≅ ̶̶ He knows that FG ≅ piece of information does he need? ∠H ≅ ∠L ∠F ≅ ∠G ∠F ≅ ∠J ∠G ≅ ∠K 31. What must the value of x be in order to prove that △EFG ≅ △EHG by SSS? 1.5 4.25 4.67 5.5 248 248 Chapter 4 Triangle Congruence �� CHALLENGE AND EXTEND 32. Given:. ∠ADC and ∠BCD are ̶̶ AD ≅ supplementary. ̶̶ CB Prove: △ADB ≅ △CBD (Hint: Draw an auxiliary line.) 33. Given: ∠QPS ≅ ∠TPR, ̶̶ PQ ≅ ̶̶ PT , ̶̶ PR ≅ ̶̶ PS Prove: △PQR ≅ △PTS Algebra Use the following information for Exercises 34 and 35. Find the value of x. Then use SSS or SAS to write a paragraph proof showing that two of the triangles are congruent. 34. m∠FKJ = 2x° m∠KFJ = (3x + 10) ° KJ = 4x + 8 HJ = 6 (x - 4) 35. ̶̶ FJ bisects ∠KFH. m∠KFJ = (2x + 6) ° m∠HFJ = (3x - 21) ° FK = 8x - 45 FH = 6x + 9 SPIRAL REVIEW Solve and graph each inequality. (Previous course) 36. x _ 2 37. 2a + 4 > 3a - 8 ≤ 5 38. -6m - 1 ≤ -13 Solve each equation. Write a justification for each step. (Lesson 2-5) 40. a _ 4 39. 4x - 7 = 21 + 5 = -8 41. 6r = 4r + 10 Given: △EFG ≅ △GHE. Find each value. (Lesson 4-3) 42. x 43. m∠FEG 44. m∠FGH Using Technology Use geometry software to complete the following. 1. Draw a triangle and label the vertices A, B, and C. Draw a point and label it D. Mark a vector from A to B and translate D by the marked vector. Label the image E. Draw   DE . Mark ∠BAC and rotate   DE about D by the marked angle. Mark ∠ABC and rotate   DE about E by the marked angle. Label the intersection F. 2. Drag A, B, and C to different locations. What do you notice about the two triangles? 3. Write a conjecture about △ABC and △DEF. 4. Test your conjecture by measuring the sides and angles of △ABC and △DEF. 4-4 Triangle Congruence: SSS and SAS 249 249 �� 4-5 Use with Lesson 4-5 Activity 1 Predict Other Triangle Congruence Relationships Geometry software can help you investigate whether certain combinations of triangle parts will make only one triangle. If a combination makes only one triangle, then this arrangement can be used to prove two triangles congruent. TEKS G.9.B Congruence and the geometry of size: formulate … conjectures about the properties and attributes of polygons … based on explorations. Also G.2.A, G.3.B, G.10.B 1 Construct ∠CAB measuring 45° and ∠EDF measuring 110°. 2 Move ∠EDF so that DE overlays AC intersect, label the BA . DF and Where point G. Measure ∠DGA. 3 Move ∠CAB to the left and right without changing the measures of the angles. Observe what happens to the size of ∠DGA. 4 Measure the distance from A to D. Try to change the shape of the triangle without changing AD and the measures of ∠A and ∠D. Try This 1. Repeat Activity 1 using angle measures of your choice. Are your results the same? Explain. 2. Do the results change if one of the given angles measures 90°? 3. What theorem proves that the measure of ∠DGA in Step 2 will always be the same? 4. In Step 3 of the activity, the angle measures in △ADG stayed the same as the size of the triangle changed. Does Angle-Angle-Angle, like Side-Side-Side, make only one triangle? Explain. 5. Repeat Step 4 of the activity but measure the length of ̶̶ AG instead of ̶̶ AD . Are your results the same? Does this lead to a new congruence postulate or theorem? 6. If you are given two angles of a triangle, what additional piece of information is needed so that only one triangle is made? Make a conjecture based on your findings in Step 5. 250 250 Chapter 4 Triangle Congruence Activity 2 1 Construct ̶̶ YZ with a length of 6.5 cm. 2 Using ̶̶ YZ as a side, construct ∠XYZ measuring 43°. 3 Draw a circle at Z with a radius of 5 cm. Construct ̶̶̶ ZW , a radius of circle Z. 4 Move W around circle Z. Observe the possible shapes of △YZW. Try This 7. In Step 4 of the activity, how many different triangles were possible? Does Side-Side-Angle make only one triangle? 8. Repeat Activity 2 using an angle measure of 90° in Step 2 and a circle with a radius of 7 cm in Step 3. How many different triangles are possible in Step 4? 9. Repeat the activity again using a measure of 90° in Step 2 and a circle with a radius of 4.25 cm in Step 3. Classify the resulting triangle by its angle measures. 10. Based on your results, complete the following conjecture. In a Side-Side-Angle combination, if the corresponding nonincluded angles are triangle is possible. ? , then only one ̶̶̶̶ 4- 5 Technology Lab 251 251 4-5 Triangle Congruence: ASA, AAS, and HL TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.1.A, G.1.B, G.2.A, G.3.B, G.3.C, G.3.E, G.9.B Objectives Apply ASA, AAS, and HL to construct triangles and to solve problems. Prove triangles congruent by using ASA, AAS, and HL. Vocabulary included side Why use this? Bearings are used to convey direction, helping people find their way to specific locations. Participants in an orienteering race use a map and a compass to find their way to checkpoints along an unfamiliar course. Directions are given by bearings, which are based on compass headings. For example, to travel along the bearing S 43° E, you face south and then turn 43° to the east. An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side. ̶̶ PQ is the included side of ∠P and ∠Q. Postulate 4-5-1 Angle-Side-Angle (ASA) Congruence POSTULATE HYPOTHESIS CONCLUSION If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. △ABC ≅ △DEF E X A M P L E 1 Problem-Solving Application Organizers of an orienteering race are planning a course with checkpoints A, B, and C. Does the table give enough information to determine the location of the checkpoints? Understand the Problem Bearing Distance A to B N 55° E 7.6 km B to C N 26° W C to A S 20° W The answer is whether the information in the table can be used to find the position of checkpoints A, B, and C. List the important information: The bearing from A to B is N 55° E. From B to C is N 26° W, and from C to A is S 20° W. The distance from A to B is 7.6 km. 252 252 Chapter 4 Triangle Congruence ��1 Make a Plan Draw the course using vertical lines to show north-south directions. Then use these parallel lines and the alternate interior angles to help find angle measures of △ABC. Solve m∠CAB = 55° - 20° = 35° m∠CBA = 180° - (26° + 55°) = 99° You know the measures of ∠CAB and ∠CBA and the length of the included side ̶̶ AB . Therefore by ASA, a unique triangle ABC is determined. Look Back One and only one triangle can be made using the information in the table, so the table does give enough information to determine the location of all the checkpoints. 1. What if...? If 7.6 km is the distance from B to C, is there enough information to determine the location of all the checkpoints? Explain. E X A M P L E 2 Applying ASA Congruence Determine if you can use ASA to prove △UVX ≅ △WVX. Explain. ∠UXV ≅ ∠WXV as given. Since ∠WVX is a right angle that forms a linear pair with ∠UVX, ∠WVX ≅ ∠UVX. Also by the Reflexive Property. Therefore △UVX ≅ △WVX by ASA. ̶̶ VX ≅ ̶̶ VX 2. Determine if you can use ASA to prove △NKL ≅ △LMN. Explain. Construction Congruent Triangles Using ASA Use a straightedge to draw a segment and two angles, or copy the given segment and angles.     ̶̶ CD congruent to Construct the given segment. Construct ∠C congruent to one of the angle
s. Construct ∠D congruent to the other angle. Label the intersection of the rays as E. △CDE 4-5 Triangle Congruence: ASA, AAS, and HL 253 253 2��34�� You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS). Theorem 4-5-2 Angle-Angle-Side (AAS) Congruence THEOREM HYPOTHESIS CONCLUSION If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. △GHJ ≅ △KLM PROOF PROOF Angle-Angle-Side Congruence ̶̶̶ LM ̶̶ HJ ≅ Given: ∠G ≅ ∠K, ∠J ≅ ∠M, Prove: △GHJ ≅ △KLM Proof: Statements Reasons 1. ∠G ≅ ∠K, ∠J ≅ ∠M 1. Given 2. ∠H ≅ ∠L ̶̶̶ LM ̶̶ HJ ≅ 3. 2. Third  Thm. 3. Given 4. △GHJ ≅ △KLM 4. ASA Steps 1, 3, and 2 E X A M P L E 3 Using AAS to Prove Triangles Congruent ̶̶ AB ǁ Use AAS to prove the triangles congruent. ̶̶ BC ≅ Given: Prove: △ABC ≅ △EDC Proof: ̶̶ ED , ̶̶ DC 3. Use AAS to prove the triangles congruent. ̶̶ JL bisects ∠KLM. ∠K ≅ ∠M Given: Prove: △JKL ≅ △JML There are four theorems for right triangles that are not used for acute or obtuse triangles. They are Leg-Leg (LL), Hypotenuse-Angle (HA), Leg-Angle (LA), and Hypotenuse-Leg (HL). You will prove LL, HA, and LA in Exercises 21, 23, and 33. 254 254 Chapter 4 Triangle Congruence �� Theorem 4-5-3 Hypotenuse-Leg (HL) Congruence THEOREM HYPOTHESIS CONCLUSION If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. △ABC ≅ △DEF You will prove the Hypotenuse-Leg Theorem in Lesson 4-8. E X A M P L E 4 Applying HL Congruence Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. A △VWX and △YXW According to the diagram, △VWX and △YXW are right triangles that share ̶̶̶ ̶̶̶ WX ≅ WX by the Reflexive hypotenuse ̶̶ ̶̶̶ WV ≅ Property. It is given that XY , therefore △VWX ≅ △YXW by HL. ̶̶̶ WX . B △VWZ and △YXZ This conclusion cannot be proved by HL. According to the diagram, △VWZ and △YXZ are right triangles, ̶̶̶ WZ and is congruent to hypotenuse ̶̶ XY . You do not know that hypotenuse ̶̶ XZ . ̶̶̶ WV ≅ 4. Determine if you can use the HL Congruence Theorem to prove △ABC ≅ △DCB. If not, tell what else you need to know. THINK AND DISCUSS 1. Could you use AAS to prove that these two triangles are congruent? Explain. 2. The arrangement of the letters in ASA matches the arrangement of what parts of congruent triangles? Include a sketch to support your answer. 3. GET ORGANIZED Copy and complete the graphic organizer. In each column, write a description of the method and then sketch two triangles, marking the appropriate congruent parts. 4-5 Triangle Congruence: ASA, AAS, and HL 255 255 �� 4-5 Exercises Exercises KEYWORD: MG7 4-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary A triangle contains ∠ABC and ∠ACB with ̶̶ BC “closed in” between them. How would this help you remember the definition of included side. 252 Surveying Use the table for Exercises 2 and 3. A landscape designer surveyed the boundaries of a triangular park. She made the following table for the dimemsions of the land. A to B B to C C to A Bearing E S 25° E N 62° W Distance 115 ft ? ? 2. Draw the plot of land described by the table. Label the measures of the angles in the triangle. 3. Does the table have enough information to determine the locations of points A, B, and C ? Explain Determine if you can use ASA to prove the triangles congruent. Explain. p. 253 4. △VRS and △VTS, given that ̶̶ VS bisects ∠RST and ∠RVT 5. △DEH and △FGH . Use AAS to prove the triangles congruent. p. 254 Given: ∠R and ∠P are right angles. ̶̶ QR ǁ ̶̶ SP Prove: △QPS ≅ △SRQ Proof. 255 Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 7. △ABC and △CDA 8. △XYV and △ZYV 256 256 Chapter 4 Triangle Congruence CBA115 ft ge07sec_04l05003aa�� Independent Practice For See Exercises Example 9–10 11–12 13 14–15 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Surveying Use the table for Exercises 9 and 10. From two different observation towers a fire is sighted. The locations of the towers are given in the following table. X to Y X to F Y to F Bearing E N 53° E N 16° W Distance 6 km ? ? 9. Draw the diagram formed by observation tower X, observation tower Y, and the fire F. Label the measures of the angles. 10. Is there enough information given in the table to pinpoint the location of the fire? Explain. Determine if you can use ASA to prove the triangles congruent. Explain. Determine if you can use ASA to prove the triangles congruent. Explain. Math History 11. 11. △MKJ and △MKL 12. △RST and △TUR 13. 13. Given: ̶̶ AB ≅ Prove: △ABC ≅ △DEF ̶̶ DE , ∠C ≅ ∠F Proof: Euclid wrote the mathematical text The Elements around 2300 years ago. It may be the second most reprinted book in history. Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 14. △GHJ and △JKG 15. △ABE and △DCE, given that E is the midpoint ̶̶ ̶̶ BC AD and of Multi-Step For each pair of triangles write a triangle congruence statement. Identify the transformation that moves one triangle to the position of the other triangle. 16. 17. 18. Critical Thinking Side-Side-Angle (SSA) cannot be used to prove two triangles congruent. Draw a diagram that shows why this is true. 4-5 Triangle Congruence: ASA, AAS, and HL 257 257 �� 19. This problem will prepare you for the Multi-Step TAKS Prep on page 280. A carpenter built a truss to support the roof of a doghouse. ̶̶ a. The carpenter knows that MJ . Can the carpenter ̶̶ KJ ≅ conclude that △KJL ≅ △MJL? Why or why not? b. Suppose the carpenter also knows that ∠JLK is a right angle. Which theorem can be used to show that △KJL ≅ △MJL? 20. /////ERROR ANALYSIS///// Two proofs that △EFH ≅ △GHF are given. Which is incorrect? Explain the error. 21. Write a paragraph proof of the Leg-Leg (LL) Congruence Theorem. If the legs of one right triangle are congruent to the corresponding legs of another right triangle, the triangles are congruent. 22. Use AAS to prove the triangles congruent. ̶̶ AD ≅ Prove: △AED ≅ △CEB ̶̶ AD ǁ Given: ̶̶ BC , ̶̶ CB Proof: Statements Reasons ̶̶̶ AD ǁ ̶̶ BC 1. 2. ∠DAE ≅ ∠BCE 3. c. 4. d. 5. e. ? ̶̶̶̶ ? ̶̶̶̶ ? ̶̶̶̶ 1. a. 2. b. ? ̶̶̶̶ ? ̶̶̶̶ 3. Vert.  Thm. 3. Given 4. f. ? ̶̶̶̶ 23. Prove the Hypotenuse-Angle (HA) Theorem. ̶̶ ̶̶̶ JM ≅ KM ⊥ Given: Prove: △JKM ≅ △LKM ̶̶̶ LM , ∠JMK ≅ ∠LMK ̶̶ JL , 24. Write About It The legs of both right △DEF and right △RST are 3 cm and 4 cm. They each have a hypotenuse 5 cm in length. Describe two different ways you could prove that △DEF ≅ △RST. 25. Construction Use the method for constructing perpendicular lines to construct a right triangle. 26. What additional congruence statement is necessary to prove △XWY ≅ △XVZ by ASA? ∠XVZ ≅ ∠XWY ∠VUY ≅ ∠WUZ ̶̶ VZ ≅ ̶̶ XZ ≅ ̶̶̶ WY ̶̶ XY 258 258 Chapter 4 Triangle Congruence �� 27. Which postulate or theorem justifies the congruence statement △STU ≅ △VUT? ASA SSS HL SAS 28. Which of the following congruence statements is true? ∠A ≅ ∠B ̶̶ ̶̶ DE CE ≅ △AED ≅ △CEB △AED ≅ △BEC 29. In △RST, RT = 6y - 2. In △UVW, UW = 2y + 7. ∠R ≅ ∠U, and ∠S ≅ ∠V. What must be the value of y in order to prove that △RST ≅ △UVW? 1.25 2.25 9.0 11.5 30. Extended Response Draw a triangle. Construct a second triangle that has the same angle measures but is not congruent. Compare the lengths of each pair of corresponding sides. Consider the relationship between the lengths of the sides and the measures of the angles. Explain why Angle-Angle-Angle (AAA) is not a congruence principle. CHALLENGE AND EXTEND 31. Sports This bicycle frame includes △VSU and △VTU, which lie in intersecting planes. From the given angle measures, can you conclude that △VSU ≅ △VTU? Explain. m∠VUS = (7y - 2) ° m∠VUT = (5 1 _ m∠USV = 5 2 _ y ° 3 m∠SVU = (3y - 6) ° m∠TVU = 2x ° ) x - 1 _ 2 2 m∠UTV = (4x + 8) ° ° � � � � 32. Given: △ABC is equilateral. C is the midpoint of ̶̶ DE . ∠DAC and ∠EBC are congruent and supplementary. Prove: △DAC ≅ △EBC �� 33. Write a two-column proof of the Leg-Angle (LA) Congruence Theorem. If a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent. (Hint: There are two cases to consider.) 34. If two triangles are congruent by ASA, what theorem could you use to prove that the triangles are also congruent by AAS? Explain. SPIRAL REVIEW Identify the x- and y-intercepts. Use them to graph each line. (Previous course) 35. y = 3x - 6 36 37. y = -5x + 5 38. Find AB and BC if AC = 10. (Lesson 1-6) 39. Find m∠C. (Lesson 4-2) 4-5 Triangle Congruence: ASA, AAS, and HL 259 259 �� 4-6 Triangle Congruence: CPCTC TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning … Also G.3.E, G.7.A, G.10.B Objective Use CPCTC to prove parts of triangles are cong
ruent. Vocabulary CPCTC Why learn this? You can use congruent triangles to estimate distances. CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent. E X A M P L E 1 Engineering Application SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. To design a bridge across a canyon, you need to find the distance from A to B. Locate points C, D, and E as shown in the figure. If DE = 600 ft, what is AB? ̶̶ CB ,because DC = CB = 500 ft. ∠D ≅ ∠B, because they are both right angles. ̶̶ DC ≅ ∠DCE ≅ ∠BCA, because vertical angles are congruent. Therefore △DCE ≅ △BCA by ASA or LA. By CPCTC, AB = ED = 600 ft. ̶̶ AB , so ̶̶ ED ≅ 1. A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? � �� E X A M P L E 2 Proving Corresponding Parts Congruent ̶̶ AB ≅ ̶̶ DC , ∠ABC ≅ ∠DCB Given: Prove: ∠A ≅ ∠D Proof: 2. Given: Prove: ̶̶ PR bisects ∠QPS and ∠QRS. ̶̶ PQ ≅ ̶̶ PS 260 260 Chapter 4 Triangle Congruence �� E X A M P L E 3 Using CPCTC in a Proof ̶̶ DF ̶̶ EG ≅ ̶̶ EG ǁ ̶̶ ED ǁ ̶̶ DF , ̶̶ GF Given: Prove: Proof: Statements Reasons ̶̶ EG ≅ ̶̶ EG ǁ ̶̶ DF ̶̶ DF 1. 2. 3. ∠EGD ≅ ∠FDG ̶̶̶ GD ≅ ̶̶̶ GD 4. 1. Given 2. Given 3. Alt. Int.  Thm. 4. Reflex. Prop. of ≅ 5. △EGD ≅ △FDG 5. SAS Steps 1, 3, and 4 6. ∠EDG ≅ ∠FGD ̶̶ GF ̶̶ ED ǁ 7. 6. CPCTC 7. Converse of Alt. Int.  Thm. 3. Given: J is the midpoint of Prove: ̶̶ KL ǁ ̶̶̶ MN ̶̶̶ KM and ̶̶ NL . You can also use CPCTC when triangles are on a coordinate plane. You use the Distance Formula to find the lengths of the sides of each triangle. Then, after showing that the triangles are congruent, you can make conclusions about their corresponding parts. E X A M P L E 4 Using CPCTC in the Coordinate Plane Given: A (2, 3) , B (5, -1) , C (1, 0) , D (-4, -1) , E (0, 2) , F (-1, -2) Prove: ∠ABC ≅ ∠DEF Step 1 Plot the points on a coordinate plane. Step 2 Use the Distance Formula to find the lengths of the sides of each triangle. D = √  ( AB = √  (5 -2) 2 + (-1 - 3) 2 = √  9 + 16 = √  25 = 5 BC =  2 √ (1 - 5) 2 + (0 - (-1) ) = √  16 + 1 = √  17 AC = √  (1 - 2) 2 + (0 - 3) 2 = √  1 + 9 = √  10 DE =  2 √ + (2 - (-1) ) (0 - (-4) ) 16 + 9 = √  25 = 5 = √  2 EF = √  (-1 - 0) 2 + (-2 - 2) 2 = √  1 + 16 = √  17 DF =  2 √ + (-2 - (-1) ) (-1 - (-4) ) 9 + 1 = √  10 = √  2 ̶̶ AB ≅ ̶̶ BC ≅ So and ∠ABC ≅ ∠DEF by CPCTC. ̶̶ EF , and ̶̶ DE , ̶̶ AC ≅ ̶̶ DF . Therefore △ABC ≅ △DEF by SSS, 4. Given: J (-1, -2) , K (2, -1) , L (-2, 0) , R (2, 3) , S (5, 2) , T (1, 1) Prove: ∠JKL ≅ ∠RST 4-6 Triangle Congruence: CPCTC 261 261 �� THINK AND DISCUSS ̶̶̶ VW ≅ 1. In the figure, ̶̶ UV ≅ ̶̶ XY , ̶̶ YZ , and ∠V ≅ ∠Y. Explain why △UVW ≅ △XYZ. By CPCTC, which additional parts are congruent? 2. GET ORGANIZED Copy and complete the graphic organizer. Write all conclusions you can make using CPCTC. 4-6 Exercises Exercises GUIDED PRACTICE 1. Vocabulary You use CPCTC after proving triangles are congruent. Which parts of congruent triangles are referred to as corresponding parts. Engineering To find the height of p. 260 a windmill, a rancher places a marker at C and steps off the distance from C to B. Then the rancher walks the same distance from C in the opposite direction and places a marker at D. If DE = 6.3 m, what is AB? KEYWORD: MG7 4-6 KEYWORD: MG7 Parent � � � � � � . 260 3. Given: X is the midpoint of ̶̶ ST . ̶̶ RX ⊥ ̶̶ ST Prove: ̶̶ RS ≅ ̶̶ RT Proof: 262 262 Chapter 4 Triangle Congruence �� . 261 4. Given: Prove: ̶̶ ̶̶ AC ≅ AD , ̶̶ AB bisects ∠CAD. ̶̶ CB ≅ ̶̶ DB Proof: Statements ̶̶ ̶̶ ̶̶̶ DB CB ≅ AD , ̶̶ AC ≅ 1. 2. b. ? ̶̶̶̶ 3. △ACB ≅ △ADB 4. ∠CAB ≅ ∠DAB ̶̶ AB bisects ∠CAD 5. Reasons 1. a. ? ̶̶̶̶ 2. Reflex. Prop. of ≅ 3. c. 4. d. 5. e. ? ̶̶̶̶ ? ̶̶̶̶ ? ̶̶̶̶ Multi-Step Use the given set of points to prove each congruence statement. p. 261 5. E (-3, 3) , F (-1, 3) , G (-2, 0) , J (0, -1) , K (2, -1) , L (1, 2) ; ∠EFG ≅ ∠JKL 6. A (2, 3) , B (4, 1) , C (1, -1) , R (-1, 0) , S (-3, -2) , T (0, -4) ; ∠ACB ≅ ∠RTS Independent Practice For See Exercises Example 7 8–9 10–11 12–13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 7. Surveying To find the distance AB across a river, a surveyor first locates point C. He measures the distance from C to B. Then he locates point D the same distance east of C. If DE = 420 ft, what is AB? 8. 8. Given: M is the midpoint of ̶̶ PQ and ̶̶ ̶̶ PS QR ≅ Prove: ̶̶ RS . 9. Given: ̶̶̶ WX ≅ ̶̶ XY ≅ ̶̶ YZ ≅ ̶̶̶ ZW Prove: ∠W ≅ ∠Y 10. Given: G is the midpoint of ̶̶ FH . 11. Given: ̶̶̶ LM bisects ∠JLK. Prove: M is the midpoint of ̶̶ KL ̶̶ JL ≅ ̶̶ JK . ̶̶ EF ≅ ̶̶ EH Prove: ∠1 ≅ ∠2 Multi-Step Use the given set of points to prove each congruence statement. 12. R (0, 0) , S (2, 4) , T (-1, 3) , U (-1, 0) , V (-3, -4) , W (-4, -1) ; ∠RST ≅ ∠UVW 13. A (-1, 1) , B (2, 3) , C (2, -2) , D (2, -3) , E (-1, -5) , F (-1, 0) ; ∠BAC ≅ ∠EDF 14. Given: △QRS is adjacent to △QTS. Prove: ̶̶ QS bisects ̶̶ RT . ̶̶ QS bisects ∠RQT. ∠R ≅ ∠T 15. Given: △ABE and △CDE with E the midpoint of ̶̶ AC and ̶̶ BD Prove: ̶̶ AB ǁ ̶̶ CD 4-6 Triangle Congruence: CPCTC 263 263 ��BC500 ftADE500 ftge07sec04l06004_A�� 16. This problem will prepare you for the Multi-Step TAKS Prep on page 280. The front of a doghouse has the dimensions shown. a. How can you prove that △ADB ≅ △ADC? ̶̶ b. Prove that CD . c. What is the length of ̶̶ BD and ̶̶ BD ≅ ̶̶ BC to the nearest tenth? �� Multi-Step Find the value of x. 17. 18. �� Use the diagram for Exercises 19–21. 19. Given: PS = RQ, m∠1 = m∠4 Prove: m∠3 = m∠2 20. Given: m∠1 = m∠2, m∠3 = m∠4 Prove: PS = RS 21. Given: PS = RQ, PQ = RS Prove: ̶̶ PQ ǁ ̶̶ RS 22. Critical Thinking Does the diagram contain enough information to allow you to conclude that ̶̶̶ ML ? Explain. ̶̶ JK ǁ 23. Write About It Draw a diagram and explain how a surveyor can set up triangles to find the distance across a lake. Label each part of your diagram. List which sides or angles must be congruent. 24. Which of these will NOT be used as a reason in a proof of ̶̶ AD ? ̶̶ AC ≅ SAS ASA CPCTC Reflexive Property 25. Given the points K (1, 2) , L (0, -4) , M (-2, -3) , and N (-1, 3) , which of these is true? ∠KNL ≅ ∠MNL ∠LNK ≅ ∠NLM 26. What is the value of y? ∠MLN ≅ ∠KLN ∠MNK ≅ ∠NKL 10 20 35 85 27. Which of these are NOT used to prove angles congruent? congruent triangles noncorresponding parts parallel lines perpendicular lines 264 264 Chapter 4 Triangle Congruence �� 28. Which set of coordinates represents the vertices of a triangle congruent to △RST ? (Hint: Find the lengths of the sides of △RST.) (3, 4) , (3, 0) , (0, 0) (3, 3) , (0, 4) , (0, 0) (3, 1) , (3, 3) , (4, 6) (3, 0) , (4, 4) , (0, 6) CHALLENGE AND EXTEND 29. All of the edges of a cube are congruent. All of the angles on each face of a cube are right angles. Use CPCTC to explain why any two diagonals on the faces of a cube (for example, must be congruent. ̶̶ AC and ̶̶ AF ) 30. Given: ̶̶ JK ≅ ̶̶̶ ML , ̶̶ JM ≅ ̶̶ KL Prove: ∠J ≅ ∠L (Hint: Draw an auxiliary line.) 31. Given: R is the midpoint of S is the midpoint of ̶̶ RS ⊥ ̶̶ AB . ̶̶ DC . ̶̶ AB , ∠ASD ≅ ∠BSC Prove: △ASD ≅ △BSC 32. △ABC is in plane M. △CDE is in plane P. Both planes have C in common and ∠A ≅ ∠E. What is the height AB to the nearest foot? �� SPIRAL REVIEW 33. Lina’s test scores in her history class are 90, 84, 93, 88, and 91. What is the minimum score Lina must make on her next test to have an average test score of 90? (Previous course) 34. One long-distance phone plan costs $3.95 per month plus $0.08 per minute of use. A second long-distance plan costs $0.10 per minute for the first 50 minutes used each month and then $0.15 per minute after that. Which plan is cheaper if you use an average of 75 long-distance minutes per month? (Previous course) A figure has vertices at (1, 3) , (2, 2) , (3, 2) , and (4, 3) . Identify the transformation of the figure that produces an image with each set of vertices. (Lesson 1-7) 35. (1, -3) , (2, -2) , (3, -2) , (4, -3) 36. (-2, -1) , (-1, -2) , (0, -2) , (1, -1) 37. Determine if you can use ASA to prove △ACB ≅ △ECD. Explain. (Lesson 4-5) 4-6 Triangle Congruence: CPCTC 265 265 �� Quadratic Equations Algebra A quadratic equation is an equation that can be written in the form a x 2 + bx + c = 0. See Skills Bank page S66 Example Given: △ABC is isosceles with ̶̶ AB ≅ ̶̶ AC . Solve for x. Step 1 Set x 2 – 5x equal to 6 to get x 2 – 5x = 6. Step 2 Rewrite the quadratic equation by subtracting 6 from each side to get x 2 – 5x – 6 = 0. Step 3 Solve for x. Method 1: Factoring Method 2: Quadratic Formula x 2 - 5x - 6 = 0 (x - 6) (x + 1) = 0 Factor. x = -b ± √  b 2 - 4ac __ 2a x - 6 = 0 or x + 1 = 0 Set each factor equal to 0. x = - (-5) ± √  (-5) 2 - 4 (1) (-6) ___ 2 (1) Substitute 1 for a, -5 for b, and -6 for c. Simplify. Find the square root. Simplify. x = 6 or x = -1 Solve. x = 5 ± √  49 _ 2 5 ± 7 _ x = 2 or x = -2 _ x = 12 _ 2 2 x = 6 or x = -1 Step 4 Check each solution in the original equation. x 2 - 5x = 6 (6 ) 2 - 5 (6 ) 36 - 30 6 6 x 2 - 5x = 6 (-1) 2 - 5 (- ✓ Try This TAKS Grades 9–11 Obj. 5, 6 Solve for x in each isosceles triangle. ̶̶ 1. Given: FG ̶̶ FE ≅ 2. Given: ̶̶ JK ≅ ̶̶ JL 3. Given: ̶̶ YX
≅ ̶̶ YZ 4. Given: ̶̶ QP ≅ ̶̶ QR 266 266 Chapter 4 Triangle Congruence �� 4-7 Introduction to Coordinate Proof TEK G.2.B Geometric structure: make conjectures about ... polygons … and determine validity of the conjectures. Also G.3.B, G.9.B, G.10.B Objectives Position figures in the coordinate plane for use in coordinate proofs. Prove geometric concepts by using coordinate proof. Vocabulary coordinate proof Who uses this? The Bushmen in South Africa use the Global Positioning System to transmit data about endangered animals to conservationists. (See Exercise 24.) You have used coordinate geometry to find the midpoint of a line segment and to find the distance between two points. Coordinate geometry can also be used to prove conjectures. A coordinate proof is a style of proof that uses coordinate geometry and algebra. The first step of a coordinate proof is to position the given figure in the plane. You can use any position, but some strategies can make the steps of the proof simpler. Strategies for Positioning Figures in the Coordinate Plane • Use the origin as a vertex, keeping the figure in Quadrant I. • Center the figure at the origin. • Center a side of the figure at the origin. • Use one or both axes as sides of the figure. E X A M P L E 1 Positioning a Figure in the Coordinate Plane Position a rectangle with a length of 8 units and a width of 3 units in the coordinate plane. Method 1 You can center the longer side of the rectangle at the origin. Method 2 You can use the origin as a vertex of the rectangle. Depending on what you are using the figure to prove, one solution may be better than the other. For example, if you need to find the midpoint of the longer side, use the first solution. 1. Position a right triangle with leg lengths of 2 and 4 units in the coordinate plane. (Hint: Use the origin as the vertex of the right angle.) 4- 7 Introduction to Coordinate Proof 267 267 �� Once the figure is placed in the coordinate plane, you can use slope, the coordinates of the vertices, the Distance Formula, or the Midpoint Formula to prove statements about the figure. E X A M P L E 2 Writing a Proof Using Coordinate Geometry Write a coordinate proof. Given: Right △ABC has vertices A (0, 6) , B (0, 0) , and C (4, 0) . D is the ̶̶ AC . midpoint of Prove: The area of △DBC is one half the area of △ABC. Proof: △ABC is a right triangle with height AB and base BC. area of △ABC = 1 __ 2 bh = 1 __ 2 (4) (6) = 12 square units , 6 + 0 ____ 2 By the Midpoint Formula, the coordinates of D = ( 0 + 4 ____ 2 of △DBC, and the base is 4 units. area of △DBC = 1 __ 2 bh ) = (2, 3) . The y-coordinate of D is the height = 1 __ 2 (4) (3) = 6 square units Since 6 = 1 __ 2 (12) , the area of △DBC is one half the area of △ABC. 2. Use the information in Example 2 to write a coordinate proof showing that the area of △ADB is one half the area of △ABC. A coordinate proof can also be used to prove that a certain relationship is always true. You can prove that a statement is true for all right triangles without knowing the side lengths. To do this, assign variables as the coordinates of the vertices. E X A M P L E 3 Assigning Coordinates to Vertices Position each figure in the coordinate plane and give the coordinates of each vertex. A a right triangle with leg B a rectangle with lengths a and b length c and width d Do not use both axes when positioning a figure unless you know the figure has a right angle. 3. Position a square with side length 4p in the coordinate plane and give the coordinates of each vertex. If a coordinate proof requires calculations with fractions, choose coordinates that make the calculations simpler. For example, use multiples of 2 when you are to find coordinates of a midpoint. Once you have assigned the coordinates of the vertices, the procedure for the proof is the same, except that your calculations will involve variables. 268 268 Chapter 4 Triangle Congruence �� E X A M P L E 4 Writing a Coordinate Proof Given: ∠B is a right angle in △ABC. D is the midpoint of Prove: The area of △DBC is one half the area of △ABC. ̶̶ AC . Step 1 Assign coordinates to each vertex. The coordinates of A are (0, 2 j) , the coordinates of B are (0, 0) , and the coordinates of C are (2n, 0) . Since you will use the Midpoint Formula to find the coordinates of D, use multiples of 2 for the leg lengths. Step 2 Position the figure in the coordinate plane. Step 3 Write a coordinate proof. Proof: △ABC is a right triangle with height 2j and base 2n. area of △ABC = 1 __ 2 bh = 1 __ 2 (2n) (2j) = 2nj square units Because the x- and y-axes intersect at right angles, they can be used to form the sides of a right triangle. By the Midpoint Formula, the coordinates of D = ( 0 + 2n 2j + 0 _____ _____ 2 , 2 The height of △DBC is j units, and the base is 2n units. area of △DBC = 1 __ 2 bh ) = (n, j) . = 1 __ 2 (2n) (j) = nj square units Since nj = 1 __ 2 (2nj) , the area of △DBC is one half the area of △ABC. 4. Use the information in Example 4 to write a coordinate proof showing that the area of △ADB is one half the area of △ABC. THINK AND DISCUSS 1. When writing a coordinate proof why are variables used instead of numbers as coordinates for the vertices of a figure? 2. How does the way you position a figure in the coordinate plane affect your calculations in a coordinate proof? 3. Explain why it might be useful to assign 2p as a coordinate instead of just p. 4. GET ORGANIZED Copy and complete the graphic organizer. In each row, draw an example of each strategy that might be used when positioning a figure for a coordinate proof. 4- 7 Introduction to Coordinate Proof 269 269 �� 4-7 Exercises Exercises KEYWORD: MG7 4-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary What is the relationship between coordinate geometry, coordinate plane, and coordinate proof ? Position each figure in the coordinate plane. p. 267 2. a rectangle with a length of 4 units and width of 1 unit 3. a right triangle with leg lengths of 1 unit and 3 units Write a proof using coordinate geometry. p. 268 4. Given: Right △PQR has coordinates P (0, 6) , Q (8, 0) , and R (0, 0) . A is the midpoint of B is the midpoint of ̶̶ QR . ̶̶ PR . Prove: AB = 1 __ 2 PQ . 268 Position each figure in the coordinate plane and give the coordinates of each vertex. 5. a right triangle with leg lengths m and n 6. a rectangle with length a and width . 269 Independent Practice For See Exercises Example 8–9 10 11–12 13 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 Multi-Step Assign coordinates to each vertex and write a coordinate proof. 7. Given: ∠R is a right angle in △PQR. A is the midpoint of ̶̶ PR . B is the midpoint of ̶̶ QR . Prove: AB = 1 __ 2 PQ PRACTICE AND PROBLEM SOLVING Position each figure in the coordinate plane. 8. a square with side lengths of 2 units 9. a right triangle with leg lengths of 1 unit and 5 units Write a proof using coordinate geometry. 10. Given: Rectangle ABCD has coordinates A (0, 0) , B (0, 10) , C (6, 10) , and D (6, 0) . E is the midpoint of ̶̶ AB , and F is the midpoint of ̶̶ CD . Prove: EF = BC Position each figure in the coordinate plane and give the coordinates of each vertex. 11. a square with side length 2m 12. a rectangle with dimensions x and 3x Multi-Step Assign coordinates to each vertex and write a coordinate proof. ̶̶ CD . ̶̶ AB in rectangle ABCD. F is the midpoint of 13. Given: E is the midpoint of Prove: EF = AD 14. Critical Thinking Use variables to write the general form of the endpoints of a segment whose midpoint is (0, 0) . 270 270 Chapter 4 Triangle Congruence �� Conservation The origin of the springbok’s name may come from its habit of pronking, or bouncing. When pronking, a springbok can leap up to 13 feet in the air. Springboks can run up to 53 miles per hour. 15. Recreation A hiking trail begins at E (0, 0) . Bryan hikes from the start of the trail to a waterfall at W (3, 3) and then makes a 90° turn to a campsite at C (6, 0) . a. Draw Bryan’s route in the coordinate plane. b. If one grid unit represents 1 mile, what is the total distance Bryan hiked? Round to the nearest tenth. Find the perimeter and area of each figure. 16. a right triangle with leg lengths of a and 2a units 17. a rectangle with dimensions s and t units Find the missing coordinates for each figure. 18. 19. 20. Conservation The Bushmen have sighted animals at the following coordinates: (-25, 31.5) , (-23.2, 31.4) , and (-24, 31.1) . Prove that the distance between two of these locations is approximately twice the distance between two other. 21. Navigation Two ships depart from a port at P (20, 10) . The first ship travels to a location at A (-30, 50) , and the second ship travels to a location at B (70, -30) . Each unit represents one nautical mile. Find the distance to the nearest nautical mile between the two ships. Verify that the port is at the midpoint between the two. Write a coordinate proof. 22. Given: Rectangle PQRS has coordinates P (0, 2) , Q (3, 2) , R (3, 0) , and S (0, 0) . ̶̶ PR and ̶̶ QS intersect at T (1.5, 1) . Prove: The area of △RST is 1 __ 4 of the area of the rectangle _____ _____ 2 , 2 23. Given ) , with midpoint M ( Prove: AM = 1 __ 2 AB 24. Plot the points on a coordinate plane and connect them to form △KLM and △MPK. Write a coordinate proof. Given: K (-2, 1) , L (-2, 3) , M (1, 3) , P (1, 1) Prove: △KLM ≅ △MPK 25. Write About It When you place two sides of a figure on the coordinate axes, what are you assuming about the figure? 26. This problem will prepare you for the Multi-Step TAKS Prep on page 280. Paul designed a doghouse to fit against the side of his house. His pla
n consisted of a right triangle on top of a rectangle. a. Find BD and CE. b. Before building the doghouse, Paul sketched his plan on a coordinate plane. He placed A at the origin and and E, assuming that each unit of the coordinate plane represents one inch. ̶̶ AB on the y-axis. Find the coordinates of B, C, D, 4- 7 Introduction to Coordinate Proof 271 271 �� 27. The coordinates of the vertices of a right triangle are (0, 0) , (4, 0) , and (0, 2) . Which is a true statement? The vertex of the right angle is at (4, 2) . The midpoints of the two legs are at (2, 0) and (0, 1) . The hypotenuse of the triangle is √  6 units. The shortest side of the triangle is positioned on the x-axis. 28. A rectangle has dimensions of 2g and 2f units. If one vertex is at the origin, which coordinates could NOT represent another vertex? (2g, 2f) (2f, 0) (2f, g) (-2f, 2g) 29. The coordinates of the vertices of a rectangle are (0, 0), (a, 0), (a, b), and (0, b). What is the perimeter of the rectangle? a + b ab 1 _ 2 ab 2a + 2b 30. A coordinate grid is placed over a map. City A is located at (-1, 2) and city C is located at (3, 5) . If city C is at the midpoint between city A and city B, what are the coordinates of city B? (1, 3.5) (-5, -1) (7, 8) (2, 7) CHALLENGE AND EXTEND Find the missing coordinates for each figure. 31. 32. 33. The vertices of a right triangle are at (-2s, 2s) , (0, 2s) , and (0, 0) . What coordinates could be used so that a coordinate proof would be easier to complete? 34. Rectangle ABCD has dimensions of 2f and 2g units. The equation of the line containing ̶̶ g __ BD is y = x, and f ̶̶ g __ x + 2g. AC is y = - the equation of the line containing f Use algebra to show that the coordinates of E are (f, g) . SPIRAL REVIEW Use the quadratic formula to solve for x. Round to the nearest hundredth if necessary. (Previous course) 35. 0 = 8 x 2 + 18x - 5 36. 0 = x 2 + 3x - 5 37. 0 = 3 x 2 - x - 10 Find each value. (Lesson 3-2) 38. x 39. y 40. Use A (-4, 3) , B (-1, 3) , C (-3, 1) , D (0, -2) , E (3, -2) , and F (2, -4) to prove ∠ABC ≅ ∠EDF. (Lesson 4-6). 272 272 Chapter 4 Triangle Congruence �� 4-8 Isosceles and Equilateral Triangles TEKS G.2.B Geometric structure: make conjectures about angles, lines, polygons … and determine the validity of the conjectures .... Also G.3.C, G.10.B Objectives Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles. Vocabulary legs of an isosceles triangle vertex angle base base angles Who uses this? Astronomers use geometric methods. (See Example 1.) Recall that an isosceles triangle has at least two congruent sides. The congruent sides are called the legs . The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base , and the base angles are the two angles that have the base as a side. ∠3 is the vertex angle. ∠1 and ∠2 are the base angles. Theorems Isosceles Triangle THEOREM HYPOTHESIS CONCLUSION 4-8-1 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent. 4-8-2 Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. ∠B ≅ ∠C ̶̶ DE ≅ ̶̶ DF Theorem 4-8-1 is proven below. You will prove Theorem 4-8-2 in Exercise 35. PROOF PROOF ̶̶ AB ≅ Isosceles Triangle Theorem ̶̶ AC Given: Prove: ∠B ≅ ∠C Proof: Statements Reasons The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent.” 1. Draw X, the mdpt. of 2. Draw the auxiliary line ̶̶ BC . ̶̶ AX . ̶̶ BX ≅ ̶̶ AB ≅ ̶̶ AX ≅ ̶̶ CX ̶̶ AC ̶̶ AX 3. 4. 5. 6. △ABX ≅ △ACX 7. ∠B ≅ ∠C 1. Every seg. has a unique mdpt. 2. Through two pts. there is exactly one line. 3. Def. of mdpt. 4. Given 5. Reflex. Prop. of ≅ 6. SSS Steps 3, 4, 5 7. CPCTC 4-8 Isosceles and Equilateral Triangles 273 273 �� E X A M P L E 1 Astronomy Application The distance from Earth to nearby stars can be measured using the parallax method, which requires observing the positions of a star 6 months apart. If the distance LM to a star in July is 4.0 × 10 13 km, explain why the distance LK to the star in January is the same. (Assume the distance from Earth to the Sun does not change.) Not drawn to scale m∠LKM = 180 - 90.4, so m∠LKM = 89.6°. Since ∠LKM ≅ ∠M, △LMK is isosceles by the Converse of the Isosceles Triangle Theorem. Thus LK = LM = 4.0 × 10 13 km. 1. If the distance from Earth to a star in September is 4.2 × 10 13 km, what is the distance from Earth to the star in March? Explain. E X A M P L E 2 Finding the Measure of an Angle Find each angle measure. A m∠C m∠C = m∠B = x° m∠C + m∠B + m∠A = 180 x + x + 38 = 180 2x = 142 x = 71 Thus m∠C = 71°. B m∠S Isosc. △ Thm. △ Sum Thm. Substitute the given values. Simplify and subtract 38 from both sides. Divide both sides by 2. m∠S = m∠R 2x° = (x + 30) ° Isosc. △ Thm. Substitute the given values. x = 30 Subtract x from both sides. Thus m∠S = 2x° = 2 (30) = 60°. Find each angle measure. 2b. m∠N 2a. m∠H The following corollary and its converse show the connection between equilateral triangles and equiangular triangles. Corollary 4-8-3 Equilateral Triangle COROLLARY HYPOTHESIS CONCLUSION If a triangle is equilateral, then it is equiangular. (equilateral △ → equiangular △) ∠A ≅ ∠B ≅ ∠C You will prove Corollary 4-8-3 in Exercise 36. 274 274 Chapter 4 Triangle Congruence �� Corollary 4-8-4 Equiangular Triangle COROLLARY HYPOTHESIS CONCLUSION If a triangle is equiangular, then it is equilateral. (equiangular △ → equilateral △) ̶̶ DE ≅ ̶̶ DF ≅ ̶̶ EF E X A M P L E 3 Using Properties of Equilateral Triangles You will prove Corollary 4-8-4 in Exercise 37. Find each value. A x △ABC is equiangular. (3x + 15) ° = 60° 3x = 45 x = 15 B t △JKL is equilateral. 4t - 8 = 2t + 1 2t = 9 Equilateral △ → equiangular △ The measure of each ∠ of an equiangular △ is 60°. Subtract 15 from both sides. Divide both sides by 3. Equiangular △ → equilateral △ Def. of equilateral △ Subtract 2t and add 8 to both sides. t = 4.5 Divide both sides by 2. 3. Use the diagram to find JL. E X A M P L E 4 Using Coordinate Proof A coordinate proof may be easier if you place one side of the triangle along the x-axis and locate a vertex at the origin or on the y-axis. Prove that the triangle whose vertices are the midpoints of the sides of an isosceles triangle is also isosceles. Given: △ABC is isosceles. X is the mdpt. of ̶̶ AC . Z is the mdpt. of ̶̶ AB . ̶̶ BC . Y is the mdpt. of Prove: △XYZ is isosceles. Proof: Draw a diagram and place the coordinates of △ABC and △XYZ as shown. By the Midpoint Formula, the coordinates of X are ( 2a + 0 ) = (a, b) , _____ 2 the coordinates of Y are ( 2a + 4a ) = (3a, b) , and the coordinates of Z ______ 2 are ( 4a + 0 _____ 2 By the Distance Formula, XZ = √ YZ = √  ̶̶ XZ ≅ Since XZ = YZ,  2 2 = √  + (0 - b) (2a - a) (2a - 3a) 2 + (0 - b) 2 = √  a 2 + b 2 . ̶̶ YZ by definition. So △XYZ is isosceles. ) = (2a, 0) . a 2 + b 2 , and , 2b + 0 _____ 2 , 2b + 0 _____ 2 , 0 + 0 ____ 2 4. What if...? The coordinates of △ABC are A (0, 2b) , B (-2a, 0) , and C (2a, 0) . Prove △XYZ is isosceles. 4- 8 Isosceles and Equilateral Triangles 275 275 �� THINK AND DISCUSS 1. Explain why each of the angles in an equilateral triangle measures 60°. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, draw and mark a diagram for each type of triangle. 4-8 Exercises Exercises KEYWORD: MG7 4-8 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Draw isosceles △JKL with ∠K as the vertex angle. Name the legs, base, and base angles of the triangle. 274 2. Surveying To find the distance QR across a river, a surveyor locates three points Q, R, and S. QS = 41 m, and m∠S = 35°. The measure of exterior ∠PQS = 70°. Draw a diagram and explain how you can find QR Find each angle measure. p. 274 3. m∠ECD 4. m∠K 5. m∠X 6. m∠ Find each value. p. 275 7. y 8. x 9. BC 10. JK . 275 11. Given: △ABC is right isosceles. X is the midpoint of ̶̶ AC . ̶̶ AB ≅ ̶̶ BC Prove: △AXB is isosceles. 276 276 Chapter 4 Triangle Congruence �� Independent Practice For See Exercises Example 12 13–16 17–20 21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 12. Aviation A plane is flying parallel � �  AC . When the to the ground along plane is at A, an air-traffic controller in tower T measures the angle to the plane as 40°. After the plane has traveled 2.4 mi to B, the angle to the plane is 80°. How can you find BT? � �� Find each angle measure. 13. m∠E 14. m∠TRU 15. m∠F �� 16. m∠A Find each value. 17. z 18. y 19. BC 20. XZ 21. Given: △ABC is isosceles. P is the midpoint ̶̶ AB . Q is the midpoint of ̶̶ AC . of ̶̶ AB ≅ ̶̶ PC ≅ ̶̶ AC ̶̶ QB Prove: Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 22. An equilateral triangle is an isosceles triangle. 23. The vertex angle of an isosceles triangle is congruent to the base angles. 24. An isosceles triangle is a right triangle. 25. An equilateral triangle and an obtuse triangle are congruent. 26. Critical Thinking Can a base angle of an isosceles triangle be an obtuse angle? Why or why not? 4- 8 Isosceles and Equilateral Triangles 277 277 ��
�� 27. This problem will prepare you for the Multi-Step TAKS � Prep on page 280. The diagram shows the inside view of the support structure of the back of a doghouse. ̶̶ PS ≅ a. Find m∠SPT. b. Find m∠PQR and m∠PRQ. ̶̶ PT , m∠PST = 71°, and m∠QPS = m∠RPT = 18°. ̶̶ PQ ≅ ̶̶ PR , � � � � Multi-Step Find the measure of each numbered angle. 28. 29. �� 30. Write a coordinate proof. Given: ∠B is a right angle in isosceles right △ABC. X is the midpoint of ̶̶ AC . ̶̶ BA ≅ ̶̶ BC Prove: △AXB ≅ △CXB 31. Estimation Draw the figure formed by (-2, 1) , (5, 5) , and (-1, -7) . Estimate the measure of each angle and make a conjecture about the classification of the figure. Then use a protractor to measure each angle. Was your conjecture correct? Why or why not? 32. How many different isosceles triangles have a perimeter of 18 and sides whose lengths are natural numbers? Explain. Multi-Step Find the value of the variable in each diagram. 33. 34. 35. Prove the Converse of the Isosceles Triangle Theorem. Navigation 36. Complete the proof of Corollary 4-8-3. ̶̶ AC ≅ ̶̶ ̶̶ BC AB ≅ Given: Prove: ∠A ≅ ∠B ≅ ∠C ̶̶ ̶̶ AC , a. AB ≅ ? by the Isosceles Triangle Theorem. ̶̶̶̶ ̶̶ BC , ∠A ≅ ∠B by b. Proof: Since ̶̶ AC ≅ Since By the Transitive Property of ≅, ∠A ≅ ∠B ≅ ∠C. ? . Therefore ∠A ≅ ∠C by c. ̶̶̶̶ ? . ̶̶̶̶ The taffrail log is dragged from the stern of a vessel to measure the speed or distance traveled during a voyage. The log consists of a rotator, recording device, and governor. 37. Prove Corollary 4-8-4. 38. Navigation The captain of a ship traveling along  AB sights an island C at an angle of 45°. The captain measures the distance the ship covers until it reaches B, where the angle to the island is 90°. Explain how to find the distance BC to the island. 39. Given: △ABC ≅ △CBA Prove: △ABC is isosceles. 40. Write About It Write the Isosceles Triangle Theorem and its converse as a biconditional. 278 278 Chapter 4 Triangle Congruence �� 41. Rewrite the paragraph proof of the Hypotenuse-Leg (HL) Congruence Theorem as a two-column proof. Given: △ABC and △DEF are right triangles. ∠C and ∠F are right angles. ̶̶ ̶̶ DF , and AC ≅ Prove: △ABC ≅ △DEF ̶̶ AB ≅ ̶̶ DE .  EF . Mark G so that FG = CB. Thus Proof: On △DEF draw ̶̶ DF and ∠C and ∠F are right angles. ̶̶ AC ≅ lines. Thus ∠DFG is a right angle, and ∠DFG ≅ ∠C. △ABC ≅ △DGF by SAS. ̶̶̶ ̶̶ DG ≅ DE by the Transitive Property. By the Isosceles Triangle Theorem ∠G ≅ ∠E. ∠DFG ≅ ∠DFE since right angles are congruent. So △DGF ≅ △DEF by AAS. Therefore △ABC ≅ △DEF by the Transitive Property. ̶̶ AB by CPCTC. ̶̶ DE as given. ̶̶̶ DG ≅ ̶̶ DF ⊥ ̶̶ AB ≅ ̶̶ FG ≅ ̶̶ CB . From the diagram, ̶̶ EG by definition of perpendicular 42. Lorena is designing a window so that ∠R, ∠S, ∠T, and ̶̶ VU ≅ ̶̶ VT , and m∠UVT = 20°. ∠U are right angles, What is m∠RUV? 10° 70° 20° 80° 43. Which of these values of y makes △ABC isosceles 15 1 _ 2 44. Gridded Response The vertex angle of an isosceles triangle measures (6t - 9) °, and one of the base angles measures (4t) °. Find t. CHALLENGE AND EXTEND 45. In the figure, ̶̶ JK ≅ Prove m∠MKL must also be x°. ̶̶ JL , and ̶̶̶ KM ≅ ̶̶ KL . Let m∠J = x°. 46. An equilateral △ABC is placed on a coordinate plane. Each side length measures 2a. B is at the origin, and C is at (2a, 0) . Find the coordinates of A. 47. An isosceles triangle has coordinates A (0, 0) and B (a, b) . What are all possible coordinates of the third vertex? SPIRAL REVIEW Find the solutions for each equation. (Previous course) 48. x 2 + 5x + 4 = 0 49. x 2 - 4x + 3 = 0 50. x 2 - 2x + 1 = 0 Find the slope of the line that passes through each pair of points. (Lesson 3-5) 51. (2, -1) and (0, 5) 52. (-5, -10) and (20, -10) 53. (4, 7) and (10, 11) 54. Position a square with a perimeter of 4s in the coordinate plane and give the coordinates of each vertex. (Lesson 4-7) 4- 8 Isosceles and Equilateral Triangles 279 279 �� SECTION 4B Proving Triangles Congruent Gone to the Dogs You are planning to build a doghouse for your dog. The pitched roof of the doghouse will be supported by four trusses. Each truss will be an isosceles triangle with the dimensions shown. To determine the materials you need to purchase and how you will construct the trusses, you must first plan carefully. 1. You want to be sure that all four trusses are exactly the same size and shape. Explain how you could measure three lengths on each truss to ensure this. Which postulate or theorem are you using? 2. Prove that the two triangular halves of the truss are congruent. 3. What can you say about ̶̶ DB ? Why is this true? and Use this to help you find the ̶̶ AC , and lengths of ̶̶ DB , ̶̶ AD , ̶̶ BC . ̶̶ AD 4. You want to make careful plans on a coordinate plane before you begin your construction of the trusses. Each unit of the coordinate plane represents 1 inch. How could you assign coordinates to vertices A, B, and C? 5. m∠ACB = 106°. What is the measure of each of the acute angles in the truss? Explain how you found your answer. 6. You can buy the wood for the trusses at the building supply store for $0.80 a foot. The store sells the wood in 6-foot lengths only. How much will you have to spend to get enough wood for the 4 trusses of the doghouse? (Hint: You need to use the Pythagorean Theorem to find the two unknown side lengths of each truss.) 280 280 Chapter 4 Triangle Congruence �� Quiz for Lessons 4-4 Through 4-8 4-4 Triangle Congruence: SSS and SAS SECTION 4B 1. The figure shows one tower and the cables of a suspension bridge. ̶̶ AC ≅ ̶̶ BC , use SAS to explain why △ACD ≅ △BCD. Given that 2. Given: ̶̶ JK bisects ∠MJN. ̶̶ MJ ≅ ̶̶ NJ Prove: △MJK ≅ △NJK 4-5 Triangle Congruence: ASA, AAS, and HL Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 3. △RSU and △TUS 4. △ABC and △DCB Observers in two lighthouses K and L spot a ship S. 5. Draw a diagram of the triangle formed by the lighthouses and the ship. Label each measure. K to L K to S L to S Bearing E N 58° E N 77° W 6. Is there enough data in the table to pinpoint Distance 12 km ? ? the location of the ship? Why? 4-6 Triangle Congruence: CPCTC 7. Given: ̶̶ CD ǁ ̶̶ BE , Prove: ∠D ≅ ∠B ̶̶ DE ǁ ̶̶ CB 4-7 Introduction to Coordinate Proof 8. Position a square with side lengths of 9 units in the coordinate plane 9. Assign coordinates to each vertex and write a coordinate proof. ̶̶ Given: ABCD is a rectangle with M as the midpoint of AB . N is the midpoint of Prove: The area of △AMN is 1 __ 8 the area of rectangle ABCD. ̶̶ AD . 4-8 Isosceles and Equilateral Triangles Find each value. 10. m∠C 11. ST 12. Given: Isosceles △JKL has coordinates J (0, 0) , K (2a, 2b) , and L (4a, 0) . M is the midpoint of ̶̶ JK , and N is the midpoint of ̶̶ KL . Prove: △KMN is isosceles. Ready to Go On? 281 281 �� EXTENSION EXTENSION Proving Constructions Valid TEK G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.3.B Objective Use congruent triangles to prove constructions valid. When performing a compass and straight edge construction, the compass setting remains the same width until you change it. This fact allows you to construct a segment congruent to a given segment. You can assume that two distances constructed with the same compass setting are congruent. The steps in the construction of a figure can be justified by combining the assumptions of compass and straightedge constructions and the postulates and theorems that are used for proving triangles congruent. You have learned that there exists exactly one midpoint on any line segment. The proof below justifies the construction of a midpoint. E X A M P L E 1 Proving the Construction of a Midpoint Given: diagram showing the steps in the construction Prove: M is the midpoint of ̶̶ AB . To construct a midpoint, see the construction of a perpendicular bisector on p. 172. Proof: Statements Reasons 1. Draw ̶̶ AC , ̶̶ BC , ̶̶̶ AD , and ̶̶ BD . ̶̶ AC ≅ ̶̶ CD ≅ ̶̶ BC ≅ ̶̶ CD 2. 3. ̶̶̶ AD ≅ ̶̶ BD 4. △ACD ≅ △BCD 5. ∠ACD ≅ ∠BCD ̶̶̶ CM ≅ ̶̶̶ CM 6. 7. △ ACM ≅ △BCM ̶̶̶ AM ≅ ̶̶̶ BM 8. 9. M is the midpt. of ̶̶ AB . 1. Through any two pts. there is exactly one line. 2. Same compass setting used 3. Reflex. Prop. of ≅ 4. SSS Steps 2, 3 5. CPCTC 6. Reflex. Prop. of ≅ 7. SAS Steps 2, 5, 6 8. CPCTC 9. Def. of mdpt. 1. Given: above diagram Prove:   CD is the perpendicular bisector of ̶̶ AB . 282 282 Chapter 4 Triangle Congruence �� E X A M P L E 2 Proving the Construction of an Angle Given: diagram showing the steps in the construction Prove: ∠A ≅ ∠D To review the construction of an angle congruent to another angle, see page 22. Proof: Since there is a straight line through any two points, you can draw ̶̶ EF . The same compass setting was used to construct ̶̶ AB , ̶̶ DE . The same compass setting was used ̶̶ DF ≅ ̶̶ AC , ̶̶ DF , ̶̶ BC ≅ ̶̶ EF . Therefore △BAC ≅ △EDF by SSS, ̶̶ BC and ̶̶ DE , so and to construct and ∠A ≅ ∠D by CPCTC. ̶̶ AC ≅ ̶̶ BC and ̶̶ AB ≅ ̶̶ EF , so 2. Prove the construction for bisecting an angle. (See page 23.) EXTENSION Exercises Exercises Use each diagram to prove the construction valid. 1. parallel lines 2. a perpendicular through a point not (See page 163 and page 170.) on the line (See page 179.) 3. constructing a triangle using SAS 4. constructing a triangle using ASA (See page 243.) (See page 253.) Extension 283 283 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary acute triangle . . . . . . . . . . . . . . 216 CPCTC . . . . . . . . . . . . . . . . . . . . . 260 isosceles triangle . . . . . . . . . . . 217 auxiliary line . . . . . . . . . . . . . . . 223 equiangular triangle . . . . . . . . 216 legs of an isosceles triangle .
. 273 base . . . . . . . . . . . . . . . . . . . . . . . 273 equilateral triangle . . . . . . . . . 217 obtuse triangle . . . . . . . . . . . . . 216 base angle . . . . . . . . . . . . . . . . . . 273 exterior . . . . . . . . . . . . . . . . . . . . 225 remote interior angle . . . . . . . 225 congruent polygons . . . . . . . . . 231 exterior angle . . . . . . . . . . . . . . 225 right triangle . . . . . . . . . . . . . . . 216 coordinate proof . . . . . . . . . . . . 267 included angle . . . . . . . . . . . . . . 242 scalene triangle . . . . . . . . . . . . . 217 corollary . . . . . . . . . . . . . . . . . . . 224 included side . . . . . . . . . . . . . . . 252 triangle rigidity . . . . . . . . . . . . . 242 corresponding angles . . . . . . . 231 interior . . . . . . . . . . . . . . . . . . . . 225 vertex angle . . . . . . . . . . . . . . . . 273 corresponding sides . . . . . . . . . 231 interior angle . . . . . . . . . . . . . . . 225 Complete the sentences below with vocabulary words from the list above. 1. A(n) ? is a triangle with at least two congruent sides. ̶̶̶̶ 2. A name given to matching angles of congruent triangles is ? . ̶̶̶̶ 3. A(n) ? is the common side of two consecutive angles in a polygon. ̶̶̶̶ 4-1 Classifying Triangles (pp. 216–221) TEKS G.1.A E X A M P L E EXERCISES ■ Classify the triangle by its angle measures and side lengths. isosceles right triangle Classify each triangle by its angle measures and side lengths. 4. 5. 4-2 Angle Relationships in Triangles (pp. 223–230) TEKS G.1.A, G.2.B E X A M P L E ■ Find m∠S. EXERCISES Find m∠N. 6. 12x = 3x + 42 + 6x 12x = 9x + 42 3x = 42 x = 14 m∠S = 6 (14) = 84° 284 284 Chapter 4 Triangle Congruence 7. In△LMN, m∠L = 8x °, m∠M = (2x + 1) °, and m∠N = (6x - 1) °. �� 4-3 Congruent Triangles (pp. 231–237) TEKS G.2.B, G.10.B E X A M P L E EXERCISES ■ Given: △DEF ≅ △JKL. Identify all pairs of Given: △PQR ≅ △XYZ. Identify the congruent congruent corresponding parts. Then find the value of x. ̶̶ DE ≅ ∠F ≅ ∠L, The congruent pairs follow: ∠D ≅ ∠J, ∠E ≅ ∠K, ̶̶ EF ≅ Since m∠E = m∠K, 90 = 8x - 22. After 22 is added to both sides, 112 = 8x. So x = 14. ̶̶ KL , and ̶̶ DF ≅ ̶̶ JK , ̶̶ JL . corresponding parts. 8. ̶̶ PR ≅ ? ̶̶̶̶ 9. ∠Y ≅ ? ̶̶̶ Given: △ABC ≅ △CDA Find each value. 10. x 11. CD 4-4 Triangle Congruence: SSS and SAS (pp. 242–249) TEKS G.2.A, G.3.B, G.3.E, G.10.B ■ Given: E X A M P L E S ̶̶ ̶̶ UT , and RS ≅ ̶̶ ̶̶ VS ≅ VT . V is the midpoint of ̶̶ RU . Prove: △RSV ≅ △UTV Proof: Statements Reasons EXERCISES 12. Given: ̶̶ AB ≅ ̶̶ DB ≅ Prove: △ADB ≅ △DAE ̶̶ DE , ̶̶ AE 13. Given: ̶̶ GJ bisects ̶̶ FH bisects and Prove: △FGK ≅ △HJK ̶̶ FH , ̶̶ GJ . ̶̶ RS ≅ ̶̶ VS ≅ ̶̶ UT ̶̶ VT 1. 2. 3. V is the mdpt. of ̶̶ UV ̶̶ RV ≅ 4. ̶̶ RU . 1. Given 2. Given 3. Given 4. Def. of mdpt. 14. Show that △ABC ≅ △XYZ when x = -6. 5. △RSV ≅ △UTV 5. SSS Steps 1, 2, 4 ■ Show that △ADB ≅ △CDB when s = 5. AB = s 2 - 4s AD = 14 - 2s 15. Show that △LMN ≅ △PQR when y = 25. = 5 2 - 4 (5 ) = 5 ̶̶ BD by the Reflexive Property. = 14 - 2 (5 ) = 4 ̶̶ CB . So △ADB ≅ △CDB by SSS. ̶̶ BD ≅ ̶̶ AB ≅ and ̶̶ AD ≅ ̶̶ CD Study Guide: Review 285 285 �� 4-5 Triangle Congruence: ASA, AAS, and HL (pp. 252–259) TEKS G.1.A, G.1.B, G.2.A, E X A M P L E S ■ Given: B is the midpoint of ̶̶ AE . ∠A ≅ ∠E, ∠ABC ≅ ∠EBD Prove: △ABC ≅ △EBD EXERCISES 16. Given: C is the midpoint ̶̶ AG . of ̶̶ HA ǁ ̶̶ GB Prove: △HAC ≅ △BGC G.3.B, G.3.C, G.3.E, G.9.B, G.10.B Proof: Statements Reasons 1. ∠A ≅ ∠E 2. ∠ABC ≅ ∠EBD 3. B is the mdpt. of ̶̶ EB ̶̶ AB ≅ 4. ̶̶ AE . 1. Given 2. Given 3. Given 4. Def. of mdpt. 5. △ABC ≅ △EBD 5. ASA Steps 1, 4, 2 17. Given: ̶̶̶ WX ⊥ ̶̶ YZ ⊥ ̶̶̶ WZ ≅ Prove: △WZX ≅ △YXZ ̶̶ XZ , ̶̶ ZX , ̶̶ YX 18. Given: ∠S and ∠V are right angles. RT = UW. m∠T = m∠W Prove: △RST ≅ △UVW 4-6 Triangle Congruence: CPCTC (pp. 260–265) TEKS G.1.A, G.3.E, G.7.A, G.10.B E X A M P L E S ■ Given: ̶̶ JL and Prove: ∠JHG ≅ ∠LKG ̶̶ HK bisect each other. EXERCISES 19. Given: M is the midpoint ̶̶ BD . of ̶̶ BC ≅ ̶̶ DC Prove: ∠1 ≅ ∠2 Proof: Statements ̶̶ HK bisect ̶̶ 1. JL and each other. 2. ̶̶ JG ≅ ̶̶̶ HG ≅ ̶̶ LG , and ̶̶ KG . Reasons 1. Given 2. Def. of bisect 20. Given: ̶̶ RQ , ̶̶ RS ̶̶ PQ ≅ ̶̶ PS ≅ ̶̶ QS bisects ∠PQR. Prove: 3. ∠JGH ≅ ∠LGK 4. △JHG ≅ △LKG 3. Vert.  Thm. 4. SAS Steps 2, 3 5. ∠JHG ≅ ∠LKG 5. CPCTC 286 286 Chapter 4 Triangle Congruence 21. Given: H is the midpoint of L is the midpoint of ̶̶ ̶̶̶ ̶̶̶ GM ≅ KM , GJ ≅ ∠G ≅ ∠K Prove: ∠GMH ≅ ∠KJL ̶̶ KJ , ̶̶ GL . ̶̶̶ MK . �� 4-7 Introduction to Coordinate Proof (pp. 267–272) TEKS G.2.B, G.3.B, G.9.B, G.10.B E X A M P L E S EXERCISES ■ Given: ∠B is a right angle in isosceles right △ABC. E is the midpoint of ̶̶ AB ≅ D is the midpoint of ̶̶ CE ≅ Prove: Proof: Use the coordinates A(0, 2a) , B(0, 0) , ̶̶ CB . ̶̶ AB . ̶̶ AD ̶̶ CB and C (2a, 0) . Draw ̶̶ AD and ̶̶ CE . Position each figure in the coordinate plane and give the coordinates of each vertex. 22. a right triangle with leg lengths r and s 23. a rectangle with length 2p and width p 24. a square with side length 8m For exercises 25 and 26 assign coordinates to each vertex and write a coordinate proof. 25. Given: In rectangle ABCD, E is the midpoint of ̶̶ BC , G is the ̶̶ CD , and H is the midpoint ̶̶ AB , F is the midpoint of midpoint of ̶̶ of AD . ̶̶ EF ≅ ̶̶̶ GH Prove: 26. Given: △PQR has a right ∠Q . ̶̶ PR . M is the midpoint of Prove: MP = MQ = MR 27. Show that a triangle with vertices at (3, 5) , (3, 2) , and (2, 5) is a right triangle. By the Midpoint Formula, E = ( D = ( 2a + + 2a _ _ , 2 2 ) = (0, a) and ) = (a, 0) = (2a - 0) 2 + (0 - a) 2 By the Distance Formula, CE = √  √  4a 2 + a 2 = a √  5 √  (a - 0) 2 + (0 - 2a) 2 √  = ̶̶ AD by the definition of congruence. a 2 + 4a 2 = a √  5 AD = ̶̶ CE ≅ Thus 4-8 Isosceles and Equilateral Triangles (pp. 273–279) TEKS G.2.B, G.3.C, G.10.B E X A M P L E ■ Find the value of x. m∠D + m∠E + m∠F = 180° by the Triangle Sum Theorem. m∠E = m∠F by the Isosceles Triangle Theorem. m∠D + 2 m∠E = 180° Substitution 42 + 2 (3x) = 180 Substitute the given 6x = 138 x = 23 values. Simplify. Divide both sides by 6. EXERCISES Find each value. 28. x 29. RS 30. Given: △ACD is isosceles with ∠D as the vertex angle. B is the midpoint of AB = x + 5, BC = 2x - 3, and CD = 2x + 6. ̶̶ AC . Find the perimeter of △ACD. Study Guide: Review 287 287 �� 1. Classify △ACD by its angle measures. Classify each triangle by its side lengths. 2. △ACD 3. △ABC 4. △ABD 5. While surveying the triangular plot of land shown, a surveyor finds that m∠S = 43°. The measure of ∠RTP is twice that of ∠RTS. What is m∠R? Given: △XYZ ≅ △JKL Identify the congruent corresponding parts. 6. ̶̶ JL ≅ ? ̶̶̶̶ 10. Given: T is the midpoint of Prove: △PTS ≅ △RTQ 7. ∠Y ≅ ̶̶ PR and ? ̶̶̶̶ ̶̶ SQ . 8. ∠L ≅ ? ̶̶̶̶ ̶̶ YZ ≅ 9. ? ̶̶̶̶ 11. The figure represents a walkway with triangular supports. Given that ∠HGK and ∠H ≅ ∠K, use AAS to prove △HGJ ≅ △KGJ ̶̶ GJ bisects 12. Given: ̶̶ AB ≅ ̶̶ AB ⊥ ̶̶ DC ⊥ Prove: △ABC ≅ △DCB ̶̶ DC , ̶̶ AC , ̶̶ DB 13. Given: Prove: ̶̶ ̶̶ PQ ǁ SR , ∠S ≅ ∠Q ̶̶ ̶̶ QR PS ǁ 14. Position a right triangle with legs 3 m and 4 m long in the coordinate plane. Give the coordinates of each vertex. 15. Assign coordinates to each vertex and write a coordinate proof. Given: Square ABCD ̶̶ AC ≅ Prove: ̶̶ BD Find each value. 16. y 17. m∠S 18. Given: Isosceles △ABC has coordinates A (2a, 0) , B (0, 2b) , and C (-2a, 0) . D is the midpoint of ̶̶ AC , and E is the midpoint of ̶̶ AB . Prove: △AED is isosceles. 288 288 Chapter 4 Triangle Congruence �� FOCUS ON ACT The ACT Mathematics Test is one of four tests in the ACT. You are given 60 minutes to answer 60 multiplechoice questions. The questions cover material typically taught through the end of eleventh grade. You will need to know basic formulas but nothing too difficult. You may want to time yourself as you take this practice test. It should take you about 5 minutes to complete. There is no penalty for guessing on the ACT. If you are unsure of the correct answer, eliminate as many answer choices as possible and make your best guess. Make sure you have entered an answer for every question before time runs out. 1. For the figure below, which of the following must be true? 3. Which of the following best describes a triangle with vertices having coordinates (-1, 0) , (0, 3) , and (1, -4) ? I. m∠EFG > m∠DEF II. m∠EDF = m∠EFD III. m∠DEF + m∠EDF > m∠EFG (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III 2. In the figure below, △ABD ≅ △CDB, m∠A = (2x + 14) °, m∠C = (3x - 15) °, and m∠DBA = 49°. What is the measure of ∠BDA? (F) 29° (G) 49° (H) 59° (J) 72° (K) 101° (A) Equilateral (B) Isosceles (C) Right (D) Scalene (E) Equiangular 4. In the figure below, what is the value of y? (F) 49 (G) 87 (H) 93 (J) 131 (K) 136 5. In △RST, RS = 2x + 10, ST = 3x - 2, and RT = 1 __ 2 x + 28. If △RST is equiangular, what is the value of x? (A) 2 (B) 5 1 _ 3 (C) 6 (D) 12 (E) 34 College Entrance Exam Practice 289 289 �� Any Question Type: Identify Key Words and Context Clues When reading a test item, you should pay attention to key words and context clues given in the problem statement. These clues will guide you in providing a correct response. Multiple Choice What is the side length of an equilateral triangle with a perimeter of 42 3 __ 4 cm? 42 3 __ 4 cm 24 3 __ 7 cm 21 3 __ 8 cm 14 1 __ 4 cm LOOK for key words and context clues and underline them. Identify what they mean. What is the side length of an equilateral triangle with a perimeter of 42 3 __ 4 in.? equilateral triangle → → perimeter perimeter = 3 (length of one side) a triangle with three congruent sides the distance around a figure = 3 (x) = 3 (x) _ 3 42 3 _ 4 42 3 __ 4 _ 3 171 _ · 1 _
4 3 14 1 _ 4 = x = x You find the perimeter of an equilateral triangle by multiplying the length of one side of the triangle by three. The correct choice is D because the length of the side of the equilateral triangle is 14 1 __ cm. 4 Gridded Response The vertex angle of an isosceles triangle measures (5t - 5) °, and one of the base angles measures (t + 5) °. Find t. → isosceles triangle → vertex angle → base angles a triangle with at least two congruent sides the angle formed by the legs The side opposite the vertex angle is called the base, and the base angles are the two angles that have the base as a side. 2(measure of the base angle) + (measure of the vertex angle) = 180° 2 (t + 5 ) + (5t - 5 ) = 180 2t + 10 + 5t - 5 = 180 7t + 5 = 180 t = 25 The correct value for t is 25. 290 290 Chapter 4 Triangle Congruence �� If you do not understand what a word means, reread the sentences that contain the word and make a logical guess. 6. How will you use the abbreviation SSS to help you answer the question? Read each test item and answer the questions that follow. Item A Multiple Choice Which value of k would make △CDE isosceles. Whether a triangle is isosceles depends on what characteristics of the triangle? 2. What do 2k + 1, 3k - 7, and 5k - 6 represent in the model? 3. How will you use the definition of an isosceles triangle to find the correct value of k? Item C Multiple Choice ∠X and ∠Y are the remote interior angles of ∠YZW in △XYZ. Which of these equations must be true? 180° - m∠X = m∠YZW m∠X = m∠Y + 90° m∠X = m∠YZW - m∠Y m∠YZW = m∠YZX - m∠YXZ 7. Create a drawing that represents the situation. Label the remote interior angles. 8. What is the relationship between the remote interior angles and an exterior angle? 9. How can you manipulate the relationship given in Problem 8 to get one of the four choices? Item D Multiple Choice Which of the following is a correct classification of △FGH? Item B Gridded Response What must the value of x be in order to prove that △MNQ ≅ △PNQ by SSS? Acute Equiangular Isosceles Scalene 4. What statement are you trying to prove? 5. Explain the meaning of the symbol ≅. 10. What are the two ways by which triangles can be classified? 11. What must be true for the triangle to be classified as acute? as equiangular? 12. What must be true for the triangle to be classified as isosceles? as scalene? TAKS Tackler 291 291 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–4 Multiple Choice Use the diagram for Items 1 and 2. 6. Which conditional statement has the same truth value as its inverse? If n < 0, then n 2 > 0. If a triangle has three congruent sides, then it is an isosceles triangle. If an angle measures less than 90°, then it is an acute angle. If n is a negative integer, then n < 0. 1. Which of these congruence statements can be proved from the information given in the figure? △AEB ≅ △CED △ABD ≅ △BCA △BAC ≅ △DAC △DEC ≅ △DEA 7. On a map, an island has coordinates (3, 5) , and a reef has coordinates (6, 8) . If each map unit represents 1 mile, what is the distance between the island and the reef to the nearest tenth of a mile? 2. What other information is needed to prove that △CEB ≅ △AED by the HL Congruence Theorem? ̶̶̶ AD ≅ ̶̶ BE ≅ ̶̶ AB ̶̶ AE ̶̶ CB ≅ ̶̶ DE ≅ ̶̶̶ AD ̶̶ CE 3. Which biconditional statement is true? Tomorrow is Monday if and only if today is not Saturday. Next month is January if and only if this month is December. Today is a weekend day if and only if yesterday was Friday. This month had 31 days if and only if last month had 30 days. 4. What must be true if   PQ intersects   ST at more than one point? P, Q, S, and T are collinear. P, Q, S, and T are noncoplanar.  PQ and  ST are opposite rays.   PQ and   ST are perpendicular. 5. △ABC ≅ △DEF, EF = x 2 - 7, and BC = 4x - 2. Find the values of x. -1 and 5 -1 and 6 1 and 5 2 and 3 292 292 Chapter 4 Triangle Congruence 4.2 miles 6.0 miles 9.0 miles 15.8 miles 8. A line has an x-intercept of -8 and a y-intercept of 3. What is the equation of the line? y = -8x + = 3x - 8 9.   JK passes through points J (1, 3) and K (-3, 11) . JK ? Which of these lines is perpendicular to   y = -2x - = 2x - 4 10. If PQ = 2 (RS) + 4 and RS = TU + 1, which equation is true by the Substitution Property of Equality? PQ = TU + 5 PQ = TU + 6 PQ = 2 (TU) + 5 PQ = 2 (TU) + 6 11. Which of the following is NOT valid for proving that triangles are congruent? AAA ASA SAS HL �� Use this diagram for Items 12 and 13. STANDARDIZED TEST PREP Short Response 20. Given ℓ ǁ m with transversal n, explain why ∠2 and ∠3 are complementary. 12. What is the measure of ∠ACD? 40° 80° 100° 140° 13. What type of triangle is △ABC? Isosceles acute Equilateral acute Isosceles obtuse Scalene acute �� Take some time to learn the directions for filling in a grid. Check and recheck to make sure you are filling in the grid properly. You will only get credit if the ovals below the boxes are filled in correctly. To check your answer, solve the problem using a different method from the one you originally used. If you made a mistake the first time, you are unlikely to make the same mistake when you solve a different way. Gridded Response 14. △CDE ≅ △JKL. m∠E = (3x + 4) °, and m∠L = (6x - 5) °. What is the value of x? 15. Lucy, Eduardo, Carmen, and Frank live on the same street. Eduardo’s house is halfway between Lucy’s house and Frank’s house. Lucy’s house is halfway between Carmen’s house and Frank’s house. If the distance between Eduardo’s house and Lucy’s house is 150 ft, what is the distance in feet between Carmen’s house and Eduardo’s house? 16. △JKL ≅ △XYZ, and JK = 10 - 2n. XY = 2, and YZ = n 2 . Find KL. 21. ∠G and ∠H are supplementary angles. m∠G = (2x + 12) °, and m∠H = x°. a. Write an equation that can be used to determine the value of x. Solve the equation and justify each step. b. Explain why ∠H has a complement but ∠G does not. 22. A manager conjectures that for every 1000 parts a factory produces, 60 are defective. a. If the factory produces 1500 parts in one day, how many of them can be expected to be defective based on the manager’s conjecture? Explain how you found your answer. b. Use the data in the table below to show that the manager’s conjecture is false. Day Parts Defective Parts 1 2 3 4 5 1000 2000 500 1500 2500 60 150 30 90 150 23. ̶̶ BD is the perpendicular bisector of ̶̶ AC . a. What are the conclusions you can make from this statement? b. Suppose ̶̶ BD intersects is the shortest path from B to ̶̶ AC . ̶̶ AC at D. Explain why ̶̶ BD Extended Response 24. △ABC and △DEF are isosceles triangles. ̶̶ AC ≅ ̶̶ DF . m∠C = 42.5°, and m∠E = 95°. and ̶̶ BC ≅ ̶̶ EF , 17. An angle is its own supplement. What is your answer. a. What is m∠D? Explain how you determined its measure? 18. The area of a circle is 154 square inches. What is its circumference to the nearest inch? 19. The measure of ∠P is 3 1 __ times the measure of ∠Q. 2 If ∠P and ∠Q are complementary, what is m∠P in degrees? b. Show that △ABC and △DEF are congruent. c. Given that EF = 2x + 7 and AB = 3x + 2, find the value for x. Explain how you determined your answer. Cumulative Assessment, Chapters 1–4 293 293 �� T E X A S TAKS Grades 9–11 Obj. 10 Addison Longview Longview Cavanaugh Flight Museum Located on the grounds of the Addison Airport in Addison, Texas, the Cavanaugh Flight Museum offers visitors a thrilling voyage through the history of U.S. ge07ts_c04psl001aa military flight. The museum exhibits aircraft and other 2nd pass military artifacts on almost 50,000 square feet of display 6/21/5 area. It also features an informative self-guided tour cmurphy and offers rides in two classic airplanes. Choose one or more strategies to solve each problem. Airplane Number Built Number Left in 2005 Year First Built N2S-4 Stearman 10,346 Fairchild PT-19 4,889 Spitfire Mk VIII 20,334 F9F-2 Panther 761 2136 272 70 9 1933 1938 1943 1947 2. Visitors to the museum can see a replica of an N2S-4 Stearman “Yellow Peril,” a plane used to train pilots during World War II. ̶̶ AB and The plane has two parallel wings that are connected by bracing wires. The wires are arranged such that m∠EFG = 29° ̶̶ GF bisects ∠EGD. What is m∠AEG? and ̶̶ CD � �� 294 294 Chapter 4 Triangle Congruence �� 1. The table gives data on some of the aircraft on display in the museum. Suppose the number of each type of aircraft declines at the same rate each year. Find the yearly rate of change for each given year. Round to the nearest whole number. � � � � � � � 3. Visitors have the opportunity to ride in either the N2S-4 Stearman or the AT-6 Texan. If the airport uses two cameras mounted 1000 ft apart to determine the position of a plane during landing and takeoff, what is the distance d that the plane in the diagram has moved along the runway since it passed camera 1? Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List The Great Texas Balloon Race The annual Great Texas Balloon Race is one of the most exciting hot air balloon events in Texas. “Balloon Glow,” in which balloons are tethered and illuminated in an evening display, was begun in Longview, the race’s starting point, in 1980. Traditionally held in July, the race attracts balloonists who compete to fly the obstacle course the most accurately. Choose one or more strategies to solve each problem. 1. The event starts in Longview, and ends near Estes, Texas. The balloons do not fly from the start to the finish in a straight line. They follow a zigzag course to take advantage of the wind. Suppose one of the balloons leaves Longview at a bearing of N 50° E and follows the course shown. At what bearing does the balloon approach Estes? � � ��
�� 2. The speed of the balloon depends on the current wind speed. One event in The Great Texas Balloon Race requires the balloonist to fly to a pole that is 2 mi from the starting point. The balloonist must drop a small ring around the pole, which is 20 ft tall. A second target is 1 mi from the first, a third target is another 3 mi from the second, and a final target is 5 mi farther. If the wind speed is 3.5 mi/h, how long will it take the balloonist to finish the course? Round to the nearest hundredth of an hour. 3. During the race, one of the balloons leaves Longview L, flies to X, and then flies to Y. The team discovers a problem with the balloon, so it must return directly to Longview. Does the table contain enough information to determine the return course to L? Explain to X X to Y Y to L Bearing Distance (mi) N 42° E S 59° E 3.1 2.4 Problem Solving on Location 295 295 Properties and Attributes of Triangles 5A Segments in Triangles 5-1 Perpendicular and Angle Bisectors 5-2 Bisectors of Triangles 5-3 Medians and Altitudes of Triangles Lab Special Points in Triangles 5-4 The Triangle Midsegment Theorem 5B Relationships in Triangles Lab Explore Triangle Inequalities 5-5 Indirect Proof and Inequalities in One Triangle 5-6 Inequalities in Two Triangles Lab Hands-on Proof of the Pythagorean Theorem 5-7 The Pythagorean Theorem 5-8 Applying Special Right Triangles Lab Graph Irrational Numbers KEYWORD: MG7 ChProj The Broken Obelisk in Houston was dedicated in 1971 as a memorial to Martin Luther King Jr. 296 296 Chapter 5 Vocabulary Match each term on the left with a definition on the right. 1. angle bisector A. the side opposite the right angle in a right triangle 2. conclusion 3. hypotenuse 4. leg of a right triangle 5. perpendicular bisector of a segment B. a line that is perpendicular to a segment at its midpoint C. the phrase following the word then in a conditional statement D. one of the two sides that form the right angle in a right triangle E. a line or ray that divides an angle into two congruent angles F. the phrase following the word if in a conditional statement Classify Triangles Tell whether each triangle is acute, right, or obtuse. 7. 6. 8. 9. Squares and Square Roots Simplify each expression. 10. 8 2 11. (-12)2 Simplify Radical Expressions Simplify each expression. 14. √9 + 16 15. √ 100 - 36 12. √49 13. - √36 16. √81_ 25 17. √ 2 2 Solve and Graph Inequalities Solve each inequality. Graph the solutions on a number line. 19. -4 ≤ w - 7 18. d + 5 < 1 20. -3s ≥ 6 21. -2 > m_ 10 Logical Reasoning Draw a conclusion from each set of true statements. 22. If two lines intersect, then they are not parallel. Lines ℓ and m intersect at P. 23. If M is the midpoint of ̶̶ AB , then AM = MB. If AM = MB, then AM = 1 __ 2 AB and MB = 1 __ 2 AB. Properties and Attributes of Triangles 297 297 �� Key Vocabulary/Vocabulario altitude of a triangle altura de un triángulo centroid of a triangle centroide de un triángulo circumcenter of a triangle circuncentro de un triángulo concurrent equidistant concurrente equidistante incenter of a triangle incentro de un triángulo median of a triangle mediana de un triángulo midsegment of a triangle segmento medio de un triángulo orthocenter of a triangle orthocentro de un triángulo Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. In Latin, co means “together with,” and currere means “to run.” How can you use these meanings to understand what concurrent lines are? 2. The endpoints of a midsegment of a triangle are on two sides of the triangle. Where on the sides do you think the endpoints are located? 3. The strip of concrete or grass in the middle of some roadways is called the median. What do you think the term median of a triangle means? 4. Think of the everyday meaning of altitude. What do you think the altitude of a triangle is? Geometry TEKS Les. 5-1 Les. 5-2 Les. 5-3 5-3 Tech. Lab 5-5 Geo. Lab Les. 5-4 Les. 5-5 Les. 5-6 5-7 Geo. Lab Les. 5-7 Les. 5-8 G.2.A Geometric structure* use constructions to ★ ★ ★ ★ explore attributes of geometric figures and to make conjectures ... G.3.B Geometric structure* construct and justify ★ ★ ★ ★ ★ ★ ★ 5-8 Geo. Lab ★ statements about geometric figures and their properties G.5.D Geometric patterns* identify and apply patterns from right triangles ... including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples G.7.B Dimensionality and the geometry of location* ... investigate geometric relationships, including ... special segments of triangles ... G.8.C Congruence and the geometry of size* derive, extend, and use the Pythagorean Theorem G.9.B Congruence and the geometry of size* formulate and test conjectures about ... polygons and their component parts ... G.11.C Similarity and the geometry of shape* develop, apply, and justify triangle similarity relationships, such as ... Pythagorean triples ... ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 298 298 Chapter 5 Reading Strategy: Learn Math Vocabulary Mathematics has a vocabulary all its own. To learn and remember new vocabulary words, use the following study strategies. • Try to figure out the meaning of a new word based on its context. • Use a dictionary to look up the root word or prefix. • Relate the new word to familiar everyday words. Once you know what a word means, write its definition in your own words. Term Study Notes Definition Polygon The prefix poly means “many” or “several.” Bisect Slope Intersection The prefix bi means “two.” Think of a ski slope. The root word intersect means “to overlap.” Think of the intersection of two roads. A closed plane figure formed by three or more line segments Cuts or divides something into two equal parts The measure of the steepness of a line The set of points that two or more lines have in common Try This Complete the table below. Term Study Notes Definition 1. Trinomial 2. Equiangular triangle 3. Perimeter 4. Deductive reasoning Use the given prefix and its meanings to write a definition for each vocabulary word. 5. circum (about, around); circumference 6. co (with, together); coplanar 7. trans (across, beyond, through); translation Properties and Attributes of Triangles 299 299 5-1 Perpendicular and Angle Bisectors TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.3.E, G.7.A, G.7.B, G.7.C, G.10.B Objectives Prove and apply theorems about perpendicular bisectors. Prove and apply theorems about angle bisectors. Vocabulary equidistant locus Who uses this? The suspension and steering lines of a parachute keep the sky diver centered under the parachute. (See Example 3.) When a point is the same distance from two or more objects, the point is said to be equidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points. Theorems Distance and Perpendicular Bisectors THEOREM HYPOTHESIS CONCLUSION 5-1-1 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. 5-1-2 Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. XA = XB ̶̶ XY ⊥ ̶̶ YA ≅ ̶̶ AB ̶̶ YB ̶̶ XY ⊥ ̶̶ YA ≅ ̶̶ AB ̶̶ YB XA = XB You will prove Theorem 5-1-2 in Exercise 30. PROOF PROOF Perpendicular Bisector Theorem Given: ℓ is the perpendicular bisector of Prove: XA = XB ̶̶ AB . The word locus comes from the Latin word for location. The plural of locus is loci, which is pronounced LOW-sigh. Proof: ̶̶ AB , ℓ ⊥ ̶̶ AB and Y is the midpoint ̶̶ AB . By the definition of perpendicular, ∠AYX and ∠BYX are right ̶̶ BY . Since ℓ is the perpendicular bisector of of angles and ∠AYX ≅ ∠BYX. By the definition of midpoint, By the Reflexive Property of Congruence, by SAS, and of congruent segments. ̶̶ XY . So △AYX ≅ △BYX ̶̶ XB by CPCTC. Therefore XA = XB by the definition ̶̶ XA ≅ ̶̶ XY ≅ ̶̶ AY ≅ A locus is a set of points that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment. 300 300 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. A YW YW = XW YW = 7.3 B BC ⊥ Bisector Thm. Substitute 7.3 for XW. ̶̶ BC , ℓ is the perpendicular Since AB = AC and ℓ ⊥ ̶̶ BC by the Converse of the bisector of Perpendicular Bisector Theorem. BC = 2CD BC = 2 (16) = 32 Def. of seg. bisector Substitute 16 for CD. C PR PR = RQ 2n + 9 = 7n - 18 9 = 5n - 18 27 = 5n 5.4 = n ⊥ Bisector Thm. Substitute the given values. Subtract 2n from both sides. Add 18 to both sides. Divide both sides by 5. So PR = 2 (5.4) + 9 = 19.8. Find each measure. 1a. Given that line ℓ is the perpendicular ̶̶ DE and EG = 14.6, find DG. bisector of 1b. Given that DE = 20.8, DG = 36.4, and EG = 36.4, find EF. Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line. Theorems Distance and Angle Bisectors THEOREM HYPOTHESIS CONCLUSION 5-1-3 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. 5-1-4 Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. ∠APC ≅ ∠BPC AC = BC ∠APC ≅ ∠BPC AC = BC You will prove these theorems in Exercises 31 and 40. 5- 1 Perpendicular and Angle Bisectors 301 301 �� Based on these theorems, an angle bisector can be defined
as the locus of all points in the interior of the angle that are equidistant from the sides of the angle. E X A M P L E 2 Applying the Angle Bisector Theorems Find each measure. A LM LM = JM LM = 12.8 ∠ Bisector Thm. Substitute 12.8 for JM. B m∠ABD, given that m∠ABC = 112° ̶̶ BA , and ̶̶ AD ⊥  BD bisects ∠ABC Since AD = DC, ̶̶ ̶̶ DC ⊥ BC , by the Converse of the Angle Bisector Theorem. m∠ABD = 1 _ 2 m∠ABD = 1 _ (112°) = 56° Substitute 112° for m∠ABC. 2 Def. of ∠ bisector m∠ABC C m∠TSU ̶̶ UT ⊥ ̶̶ SR , and ̶̶ ̶̶ Since RU = UT, ST , RU ⊥  SU bisects ∠RST by the Converse of the Angle Bisector Theorem. m∠RSU = m∠TSU 6z + 14 = 5z + 23 z + 14 = 23 Def. of ∠ bisector Substitute the given values. Subtract 5z from both sides. z = 9 Subtract 14 from both sides. ⎤ ⎦ ⎡ ⎣ ° = 68°. 5 (9) + 23 So m∠TSU = Find each measure. 2a. Given that  YW bisects ∠XYZ and WZ = 3.05, find WX. 2b. Given that m∠WYZ = 63°, XW = 5.7, and ZW = 5.7, find m∠XYZ. E X A M P L E 3 Parachute Application Each pair of suspension lines on a parachute are the same length and are equally spaced from the center of the chute. How do these lines keep the sky diver centered under the parachute? ̶̶ PQ ≅ ̶̶ RQ . So Q is It is given that on the perpendicular bisector ̶̶ PR by the Converse of the of Perpendicular Bisector Theorem. Since S is the midpoint of ̶̶ QS is the perpendicular bisector ̶̶ PR . Therefore the sky diver of remains centered under the chute. ̶̶ PR , 302 302 Chapter 5 Properties and Attributes of Triangles �� 3. S is equidistant from each pair of suspension lines. What can you conclude about  QS ? E X A M P L E 4 Writing Equations of Bisectors in the Coordinate Plane Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints A (-1, 6) and B (3, 4) . ̶̶ AB . Step 1 Graph The perpendicular bisector of perpendicular to ̶̶ AB is ̶̶ AB at its midpoint. ̶̶ AB . Step 2 Find the midpoint of AB = ( 6 + 4 -1 + 3 _ _ , 2 2 mdpt. of ̶̶ ) = (1, 5) Midpoint formula Step 3 Find the slope of the perpendicular bisector. slope = slope of ̶̶ AB = 4 - 6 _ 3 - (-1) Slope formula = -2 _ = - 1 _ 4 2 Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is 2. Step 4 Use point-slope form to write an equation. The perpendicular bisector of ̶̶ AB has slope 2 and passes through (1, 5) . y - y 1 = m (x - 1 ) Point-slope form Substitute 5 for y 1 , 2 for m, and 1 for x 1 . 4. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints P (5, 2) and Q (1, -4) . THINK AND DISCUSS ̶̶ PQ ? Is it a perpendicular 1. Is line ℓ a bisector of bisector of ̶̶ PQ ? Explain. 2. Suppose that M is in the interior of ∠JKL and MJ = ML. Can you conclude that is the bisector of ∠JKL? Explain.  KM 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the theorem or its converse in your own words. 5- 1 Perpendicular and Angle Bisectors 303 303 �� 5-1 Exercises Exercises KEYWORD: MG7 5-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary A from the endpoints of a segment. (perpendicular bisector or angle bisector) ? is the locus of all points in a plane that are equidistant ̶̶̶̶ Use the diagram for Exercises 2–4. p. 301 2. Given that PS = 53.4, QT = 47.7, and QS = 53.4, find PQ. 3. Given that m is the perpendicular bisector ̶̶ PQ and SQ = 25.9, find SP. of 4. Given that m is the perpendicular bisector ̶̶ PQ , PS = 4a, and QS = 2a + 26, find QS. of Use the diagram for Exercises 5–7. p. 302 5. Given that  BD bisects ∠ABC and CD = 21.9, find AD. 6. Given that AD = 61, CD = 61, and m∠ABC = 48°, find m∠CBD. 7. Given that DA = DC, m∠DBC = (10y + 3) °, and m∠DBA = (8y + 10) °, find m∠DBC. 302 8. Carpentry For a king post truss to be constructed correctly, P must lie on the bisector of ∠JLN. How can braces and the proper location? ̶̶ PK ̶̶̶ PM be used to ensure that P is in . 303 Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 9. M (-5, 4) , N (1, -2) 10. U (2, -6) , V (4, 0) 11. J (-7, 5) , K (1, -1) Independent Practice Use the diagram for Exercises 12–14. PRACTICE AND PROBLEM SOLVING For See Exercises Example 12–14 15–17 18 19–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 12. Given that line t is the perpendicular bisector ̶̶ JK and GK = 8.25, find GJ. of 13. Given that line t is the perpendicular bisector ̶̶ JK , JG = x + 12, and KG = 3x - 17, find KG. of 14. Given that GJ = 70.2, JH = 26.5, and GK = 70.2, find JK. Use the diagram for Exercises 15–17. 15. Given that m∠RSQ = m∠TSQ and TQ = 1.3, find RQ. 16. Given that m∠RSQ = 58°, RQ = 49, and TQ = 49, find m∠RST. 17. Given that RQ = TQ, m∠QSR = (9a + 48) °, and m∠QST = (6a + 50) °, find m∠QST. 304 304 Chapter 5 Properties and Attributes of Triangles ��ge07sec05l01003aa1st pass4/4/5cmurphyLMNKJP�� 18. City Planning The planners for a new section of the city want every location on Main Street to be equidistant from Elm Street and Grove Street. How can the planners ensure that this is the case? Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 19. E (-4, -7) , F (0, 1) 20. X (-7, 5) , Y (-1, -1) ̶̶ ST . 22. ̶̶ PQ is the perpendicular bisector of Find the values of m and n. 21. M (-3, -1) , N (7, -5) Shuffleboard One of the first recorded shuffleboard games was played in England in 1532. In this game, Henry VIII supposedly lost £9 to Lord William. Shuffleboard Use the diagram of a shuffleboard and the following information to find each length in Exercises 23–28. ̶̶ KZ is the perpendicular bisector of ̶̶̶ HM , and ̶̶̶ GN , ̶̶ JL . 23. JK 24. GN 25. ML 27. JL 26. HY 29. Multi-Step The endpoints of 28. NM ̶̶ AB are A (-2, 1) and B (4, -3) . Find the coordinates of a point C other than the midpoint of you know it is on the perpendicular bisector? � �� ̶̶ AB that is on the perpendicular bisector of ̶̶ AB . How do 30. Write a paragraph proof of the Converse of the Perpendicular Bisector Theorem. Given: AX = BX Prove: X is on the perpendicular bisector of ̶̶ AB . Plan: Draw ℓ perpendicular to ̶̶ △AYX ≅ △BYX and thus AY ≅ the perpendicular bisector of ̶̶ AB through X. Show that ̶̶ BY . By definition, ℓ is ̶̶ AB . 31. Write a two-column proof of the Angle Bisector Theorem. ̶̶ SQ ⊥  PS bisects ∠QPR. ̶̶ SR ⊥  PQ ,  PR Given: Prove: SQ = SR Plan: Use the definitions of angle bisector and perpendicular to identify two pairs of congruent angles. Show that △PQS ≅ △PRS and thus ̶̶ SQ ≅ ̶̶ SR . 32. Critical Thinking In the Converse of the Angle Bisector Theorem, why is it important to say that the point must be in the interior of the angle? 33. This problem will prepare you for the Multi-Step TAKS Prep on page 328. A music company has stores in Abby (-3, -2) and Cardenas (3, 6) . Each unit in the coordinate plane represents 1 mile. a. The company president wants to build a warehouse that is equidistant from the two stores. Write an equation that describes the possible locations. b. A straight road connects Abby and Cardenas. The warehouse will be located exactly 4 miles from the road. How many locations are possible? c. To the nearest tenth of a mile, how far will the warehouse be from each store? 5- 1 Perpendicular and Angle Bisectors 305 305 Elm StreetMain StreetGrove StreetHolt, Rinehart & WinstonGeometry 2007ge07se_c05l01004a 1st proofCity Planning map�� 34. Write About It How is the construction of the perpendicular bisector of a segment related to the Converse of the Perpendicular Bisector Theorem? 35. If   JK is perpendicular to JX = KY ̶̶ XY at its midpoint M, which statement is true? JX = KX JM = KM JX = JY 36. What information is needed to conclude that m∠DEF = m∠DEG m∠FEG = m∠DEF  EF is the bisector of ∠DEG? m∠GED = m∠GEF m∠DEF = m∠EFG 37. Short Response The city wants to build a visitor center in the park so that it is equidistant from Park Street and Washington Avenue. They also want the visitor center to be equidistant from the museum and the library. Find the point V where the visitor center should be built. Explain your answer. CHALLENGE AND EXTEND 38. Consider the points P (2, 0) , A (-4, 2) , B (0, -6) , and C (6, -3) . a. Show that P is on the bisector of ∠ABC. b. Write an equation of the line that contains the bisector of ∠ABC. 39. Find the locus of points that are equidistant from the x-axis and y-axis. 40. Write a two-column proof of the Converse of the Angle Bisector Theorem. Given: Prove: ̶̶ ̶̶ VZ ⊥  YX , VX ⊥  YV bisects ∠XYZ.  YZ , VX = VZ 41. Write a paragraph proof. Given: ̶̶ KN is the perpendicular bisector of ̶̶ LN is the perpendicular bisector of ̶̶ JR ≅ Prove: ∠JKM ≅ ∠MLJ ̶̶̶ MT ̶̶ JL . ̶̶̶ KM . SPIRAL REVIEW 42. Lyn bought a sweater for $16.95. The change c that she received can be described by c = t - 16.95, where t is the amount of money Lyn gave the cashier. What is the dependent variable? (Previous course) For the points R (-4, 2) , S (1, 4) , T (3, -1) , and V (-7, -5) , determine whether the lines are parallel, perpendicular, or neither. (Lesson 3-5) 43.   RS and   VT 44.   RV and   ST 45.   RT and   VR Write the equation of each line in slope-intercept form. (Lesson 3-6) 46. the line through the points (1, -1) and (2, -9) 47. the line with slope -0.5 through (10, -15) 48. the line with x-intercept -4 and y-intercept 5 306 306 Chapter 5 Properties and Attributes of Triangles ge07sec05l01006a3rd Pass2/24/05C MurphyMuseumPark StreetWashington AvenueLibrary�� 5-2 Bisectors of Triangles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.7.A, G.7.B Objectives Prove and apply properti
es of perpendicular bisectors of a triangle. Prove and apply properties of angle bisectors of a triangle. Vocabulary concurrent point of concurrency circumcenter of a triangle circumscribed incenter of a triangle inscribed The perpendicular bisector of a side of a triangle does not always pass through the opposite vertex. Who uses this? An event planner can use perpendicular bisectors of triangles to find the best location for a fireworks display. (See Example 4.) Since a triangle has three sides, it has three perpendicular bisectors. When you construct the perpendicular bisectors, you find that they have an interesting property. Construction Circumcenter of a Triangle    Draw a large scalene acute triangle ABC on a piece of patty paper. Fold the perpendicular bisector of each side. Label the point where the three perpendicular bisectors intersect as P. When three or more lines intersect at one point, the lines are said to be concurrent . The point of concurrency is the point where they intersect. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle . Theorem 5-2-1 Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle. PA = PB = PC The circumcenter can be inside the triangle, outside the triangle, or on the triangle. 5- 2 Bisectors of Triangles 307 307 �� The circumcenter of △ABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon. PROOF PROOF Circumcenter Theorem Given: Lines ℓ, m, and n are the perpendicular ̶̶ ̶̶ AC , respectively. BC , and ̶̶ AB , bisectors of Prove: PA = PB = PC Proof: P is the circumcenter of △ABC. Since P lies on the perpendicular bisector of by the Perpendicular Bisector Theorem. Similarly, P also lies on the perpendicular bisector of by the Transitive Property of Equality. ̶̶ BC , so PB = PC. Therefore PA = PB = PC ̶̶ AB , PA = PB E X A M P L E 1 Using Properties of Perpendicular Bisectors ̶̶ LZ , and ̶̶ KZ , of △GHJ. Find HZ. ̶̶ MZ are the perpendicular bisectors Z is the circumcenter of △GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of △GHJ. HZ = GZ HZ = 19.9 Circumcenter Thm. Substitute 19.9 for GZ. Use the diagram above. Find each length. 1a. GM 1b. GK 1c. JZ E X A M P L E 2 Finding the Circumcenter of a Triangle Find the circumcenter of △RSO with vertices R (-6, 0) , S (0, 4) , and O (0, 0) . Step 1 Graph the triangle. Step 2 Find equations for two perpendicular bisectors. Since two sides of the triangle lie along the axes, use the graph to find the perpendicular bisectors of these two sides. The perpendicular bisector of x = -3, and the perpendicular bisector of ̶̶ OS is y = 2. ̶̶ RO is Step 3 Find the intersection of the two equations. The lines x = -3 and y = 2 intersect at (-3, 2 ) , the circumcenter of △RSO. 308 308 Chapter 5 Properties and Attributes of Triangles �� 2. Find the circumcenter of △GOH with vertices G (0, -9) , O (0, 0) , and H (8, 0) . A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle . Theorem 5-2-2 Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle. PX = PY = PZ You will prove Theorem 5-2-2 in Exercise 35. Unlike the circumcenter, the incenter is always inside the triangle. The distance between a point and a line is the length of the perpendicular segment from the point to the line. The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point. E X A M P L E 3 Using Properties of Angle Bisectors ̶̶ KV are angle bisectors of △JKL. ̶̶ JV and Find each measure. A the distance from V to ̶̶ KL V is the incenter of △JKL. By the Incenter Theorem, V is equidistant from the sides of △JKL. The distance from V to So the distance from V to ̶̶ JK is 7.3. ̶̶ KL is also 7.3. B m∠VKL m∠KJL = 2m∠VJL m∠KJL = 2 (19°) = 38° m∠KJL + m∠JLK + m∠JKL = 180° 38 + 106 + m∠JKL = 180 m∠JKL = 36° m∠JKL m∠VKL = 1 _ 2 m∠VKL = 1 _ (36°) = 18° 2 ̶̶ JV is the bisector of ∠KJL. Substitute 19° for m∠VJL. △ Sum Thm. Substitute the given values. Subtract 144° from both sides. ̶̶ KV is the bisector of ∠JKL. Substitute 36° for m∠JKL. 5- 2 Bisectors of Triangles 309 309 �� ̶̶ RX are angle bisectors ̶̶ QX and of △PQR. Find each measure. ̶̶ PQ 3a. the distance from X to 3b. m∠PQX E X A M P L E 4 Community Application The city of Odessa will host a fireworks display for the next Fourth of July celebration. Draw a sketch to show where the display should be positioned so that it is the same distance from all three viewing locations A, B, and C on the map. Justify your sketch. Let the three viewing locations be vertices of a triangle. By the Circumcenter Theorem, the circumcenter of the triangle is equidistant from the vertices. Trace the map. Draw the triangle formed by the viewing locations. To find the circumcenter, find the perpendicular bisectors of each side. The position of the display is the circumcenter, F. 4. A city plans to build a � � � � firefighters’ monument in the park between three streets. Draw a sketch to show where the city should place the monument so that it is the same distance from all three streets. Justify your sketch. � � � � � � � � �� THINK AND DISCUSS 1. Sketch three lines that are concurrent. 2. P and Q are the circumcenter and incenter of △RST, but not necessarily in that order. Which point is the circumcenter? Which point is the incenter? Explain how you can tell without constructing any of the bisectors. 3. GET ORGANIZED Copy and complete the graphic organizer. Fill in the blanks to make each statement true. 310 310 Chapter 5 Properties and Attributes of Triangles BAC38520print-ready file 7/12/05ge07ts_c05l02005aGeometry SE 2007 Texasmap of OdessaHolt Rinehart WinstonKaren Minot(415)883-6560�� 5-2 Exercises Exercises KEYWORD: MG7 5-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Explain why lines ℓ, m, and n are NOT concurrent. 2. A circle that contains all the vertices of a polygon is . 308 ? the polygon. (circumscribed about or inscribed in) ̶̶̶̶ ̶̶ TN , and ̶̶ VN are the perpendicular bisectors ̶̶ SN , of △PQR. Find each length. 3. NR 5. TR 4. RV 6. QN Multi-Step Find the circumcenter of a triangle with the given vertices. p. 308 7. O (0, 0) , K (0, 12) , L (4, 0. 309 8. A (-7, 0) , O (0, 0) , B (0, -10) ̶̶ CF and Find each measure. ̶̶ EF are angle bisectors of △CDE. 9. the distance from F to ̶̶ CD 10. m∠FED 11. Design The designer of the p. 310 Newtown High School pennant wants the circle around the bear emblem to be as large as possible. Draw a sketch to show where the center of the circle should be located. Justify your sketch. Independent Practice For See Exercises Example 12–15 16–17 18–19 20 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 PRACTICE AND PROBLEM SOLVING ̶̶ DY , of △ABC. Find each length. ̶̶ FY are the perpendicular bisectors ̶̶ EY , and 12. CF 14. DB 13. YC 15. AY Multi-Step Find the circumcenter of a triangle with the given vertices. 16. M (-5, 0) , N (0, 14) , O (0, 0) 17. O (0, 0) , V (0, 19) , W (-3, 0) ̶̶ SJ are angle bisectors of △RST. ̶̶ TJ and Find each measure. 18. the distance from J to ̶̶ RS 19. m∠RTJ 5- 2 Bisectors of Triangles 311 311 �� 20. Business A company repairs photocopiers in Harbury, Gaspar, and Knowlton. Draw a sketch to show where the company should locate its office so that it is the same distance from each city. Justify your sketch. 21. Critical Thinking If M is the incenter of △JKL, explain why ∠JML cannot be a right angle. Tell whether each segment lies on a perpendicular bisector, an angle bisector, or neither. Justify your answer. 22. 25. ̶̶ AE ̶̶ CR 23. 26. ̶̶̶ DG ̶̶ FR 24. 27. ̶̶ BG ̶̶ DR Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 28. The angle bisectors of a triangle intersect at a point outside the triangle. 29. An angle bisector of a triangle bisects the opposite side. 30. A perpendicular bisector of a triangle passes through the opposite vertex. 31. The incenter of a right triangle is on the triangle. 32. The circumcenter of a scalene triangle is inside the triangle. Algebra Find the circumcenter of the triangle with the given vertices. 33. O (0, 0) , A (4, 8) , B (8, 0) 34. O (0, 0) , Y (0, 12) , Z (6, 6) 35. Complete this proof of the Incenter Theorem by filling in the blanks. Given:  AP , ̶̶ PX ⊥ Prove: PX = PY = PZ  BP , and ̶̶ ̶̶ PY ⊥ AC , ̶̶ AB , ̶̶ PZ ⊥ ̶̶ BC  CP bisect ∠A, ∠B, and ∠C, respectively. Proof: Let P be the incenter of △ABC. Since P lies on the bisector of ∠A, PX = PY by a. Similarly, P also lies on b. Therefore c. ? . ̶̶̶̶ ? , so PY = PZ. ̶̶̶̶ ? by the Transitive Property of Equality. ̶̶̶̶ 36. Prove that the bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. Given:   QS bisects ∠PQR. Prove:   QS is the perpendicular bisector of ̶̶ PR . ̶̶ PQ ≅ ̶̶ RQ Plan: Show that △PQS ≅ △RQS. Then use CPCTC to show that S is the midpoint of ̶̶ PR and that   QS ⊥ ̶̶ PR . 37. Thi
s problem will prepare you for the Multi-Step TAKS Prep on page 328. A music company has stores at A (0, 0) , B (8, 0) , and C (4, 3) , where each unit of the coordinate plane represents one mile. a. A new store will be built so that it is equidistant from the three existing stores. Find the coordinates of the new store’s location. b. Where will the new store be located in relation to △ABC? c. To the nearest tenth of a mile, how far will the new store be from each of the existing stores? 312 312 Chapter 5 Properties and Attributes of Triangles HarburyGasparKnowltonHarburyGasparKnowltonHolt, Rinehart & WinstonGeometry 2007ge07se_c05l02008a 1st pass3/9/5C MurphyBisector map�� 38. Write About It How are the inscribed circle and the circumscribed circle of a triangle alike? How are they different? 39. Construction Draw a large scalene acute triangle. a. Construct the angle bisectors to find the incenter. Inscribe a circle in the triangle. b. Construct the perpendicular bisectors to find the circumcenter. Circumscribe a circle around the triangle. 40. P is the incenter of △ABC. Which must be true? PA = PB PX = PY YA = YB AX = BZ 41. Lines r, s, and t are concurrent. The equation of line r is x = 5, and the equation of line s is y = -2. Which could be the equation of line t 42. Gridded Response Lines a, b, and c are the perpendicular bisectors of △KLM. Find LN. CHALLENGE AND EXTEND 43. Use the right triangle with the given coordinates. a. Prove that the midpoint of the hypotenuse of a right triangle is equidistant from all three vertices. b. Make a conjecture about the circumcenter of a right triangle. 44. Design A trefoil is created by constructing a circle at each vertex of an equilateral triangle. The radius of each circle equals the distance from each vertex to the circumcenter of the triangle. If the distance from one vertex to the circumcenter is 14 cm, what is the distance AB across the trefoil? Design The trefoil shape, as seen in this stained glass window, has been used in design for centuries. SPIRAL REVIEW Solve each proportion. (Previous course) 45. t _ 26 46. 2.5 _ 1.75 = 10 _ 65 = 6 _ x 47. 420 _ y = 7 _ 2 Find each angle measure. (Lesson 2-6) 48. m∠BFE 49. m∠BFC 50. m∠CFE Determine whether each point is on the perpendicular bisector of the segment with endpoints S (0, 8) and T (4, 0) . (Lesson 5-1) 51. X (0, 3) 53. Z (-8, -2) 52. Y (-4, 1) 5- 2 Bisectors of Triangles 313 313 �� 5-3 Medians and Altitudes of Triangles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.7.A, G.7.B, G.7.C Objectives Apply properties of medians of a triangle. Apply properties of altitudes of a triangle. Vocabulary median of a triangle centroid of a triangle altitude of a triangle orthocenter of a triangle Who uses this? Sculptors who create mobiles of moving objects can use centers of gravity to balance the objects. (See Example 2.) A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Every triangle has three medians, and the medians are concurrent, as shown in the construction below. Construction Centroid of a Triangle    Draw △ABC. Construct the ̶̶ midpoints of BC , and Label the midpoints of the sides X, Y, and Z, respectively. ̶̶ AB , ̶̶ AC . ̶̶ AY , ̶̶ Draw BZ , and the three medians of △ABC. ̶̶ CX . These are Label the point where and ̶̶ CX intersect as P. ̶̶ AY , ̶̶ BZ , The point of concurrency of the medians of a triangle is the centroid of the triangle . The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance. Theorem 5-3-1 Centroid Theorem The centroid of a triangle is located 2 __ of the distance 3 from each vertex to the midpoint of the opposite side. AP = 2 _ 3 AY BP = 2 _ 3 BZ CP = 2 _ 3 CX 314 314 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Using the Centroid to Find Segment Lengths In △ABC, AF = 9, and GE = 2.4. Find each length. A AG AF AG = 2 _ 3 AG = 2 _ (9) 3 AG = 6 B CE CG = 2 _ 3 CE Centroid Thm. Substitute 9 for AF. Simplify. Centroid Thm. CG + GE = CE 2 _ CE + GE = CE 3 GE = 1 _ 3 2.4 = 1 _ 3 CE CE Seg. Add. Post. Substitute 2 _ 3 Subtract 2 _ 3 CE for CG. CE from both sides. Substitute 2.4 for GE. 7.2 = CE Multiply both sides by 3. In △JKL, ZW = 7, and LX = 8.1. Find each length. 1a. KW 1b. LZ E X A M P L E 2 Problem-Solving Application The diagram shows the plan for a triangular piece of a mobile. Where should the sculptor attach the support so that the triangle is balanced? Understand the Problem The answer will be the coordinates of the centroid of △PQR. The important information is the location of the vertices, P (3, 0) , Q (0, 8) , and R (6, 4) . Make a Plan The centroid of the triangle is the point of intersection of the three medians. So write the equations for two medians and find their point of intersection. Solve Let M be the midpoint of ̶̶ QR and N be the midpoint of ̶̶ QP . 3, 61.5, 4) ̶̶̶ PM is vertical. Its equation is x = 3. Its equation is y = 4. The coordinates of the centroid are S (3, 4) . ̶̶ RN is horizontal. 5- 3 Medians and Altitudes of Triangles 315 315 ��123 Look Back Let L be the midpoint of intersects x = 3 at S (3, 4) . ̶̶ PR . The equation for ̶̶ QL is y = - 4 _ 3 x + 8, which The height of a triangle is the length of an altitude. 2. Find the average of the x-coordinates and the average of the y-coordinates of the vertices of △PQR. Make a conjecture about the centroid of a triangle. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle. ̶̶ QY is inside the triangle, but ̶̶ SZ are not. Notice that the lines containing In △QRS, altitude and the altitudes are concurrent at P. This point of concurrency is the orthocenter of the triangle . ̶̶ RX E X A M P L E 3 Finding the Orthocenter Find the orthocenter of △JKL with vertices J (-4, 2) , K (-2, 6) , and L (2, 2) . Step 1 Graph the triangle. Step 2 Find an equation of the line containing the altitude from K to ̶̶ JL . JL is horizontal, the altitude is Since   vertical. The line containing it must pass through K (-2, 6) , so the equation of the line is x = -2. Step 3 Find an equation of the line containing the altitude from J to ̶̶ KL . slope of   KL = 2 - 6 _ = -1 2 - (-2) The slope of a line perpendicular to   KL is 1. This line must pass through J (-4, 2) . y - y 1 = m ( - (-4 Point-slope form Substitute 2 for y 1 , 1 for m, and -4 for x 1 . Distribute 1. Add 2 to both sides. Step 4 Solve the system to find the coordinates of the orthocenter. ⎧ x = -2 + 6 = 4 Substitute -2 for x. The coordinates of the orthocenter are (-2, 4) . 3. Show that the altitude to ̶̶ JK passes through the orthocenter of △JKL. 316 316 Chapter 5 Properties and Attributes of Triangles 4�� THINK AND DISCUSS 1. Draw a triangle in which a median and an altitude are the same segment. What type of triangle is it? 2. Draw a triangle in which an altitude is also a side of the triangle. What type of triangle is it? 3. The centroid of a triangle divides each median into two segments. What is the ratio of the two lengths of each median? 4. GET ORGANIZED Copy and complete the graphic organizer. Fill in the blanks to make each statement true. 5-3 Exercises Exercises KEYWORD: MG7 5-3 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. ? of a triangle is located 2 __ 3 of the distance from each vertex to the ̶̶̶̶ 1. The midpoint of the opposite side. (centroid or orthocenter) 2. The ? of a triangle is perpendicular to the line containing a side. ̶̶̶̶ (altitude or median VX = 204, and RW = 104. Find each length. p. 315 3. VW 5. RY 4. WX 6. WY . Design The diagram shows a plan for p. 315 a piece of a mobile. A chain will hang from the centroid of the triangle. At what coordinates should the artist attach the chain Multi-Step Find the orthocenter of a triangle with the given vertices. p. 316 8. K (2, -2) , L (4, 6) , M (8, -2) 9. U (-4, -9) , V (-4, 6) , W (5, -3) 10. P (-5, 8) , Q (4, 5) , R (-2, 5) 11. C (-1, -3) , D (-1, 2) , E (9, 2) 5- 3 Medians and Altitudes of Triangles 317 317 �� Independent Practice For See Exercises Example 12–15 16 17–20 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 PRACTICE AND PROBLEM SOLVING PA = 2.9, and HC = 10.8. Find each length. 12. PC 14. JA 13. HP 15. JP 16. Design In the plan for a table, the triangular top has coordinates (0, 10) , (4, 0) , and (8, 14) . The tabletop will rest on a single support placed beneath it. Where should the support be attached so that the table is balanced? Multi-Step Find the orthocenter of a triangle with the given vertices. 17. X (-2, -2) , Y (6, 10) , Z (6, -6) 18. G (-2, 5) , H (6, 5) , J (4, -1) 19. R (-8, 9) , S (-2, 9) , T (-2, 1) 20. A (4, -3) , B (8, 5) , C (8, -8) Find each measure. 21. GL 23. HL 22. PL 24. GJ 25. perimeter of △GHJ 26. area of △GHJ Algebra Find the centroid of a triangle with the given vertices. 27. A (0, -4) , B (14, 6) , C (16, -8) 28. X (8, -1) , Y (2, 7) , Z (5, -3) Find each length. 29. PZ 31. QZ 30. PX 32. YZ Math History 33. Critical Thinking Draw an isosceles triangle and its line of symmetry. What are four other names for this segment? Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 34. A median of a triangle bis
ects one of the angles. 35. If one altitude of a triangle is in the triangle’s exterior, then a second altitude is also in the triangle’s exterior. 36. The centroid of a triangle lies in its exterior. 37. In an isosceles triangle, the altitude and median from the vertex angle are the same line as the bisector of the vertex angle. 38. Write a two-column proof. ̶̶ PS and Given: Prove: △PQR is an isosceles triangle. ̶̶ RT are medians of △PQR. ̶̶ PS ≅ ̶̶ RT In 1678, Giovanni Ceva published his famous theorem that states the conditions necessary for three Cevians (segments from a vertex of a triangle to the opposite side) to be concurrent. The medians and altitudes of a triangle meet these conditions. Plan: Show that △PTR ≅ △RSP and use CPCTC to conclude that ∠QPR ≅ ∠QRP. 39. Write About It Draw a large triangle on a sheet of paper and cut it out. Find the centroid by paper folding. Try to balance the shape on the tip of your pencil at a point other than the centroid. Now try to balance the shape at its centroid. Explain why the centroid is also called the center of gravity. 318 318 Chapter 5 Properties and Attributes of Triangles �� 40. This problem will prepare you for the Multi-Step TAKS Prep on page 328. The towns of Davis, El Monte, and Fairview have the coordinates shown in the table, where each unit of the coordinate plane represents one mile. A music company has stores in each city and a distribution warehouse at the centroid of △DEF. a. What are the coordinates of the warehouse? b. Find the distance from the warehouse to the City Location Davis El Monte Fairview D (0, 0) E (0, 8) F (8, 0) Davis store. Round your answer to the nearest tenth of a mile. c. A straight road connects El Monte and Fairview. What is the distance from the warehouse to the road? 41. ̶̶ RV , and ̶̶ QT , NOT necessarily true? ̶̶̶ SW are medians of △QRS. Which statement is QP = 2 _ 3 RP = 2PV QT RT = ST QT = SW 42. Suppose that the orthocenter of a triangle lies outside the triangle. Which points of concurrency are inside the triangle? I. incenter II. circumcenter III. centroid I and II only I and III only II and III only I, II, and III 43. In the diagram, which of the following correctly describes ̶̶ LN ? Altitude Median Angle bisector Perpendicular bisector CHALLENGE AND EXTEND 44. Draw an equilateral triangle. a. Explain why the perpendicular bisector of any side contains the vertex opposite that side. b. Explain why the perpendicular bisector through any vertex also contains the median, the altitude, and the angle bisector through that vertex. c. Explain why the incenter, circumcenter, centroid, and orthocenter are the same point. 45. Use coordinates to show that the lines containing ̶̶ RS , ̶̶ ST , and the altitudes of a triangle are concurrent. a. Find the slopes of b. Find the slopes of lines ℓ, m, and n. c. Write equations for lines ℓ, m, and n. d. Solve a system of equations to find the point P where lines ℓ and m intersect. ̶̶ RT . e. Show that line n contains P. f. What conclusion can you draw? 5- 3 Medians and Altitudes of Triangles 319 319 �� SPIRAL REVIEW 46. At a baseball game, a bag of peanuts costs $0.75 more than a bag of popcorn. If a family purchases 5 bags of peanuts and 3 bags of popcorn for $21.75, how much does one bag of peanuts cost? (Previous course) Determine if each biconditional is true. If false, give a counterexample. (Lesson 2-4) 47. The area of a rectangle is 40 cm 2 if and only if the length of the rectangle is 4 cm and the width of the rectangle is 10 cm. 48. A nonzero number n is positive if and only if -n is negative. ̶̶ QP , and ̶̶ NQ , Find each measure. (Lesson 5-2) ̶̶̶ QM are perpendicular bisectors of △JKL. 49. KL 50. QJ 51. m∠JQL Construction Orthocenter of a Triangle    Draw a large scalene acute triangle ABC on a piece of patty paper. Find the altitude of each side by folding the side so that it overlaps itself and so that the fold intersects the opposite vertex. Mark the point where the three lines containing the altitudes intersect and label it P. P is the orthocenter of △ABC. 1. Repeat the construction for a scalene obtuse triangle 2. Make a conjecture about the location of the and a scalene right triangle. orthocenter in an acute, an obtuse, and a right triangle. Q: What high school math classes did you take? A: Algebra 1, Geometry, and Statistics. KEYWORD: MG7 Career Q: What type of training did you receive? A: In high school, I took classes in electricity, electronics, and drafting. I began an apprenticeship program last year to prepare for the exam to get my license. Q: How do you use math? A: Determining the locations of outlets and circuits on blueprints requires good spatial sense. I also use ratios and proportions, calculate distances, work with formulas, and estimate job costs. Alex Peralta Electrician 320 320 Chapter 5 Properties and Attributes of Triangles �� 5-3 Special Points in Triangles In this lab you will use geometry software to explore properties of the four points of concurrency you have studied. Use with Lesson 5-3 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.9.B KEYWORD: MG7 Lab5 Activity 1 Construct a triangle. 2 Construct the perpendicular bisector of each side of the triangle. Construct the point of intersection of these three lines. This is the circumcenter of the triangle. Label it U and hide the perpendicular bisectors. 3 4 5 In the same triangle, construct the bisector of each angle. Construct the point of intersection of these three lines. This is the incenter of the triangle. Label it I and hide the angle bisectors. In the same triangle, construct the midpoint of each side. Then construct the three medians. Construct the point of intersection of these three lines. Label the centroid C and hide the medians. In the same triangle, construct the altitude to each side. Construct the point of intersection of these three lines. Label the orthocenter O and hide the altitudes. 6 Move a vertex of the triangle and observe the positions of the four points of concurrency. In 1765, Swiss mathematician Leonhard Euler showed that three of these points are always collinear. The line containing them is called the Euler line. Try This 1. Which three points of concurrency lie on the Euler line? 2. Make a Conjecture Which point on the Euler line is always between the other two? Measure the distances between the points. Make a conjecture about the relationship of the distances between these three points. 3. Make a Conjecture Move a vertex of the triangle until all four points of concurrency are collinear. In what type of triangle are all four points of concurrency on the Euler line? 4. Make a Conjecture Find a triangle in which all four points of concurrency coincide. What type of triangle has this special property? 5- 3 Technology Lab 321 321 5-4 The Triangle Midsegment Theorem TEKS G.7.B Dimensionality and the geometry of location: use slopes and equations of lines to investigate... special segments of triangles and other polygons. Objective Prove and use properties of triangle midsegments. Vocabulary midsegment of a triangle Why learn this? You can use triangle midsegments to make indirect measurements of distances, such as the distance across a volcano. (See Example 3.) Also G.2.A, G.2.B, G.3.B, G.5.A, G.9.B A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle. E X A M P L E 1 Examining Midsegments in the Coordinate Plane In △GHJ, show that midsegment parallel to GJ and that KL = 1 __ ̶̶ Step 1 Find the coordinates of K and L. GJ. 2 ̶̶ KL is mdpt. of ̶̶̶ GH = ( ) -7 + (-5) -2 + 6 _ _ , 2 2 = (-6, 2) mdpt. of ̶̶ HJ = ( 6 + 2 -5 + 1 _ _ , 2 2 ) = (-2, 4) Step 2 Compare the slopes of slope of ̶̶ KL = ̶̶ GJ . ̶̶ KL and = 1 _ 2 4 - 2 _ -2 - (-6) slope of ̶̶ GJ = 2 - (-2) _ 1 - (-7) = 1 _ 2 Since the slopes are the same, ̶̶ KL and Step 3 Compare the lengths of ̶̶ KL ǁ ̶̶ GJ . ̶̶ GJ . 2 + (4 - 2) 2 = 2 √  5 KL = √  ⎤ ⎦ ⎡ ⎣ -2 - (-6) GJ = √  ⎤ ⎦ ⎤ ⎦ ⎡ ⎣ ⎡ ⎣ 2 - (-2) 1 - (-7) + (4 √  5 ) , KL = 1 _ Since 2 √  5 = 1 _ 2 2 GJ. 2 2 = 4 √  5 1. The vertices of △RST are R (-7, 0) , S (-3, 6) , and T (9, 2) . ̶̶ RT , and N is the midpoint of ̶̶ ST . M is the midpoint of Show that ̶̶̶ MN ǁ ̶̶ RS and MN = 1 __ 2 RS. 322 322 Chapter 5 Properties and Attributes of Triangles �� The relationship shown in Example 1 is true for the three midsegments of every triangle. Theorem 5-4-1 Triangle Midsegment Theorem A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. ̶̶ DE ǁ ̶̶ AC , DE = 1_ 2 AC You will prove Theorem 5-4-1 in Exercise 38. E X A M P L E 2 Using the Triangle Midsegment Theorem Find each measure. A UW ST UW = 1 _ 2 UW = 1 _ (7.4) 2 UW = 3.7 △ Midsegment Thm. Substitute 7.4 for ST. Simplify. B m∠SVU ̶̶ ̶̶̶ UW ǁ ST m∠SVU = m∠VUW m∠SVU = 41° △ Midsegment Thm. Alt. Int.  Thm. Substitute 41° for m∠VUW. Find each measure. 2a. JL 2b. PM 2c. m∠MLK E X A M P L E 3 Indirect Measurement Application Anna wants to find the distance across the base of Capulin Volcano, an extinct volcano in New Mexico. She measures a triangle at one side of the volcano as shown in the diagram. What is AE ? BD = 1 _ 2 775 = 1 _ 2 AE AE △ Midsegment Thm. Substitute 775 for BD. 1550 = AE Multiply both sides by 2. The distance AE across the base of the volcano is about 1550 meters. 3. What if…? Suppose Anna’s result in Example 3 is correct. To check it, she measures a second triangle. How many meters will she measure between H and F? 323 5- 4 The Triangle Midsegment Theorem 323 640 m1005 m640 m1005 mge07se_c05l04007aGFHAE��
��700 m920 m920 m775 m700 mge07se_c05l04006aCDBAE THINK AND DISCUSS 1. Explain why ̶̶ XY is NOT a midsegment of the triangle. 2. GET ORGANIZED Copy and complete the graphic organizer. Write the definition of a triangle midsegment and list its properties. Then draw an example and a nonexample. 5-4 Exercises Exercises KEYWORD: MG7 5-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary The midsegment of a triangle joins the triangle. (endpoints or midpoints) ? of two sides of the ̶̶̶̶ . 322 2. The vertices of △PQR are P (-4, -1) , Q (2, 9) , and R (6, 3) . S is the midpoint of and T is the midpoint of ̶̶ QR . Show that ̶̶ ST ǁ ̶̶ PR and ST = 1 __ 2 PR. ̶̶ PQ , Find each measure. p. 323 3. NM 5. NZ 7. m∠YXZ 4. XZ 6. m∠LMN 8. m∠XLM . 323 Independent Practice For See Exercises Example 10 11–16 17 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S12 Application Practice p. S32 9. Architecture In this A-frame house, ̶̶ XZ is 30 feet. ̶̶ CD is slightly above the width of the first floor The second floor and parallel to the midsegment of △XYZ. Is the width of the second floor more or less than 5 yards? Explain. � � � � � PRACTICE AND PROBLEM SOLVING 10. The vertices of △ABC are A (-6, 11) , B (6, -3) , and C (-2, -5) . D is the midpoint ̶̶ AB . Show that ̶̶ AC , and E is the midpoint of ̶̶ DE ǁ ̶̶ CB and DE = 1 __ 2 CB. of Find each measure. 11. GJ 13. RJ 12. RQ 14. m∠PQR 15. m∠HGJ 16. m∠GPQ 324 324 Chapter 5 Properties and Attributes of Triangles �� 17. Carpentry In each support for the garden swing, ̶̶ BA and ̶̶ DE is attached at the midpoints ̶̶ BC . The distance AC is 4 1 __ 2 feet. the crossbar of legs The carpenter has a timber that is 30 inches long. Is this timber long enough to be used as one of the crossbars? Explain. △KLM is the midsegment triangle of △GHJ. 18. What is the perimeter of △GHJ? 19. What is the perimeter of △KLM? 20. What is the relationship between the perimeter of △GHJ and the perimeter of △KLM? Algebra Find the value of n in each triangle. 21. 24. 22. 25. 23. 26. 27. /////ERROR ANALYSIS///// Below are two solutions for finding BC. Which is incorrect? Explain the error. 28. Critical Thinking Draw scalene △DEF. Label X as the midpoint of Y as the midpoint of midpoints. List all of the congruent angles in your drawing. ̶̶ EF , and Z as the midpoint of ̶̶ DE , ̶̶ DF . Connect the three 29. Estimation The diagram shows the sketch for a new street. Parallel parking spaces will be painted on both sides of the street. Each parallel parking space is 23 feet long. About how many parking spaces can the city accommodate on both sides of the new street? Explain your answer. ̶̶ EH , and ̶̶ CG , and △GHE, respectively. Find each measure. ̶̶ FJ are midsegments of △ABD, △GCD, 30. CG 31. EH 32. FJ 33. m∠DCG 34. m∠GHE 35. m∠FJH 36. Write About It An isosceles triangle has two congruent sides. Does it also have two congruent midsegments? Explain. 325 5- 4 The Triangle Midsegment Theorem 325 ��ge07sec05l04004aABoehmMarket Street (440 ft)Springfield RoadLake AvenueNew street�� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 328. The figure shows the roads connecting towns A, B, and C. A music company has a store in each town and a distribution warehouse W at the midpoint of road a. What is the distance from the warehouse to point X? b. A truck starts at the warehouse, delivers instruments to the stores in towns A, B, and C (in this order) and then returns to the warehouse. What is the total length of the trip, assuming the driver takes the shortest possible route? ̶̶ XY . 38. Use coordinates to prove the Triangle Midsegment Theorem. a. M is the midpoint of b. N is the midpoint of ̶̶ c. Find the slopes of PR and ̶̶ PQ . What are its coordinates? ̶̶ QR . What are its coordinates? ̶̶̶ MN . What can you conclude? d. Find PR and MN. What can you conclude? 39. ̶̶ PQ is a midsegment of △RST. What is the length of ̶̶ RT ? 9 meters 21 meters 45 meters 63 meters 40. In △UVW, M is the midpoint of ̶̶̶ VW . Which statement is true? midpoint of ̶̶ VU , and N is the VM = VN MN = UV VU = 2VM VW = 1 _ 2 VN 41. △XYZ is the midsegment triangle of △JKL, XY = 8, YK = 14, and m∠YKZ = 67°. Which of the following measures CANNOT be determined? KL JY m∠XZL m∠KZY CHALLENGE AND EXTEND 42. Multi-Step The midpoints of the sides of a triangle are A (-6, 3) , B (2, 1) , and C (0, -3) . Find the coordinates of the vertices of the triangle. 43. Critical Thinking Classify the midsegment triangle of an equilateral triangle by its side lengths and angle measures. Algebra Find the value of n in each triangle. 44. 45. 326 326 Chapter 5 Properties and Attributes of Triangles �� 46. △XYZ is the midsegment triangle of △PQR. Write a congruence statement involving all four of the smaller triangles. What is the relationship between the area of △XYZ and △PQR? ̶̶ AB is a midsegment of △XYZ. is a midsegment of △CDZ, and a. Copy and complete the table. 47. ̶̶ CD is a midsegment of △ABZ. ̶̶̶ GH is a midsegment of △EFZ. ̶̶ EF 1 2 3 4 Number of Midsegment Length of Midsegment b. If this pattern continues, what will be the length of midsegment 8? c. Write an algebraic expression to represent the length of midsegment n. (Hint: Think of the midsegment lengths as powers of 2.) SPIRAL REVIEW Suppose a 2% acid solution is mixed with a 3% acid solution. Find the percent of acid in each mixture. (Previous course) 48. a mixture that contains an equal amount of 2% acid solution and 3% acid solution 49. a mixture that contains 3 times more 2% acid solution than 3% acid solution A figure has vertices G (-3, -2) , H (0, 0) , J (4, 1) , and K (1, -2) . Given the coordinates of the image of G under a translation, find the coordinates of the images of H, J, and K. (Lesson 1-7) 50. (-3, 2) 51. (1, -4) 52. (3, 0) Find each length. (Lesson 5-3) 53. NX 54. MR 55. NP Construction Midsegment of a Triangle    Draw a large triangle. Label the vertices A, B, and C. ̶̶ AB ̶̶ BC . Label the midpoints X Construct the midpoints of and and Y, respectively. Draw the midsegment ̶̶ XY . 1. Using a ruler, measure ̶̶ XY and ̶̶ AC . How are the two lengths related? 2. How can you use a protractor to verify that ̶̶ XY is parallel to ̶̶ AC ? 327 5- 4 The Triangle Midsegment Theorem 327 �� SECTION 5A Segments in Triangles Location Contemplation A chain of music stores has locations in Ashville, Benton, and Carson. The directors of the company are using a coordinate plane to decide on the location for a new distribution warehouse. Each unit on the plane represents one mile. 1. A plot of land is available at the centroid of the triangle formed by the three cities. What are the coordinates for this location? 2. If the directors build the warehouse at the centroid, about how far will it be from each of the cities? 3. Another plot of land is available at the orthocenter of the triangle. What are the coordinates for this location? 4. About how far would the warehouse be from each city if it were built at the orthocenter? 5. A third option is to build the warehouse at the circumcenter of the triangle. What are the coordinates for this location? 6. About how far would the warehouse be from each city if it were built at the circumcenter? 7. The directors decide that the warehouse should be equidistant from each city. Which location should they choose? 328 328 Chapter 5 Properties and Attributes of Triangles �� SECTION 5A Quiz for Lessons 5-1 Through 5-4 5-1 Perpendicular and Angle Bisectors Find each measure. 1. PQ 2. JM 3. AC 4. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints M (-1, -3) and N (7, 1) . 5-2 Bisectors of Triangles 5. ̶̶ PY , and ̶̶ PX , bisectors of △RST. Find PS and XT. ̶̶ PZ are the perpendicular 6. ̶̶ JK and ̶̶ HK are angle bisectors of △GHJ. Find m∠GJK and the distance from K to ̶̶ HJ . 7. Find the circumcenter of △TVO with vertices T (9, 0) , V (0, -4) , and O (0, 0) . 5-3 Medians and Altitudes of Triangles 8. In △DEF, BD = 87, and WE = 38. Find BW, CW, and CE. 9. Paula cuts a triangle with vertices at coordinates (0, 4) , (8, 0) , and (10, 8) from grid paper. At what coordinates should she place the tip of a pencil to balance the triangle? 10. Find the orthocenter of △PSV with vertices P (2, 4) , S (8, 4) , and V (4, 0) . 5-4 The Triangle Midsegment Theorem 11. Find ZV, PM, and m∠RZV in △JMP. 12. What is the distance XZ across the pond? � �� Ready to Go On? 329 329 �� Solving Compound Inequalities Algebra To solve an inequality, you use the Properties of Inequality and inverse operations to undo the operations in the inequality one at a time. See Skills Bank page S60 Properties of Inequality PROPERTY ALGEBRA Addition Property Subtraction Property Multiplication Property Division Property Transitive Property Comparison Property If a < b, then a + c < b + c. If a < b, then a - c < b - c. If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc. If a < b and c > 0, then a _ c < b _ c . If a < b and c < 0, then a _ c > b _ c . If a < b and b < c, then a < c. If a + b = c and b > 0, then a < c. A compound inequality is formed when two simple inequalities are combined into one statement with the word and or or. To solve a compound inequality, solve each simple inequality and find the intersection or union of the solutions. The graph of a compound inequality may represent a line, a ray, two ray
s, or a segment. Example Solve the compound inequality 5 < 20 - 3a ≤ 11. What geometric figure does the graph represent? 5 < 20 - 3a AND 20 - 3a ≤ 11 Rewrite the compound inequality as two -15 < -3a 5 > a AND AND -3a ≤ -9 Subtract 20 from both sides. simple inequalities. a ≥ 3 Divide both sides by -3 and reverse the inequality symbols. 3 ≤ a < 5 Combine the two solutions into a single statement. The graph represents a segment. Try This TAKS Grades 9–11 Obj. 4 Solve. What geometric figure does each graph represent? 1. -4 + x > 1 OR -8 + 2x < -6 2. 2x - 3 ≥ -5 OR x - 4 > -1 3. -6 < 7 - x ≤ 12 5. 3x ≥ 0 OR x + 5 < 7 4. 22 < -2 - 2x ≤ 54 6. 2x - 3 ≤ 5 OR -2x + 3 ≤ -9 330 330 Chapter 5 Properties and Attributes of Triangles �� 5-5 Use with Lesson 5-5 Activity 1 Explore Triangle Inequalities Many of the triangle relationships you have learned so far involve a statement of equality. For example, the circumcenter of a triangle is equidistant from the vertices of the triangle, and the incenter is equidistant from the sides of the triangle. Now you will investigate some triangle relationships that involve inequalities. TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about ... polygons and their component parts based on explorations and concrete models. Also G.5.B 1 Draw a large scalene triangle. Label the vertices A, B, and C. 2 Measure the sides and the angles. Copy the table below and record the measures in the first row. BC AC AB m∠A m∠B m∠C Triangle 1 Triangle 2 Triangle 3 Triangle 4 Try This 1. In the table, draw a circle around the longest side length, and draw a circle around the greatest angle measure of △ABC. Draw a square around the shortest side length, and draw a square around the least angle measure. 2. Make a Conjecture Where is the longest side in relation to the largest angle? Where is the shortest side in relation to the smallest angle? 3. Draw three more scalene triangles and record the measures in the table. Does your conjecture hold? Activity 2 1 Cut three sets of chenille stems to the following lengths. 3 inches, 4 inches, 6 inches 3 inches, 4 inches, 7 inches 3 inches, 4 inches, 8 inches 2 Try to make a triangle with each set of chenille stems. Try This 4. Which sets of chenille stems make a triangle? 5. Make a Conjecture For each set of chenille stems, compare the sum of any two lengths with the third length. What is the relationship? 6. Select a different set of three lengths and test your conjecture. Does your conjecture hold? 5- 5 Geometry Lab 331 331 5-5 Indirect Proof and Inequalities in One Triangle TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.3.C, G.3.E, G.5.B Objectives Write indirect proofs. Apply inequalities in one triangle. Why learn this? You can use a triangle inequality to find a reasonable range of values for an unknown distance. (See Example 5.) Vocabulary indirect proof So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof , you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Writing an Indirect Proof 1. Identify the conjecture to be proven. 2. Assume the opposite (the negation) of the conclusion is true. 3. Use direct reasoning to show that the assumption leads to a contradiction. 4. Conclude that since the assumption is false, the original conjecture must be true. E X A M P L E 1 Writing an Indirect Proof Write an indirect proof that a right triangle cannot have an obtuse angle. Step 1 Identify the conjecture to be proven. Given: △JKL is a right triangle. Prove: △JKL does not have an obtuse angle. Step 2 Assume the opposite of the conclusion. Assume △JKL has an obtuse angle. Let ∠K be obtuse. Step 3 Use direct reasoning to lead to a contradiction. m∠K + m∠L = 90° The acute  of a rt. △ are comp. m∠K = 90° - m∠L m∠K > 90° Subtr. Prop. of = Def. of obtuse ∠ 90° - m∠L > 90° m∠L < 0° Substitute 90° - m∠L for m∠K. Subtract 90° from both sides and solve for m∠L. However, by the Protractor Postulate, a triangle cannot have an angle with a measure less than 0°. Step 4 Conclude that the original conjecture is true. The assumption that △JKL has an obtuse angle is false. Therefore △JKL does not have an obtuse angle. 1. Write an indirect proof that a triangle cannot have two right angles. 332 332 Chapter 5 Properties and Attributes of Triangles �� The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles. Theorems Angle-Side Relationships in Triangles THEOREM HYPOTHESIS CONCLUSION 5-5-1 If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. (In △, larger ∠ is opp. longer side.) 5-5-2 If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. (In △, longer side is opp. larger ∠.) m∠C > m∠A XY > XZ AB > BC m∠Z > m∠Y You will prove Theorem 5-5-1 in Exercise 67. PROOF PROOF Theorem 5-5-2 Given: m∠P > m∠R Prove: QR > QP Consider all cases when you assume the opposite. If the conclusion is QR > QP, the negation includes QR < QP and QR = QP. Indirect Proof: Assume QR ≯ QP. This means that either QR < QP or QR = QP. Case 1 the longer side. This contradicts the given information. So QR ≮ QP. If QR < QP, then m∠P < m∠R because the larger angle is opposite Case 2 This also contradicts the given information, so QR ≠ QP. If QR = QP, then m∠P = m∠R by the Isosceles Triangle Theorem. The assumption QR ≯ QP is false. Therefore QR > QP. E X A M P L E 2 Ordering Triangle Side Lengths and Angle Measures A Write the angles in order from smallest to largest. ̶̶ GJ , so the smallest angle is ∠H. ̶̶ HJ , so the largest angle is ∠G. The shortest side is The longest side is The angles from smallest to largest are ∠H, ∠J, and ∠G. B Write the sides in order from shortest to longest. m∠M = 180° - (39° + 54°) = 87° △ Sum Thm. ̶̶̶ The smallest angle is ∠L, so the shortest side is KM . ̶̶ KL . The largest angle is ∠M, so the longest side is ̶̶̶ LM , and The sides from shortest to longest are ̶̶̶ KM , ̶̶ KL . 2a. Write the angles in order from smallest to largest. 2b. Write the sides in order from shortest to longest. 5- 5 Indirect Proof and Inequalities in One Triangle 333 333 �� A triangle is formed by three segments, but not every set of three segments can form a triangle. Segments with lengths of 7, 4, and 4 can form a triangle. Segments with lengths of 7, 3, and 3 cannot form a triangle. A certain relationship must exist among the lengths of three segments in order for them to form a triangle. Theorem 5-5-3 Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length. AB + BC > AC BC + AC > AB AC + AB > BC You will prove Theorem 5-5-3 in Exercise 68. E X A M P L E 3 Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. A 3, 5 ✓ 10 > 5 ✓ 12 > 3 ✓ Yes—the sum of each pair of lengths is greater than the third length. B 4, 6.5, 11 4 + 6.5  11 10.5 ≯ 11 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths. C n + 5, n 2 , 2n, when n = 3 Step 1 Evaluate each expression when n = 3 2n 2 (3) 6 To show that three lengths cannot be the side lengths of a triangle, you only need to show that one of the three triangle inequalities is false. Step 2 Compare the lengths 17 > 6 ✓ 14 > 9 ✓ 15 > 8 ✓ Yes—the sum of each pair of lengths is greater than the third length. Tell whether a triangle can have sides with the given lengths. Explain. 3a. 8, 13, 21 3c. t - 2, 4t, t 2 + 1, when t = 4 3b. 6.2, 7, 9 334 334 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 4 Finding Side Lengths The lengths of two sides of a triangle are 6 centimeters and 11 centimeters. Find the range of possible lengths for the third side. Let s represent the length of the third side. Then apply the Triangle Inequality Theorem. s + 6 > 11 s + 11 > 6 s > 5 s > -5 6 + 11 > s 17 > s Combine the inequalities. So 5 < s < 17. The length of the third side is greater than 5 centimeters and less than 17 centimeters. 4. The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side. E X A M P L E 5 Travel Application The map shows the approximate distances from San Antonio to Mason and from San Antonio to Austin. What is the range of distances from Mason to Austin? Let d be the distance from Mason to Austin. d + 111 > 78 d + 78 > 111 111 + 78 > d △ Inequal. Thm. d > -33 d > 33 189 > d Subtr. Prop. of Inequal. 33 < d < 189 Combine the inequalities. The distance from Mason to Austin is greater than 33 miles and less than 189 miles. 5. The distance from San Marcos to Johnson City is 50 miles, and the distance from Seguin to San Marcos is 22 miles. What is the range of distances from Seguin to Johnson City? THINK AND DISCUSS 1. To write an indirect proof that an angle is obtuse, a student assumes that the angle is acute. Is this the correct assumption? Explain. 2. Give an example of three measures that can be the lengths of the sides of a triangle. Give an example of three lengths that cannot be the sides of a triangle. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, explain what you know about △ABC as a result of the theorem. 5- 5 Indirect Proof and Inequalities in One Triangle 335 335 AustinSanMarcosJohnsonCitySeguinSan AntonioMason10103535290281377183879078 mi111 mige07se_c05l05002aABeckmann�� 5-5 Exercises E
xercises KEYWORD: MG7 5-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Describe the process of an indirect proof in your own words Write an indirect proof of each statement. p. 332 2. A scalene triangle cannot have two congruent angles. 3. An isosceles triangle cannot have a base angle that is a right angle. Write the angles in order p. 333 from smallest to largest. 5. Write the sides in order from shortest to longest Tell whether a triangle can have sides with the given lengths. Explain. p. 334 6. 4, 7, 10 7. 2, 9, 12 , 3 1 _ 8. 3 1 _ 2 2 , 6 9. 3, 1.1, 1.7 10. 3x, 2x - 1, x 2 , when x = 5 11. 7c + 6, 10c - 7, 3 c 2 , when . 335 The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side. 12. 8 mm, 12 mm 13. 16 ft, 16 ft 14. 11.4 cm, 12 cm . 335 15. Design The refrigerator, stove, and sink in a kitchen are at the vertices of a path called the work triangle. a. If the angle at the sink is the largest, which side of the work triangle will be the longest? b. The designer wants the longest side of this triangle to be 9 feet long. Can the lengths of the other sides be 5 feet and 4 feet? Explain. �� Independent Practice Write an indirect proof of each statement. PRACTICE AND PROBLEM SOLVING 16. A scalene triangle cannot have two congruent midsegments. 17. Two supplementary angles cannot both be obtuse angles. 18. Write the angles in order from smallest to largest. �� 19. Write the sides in order �� from shortest to longest. For See Exercises Example 16–17 18–19 20–25 26–31 32 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 Tell whether a triangle can have sides with the given lengths. Explain. 20. 6, 10, 15 21. 14, 18, 32 22. 11.9, 5.8, 5.8 23. 103, 41.9, 62.5 24. z + 8, 3z + 5, 4z - 11, when z = 6 25. m + 11, 8m, m 2 + 1, when m = 3 336 336 Chapter 5 Properties and Attributes of Triangles �� The lengths of two sides of a triangle are given. Find the range of possible lengths The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side. for the third side. Bicycles 26. 26. 4 yd, 19 yd 29. 3.07 m, 1.89 m 27. 28 km, 23 km in., 3 5_ 30. 2 1_ 8 8 in. 28. 9.2 cm, 3.8 cm 31. 3 5_ ft, 6 1_ 2 6 ft 32. Bicycles The five steel tubes of this mountain bike frame form two triangles. List the five tubes in order from shortest to longest. Explain your answer. 33. Critical Thinking The length of the base of an isosceles triangle is 15. What is the range of possible lengths for each leg? Explain. List the sides of each triangle in order from shortest to longest. 34. 35. Lance Armstrong, of Austin, broke his own record in 2005 and became the first person to win seven consecutive titles in the Tour de France cycling competition. Only one other person has won five consecutive times. In each set of statements, name the two that contradict each other. 36. △PQR is a right triangle. 37. ∠Y is supplementary to ∠Z. △PQR is a scalene triangle. △PQR is an acute triangle. 38. △JKL is isosceles with base ̶̶ JL . In △JKL, m∠K > m∠J In △JKL, JK > LK 40. Figure A is a polygon. Figure A is a triangle. Figure A is a quadrilateral. m∠Y < 90° ∠Y is an obtuse angle. 39. ̶̶ AB ⊥ ̶̶ AB ≅ ̶̶ AB ǁ ̶̶ BC ̶̶ CD ̶̶ BC 41. x is even. x is a multiple of 4. x is prime. Compare. Write <, >, or =. PS 42. QS 44. QS 46. PQ QR RS 43. PQ QS 45. QS 47. RS RS PS 48. m∠ABE m∠BEA 49. m∠CBE m∠CEB 50. m∠DCE m∠DEC 51. m∠DCE m∠CDE 52. m∠ABE m∠EAB 53. m∠EBC m∠ECB List the angles of △JKL in order from smallest to largest. 54. J (-3, -2) , K (3, 6) , L (8, -2) 55. J (-5, -10) , K (-5, 2) , L (7, -5) 56. J (-4, 1) , K (-3, 8) , L (3, 4) 57. J (-10, -4) , K (0, 3) , L (2, -8) 58. Critical Thinking An attorney argues that her client did not commit a burglary because a witness saw her client in a different city at the time of the burglary. Explain how this situation is an example of indirect reasoning. 5- 5 Indirect Proof and Inequalities in One Triangle 337 337 B54.1 cm50.8 cmADCge07se_ c05105005aa1st Pass2/7/05N Patel64º66º56º50º�� 59. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The figure shows an airline’s routes between four cities. a. The airline’s planes fly at an average speed of 500 mi/h. What is the range of time it might take to fly from Auburn (A) to Raymond (R)? b. The airline offers one frequent-flier mile for every mile flown. Is it possible to earn 1800 miles by flying from Millford (M) to Auburn (A)? Explain. Multi-Step Each set of expressions represents the lengths of the sides of a triangle. Find the range of possible values of n. 60. n, 6, 8 61. 2n, 5, 7 62. n + 1, 3, 6 63. n + 1, n + 2, n + 3 64. n + 2, n + 3, 3n - 2 65. n, n + 2, 2n + 1 66. Given that P is in the interior of △XYZ, prove that XY + XP + PZ > YZ. 67. Complete the proof of Theorem 5-5-1 by filling in the blanks. Given: RS > RQ Prove: m∠RQS > m∠S Proof: ̶̶ RS so that RP = RQ. So ? , and m∠1 = m∠2 by c. ̶̶̶̶ Locate P on by b. m∠RQS = d. Inequality. Then m∠RQS > m∠2 by e. m∠2 = m∠3 + f. Inequality. Therefore m∠RQS > m∠S by g. ? . So m∠RQS > m∠1 by the Comparison Property of ̶̶̶̶ ? . So m∠2 > m∠S by the Comparison Property of ̶̶̶̶ ? . By the Exterior Angle Theorem, ̶̶̶̶ ̶̶ RQ by a. ̶̶ RP ≅ ? . By the Angle Addition Postulate, ̶̶̶̶ ? . Then ∠1 ≅ ∠2 ̶̶̶̶ ? . ̶̶̶̶ 68. Complete the proof of the Triangle Inequality Theorem. Given: △ABC Prove: AB + BC > AC, AB + AC > BC, AC + BC > AB Proof: One side of △ABC is as long as or longer than each of the other sides. Let this side be remains to be proved is AC + BC > AB. ̶̶ AB . Then AB + BC > AC, and AB + AC > BC. Therefore what Statements Reasons 1. a. ? ̶̶̶̶ 2. Locate D on  AC so that BC = DC. 3. AC + DC = b. ? ̶̶̶̶ 4. ∠1 ≅ ∠2 5. m∠1 = m∠2 6. m∠ABD = m∠2 + e. ? ̶̶̶̶ 7. m∠ABD > m∠2 8. m∠ABD > m∠1 9. AD > AB 10. AC + DC > AB 11. i. ? ̶̶̶̶ 1. Given 2. Ruler Post. 3. Seg. Add. Post. 4. c. 5. d. ? ̶̶̶̶ ? ̶̶̶̶ 6. ∠ Add. Post. 7. Comparison Prop. of Inequal. 8. f. 9. g. 10. h. ? ̶̶̶̶ ? ̶̶̶̶ ? ̶̶̶̶ 11. Subst. 69. Write About It Explain why the hypotenuse is always the longest side of a right triangle. Explain why the diagonal of a square is longer than each side. 338 338 Chapter 5 Properties and Attributes of Triangles �� 70. The lengths of two sides of a triangle are 3 feet and 5 feet. Which could be the length of the third side? 3 feet 8 feet 15 feet 16 feet 71. Which statement about △GHJ is false? GH < GJ m∠H > m∠ J GH + HJ < GJ △GHJ is a scalene triangle. 72. In △RST, m∠S = 92°. Which is the longest side of △RST? ̶̶ RS ̶̶ ST ̶̶ RT Cannot be determined CHALLENGE AND EXTEND 73. Probability A bag contains five sticks. The lengths of the sticks are 1 inch, 3 inches, 5 inches, 7 inches, and 9 inches. Suppose you pick three sticks from the bag at random. What is the probability you can form a triangle with the three sticks? 74. Complete this indirect argument that √  2 is irrational. Assume that a. ? . ̶̶̶̶ __ q , where p and q are positive integers that have no common factors. ? , and p 2 = c. ̶̶̶̶ p Then √  2 = Thus 2 = b. p is even. Since p 2 is the square of an even number, p 2 is divisible by 4 because d. ? , and so q is even. Then p and ̶̶̶̶ q have a common factor of 2, which contradicts the assumption that p and q have no common factors. ? . But then q 2 must be even because e. ̶̶̶̶ ? . This implies that p 2 is even, and thus ̶̶̶̶ 75. Prove that the perpendicular segment from a point to a line is the shortest segment from the point to the line. ̶̶ PX ⊥ ℓ. Y is any point on ℓ other than X. Given: Prove: PY > PX Plan: Show that ∠2 and ∠P are complementary. Use the Comparison Property of Inequality to show that 90° > m∠2. Then show that m∠1 > m∠2 and thus PY > PX. SPIRAL REVIEW Write the equation of each line in standard form. (Previous course) 76. the line through points (-3, 2) and (-1, -2) 77. the line with slope 2 and x-intercept of -3 Show that the triangles are congruent for the given value of the variable. (Lesson 4-4) 78. △PQR ≅ △TUS, when x = -1 79. △ABC ≅ △EFD, when p = 6 Find the orthocenter of a triangle with the given vertices. (Lesson 5-3) 80. R (0, 5) , S (4, 3) , T (0, 1) 81. M (0, 0) , N (3, 0) , P (0, 5) 5- 5 Indirect Proof and Inequalities in One Triangle 339 339 �� 5-6 Inequalities in Two Triangles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.3.E Objective Apply inequalities in two triangles. Who uses this? Designers of this circular swing ride can use the angle of the swings to determine how high the chairs will be at full speed. (See Example 2.) In this lesson, you will apply inequality relationships between two triangles. Theorems Inequalities in Two Triangles THEOREM HYPOTHESIS CONCLUSION 5-6-1 Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle. 5-6-2 Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side. m∠A > m∠D BC > EF m∠ J > m∠M GH > KL You will prove Theorem 5-6-1 in Exercise 35. PROOF PROOF Converse of the Hinge Theorem ̶̶ PR ≅ ̶̶ XZ , QR > YZ ̶̶ PQ ≅ Given: Prove: m∠P > m∠X ̶̶ XY , Indirect Proof: Assume m∠P ≯ m∠X. So either m∠P < m∠X, or m∠P = m∠X. Case 1 If m∠P < m∠X, then QR < YZ by the Hinge Theorem. This contradicts the given information that QR > YZ. So m∠P ≮ m∠X. Case 2 If m∠P = m∠X, then ∠P ≅ ∠X. So △PQR ≅ △XYZ by SAS. Then ̶̶ YZ by CPCTC, and QR = YZ. This also contradicts ̶̶ QR ≅ the given information. So m∠P ≠ m∠X. The assumption m∠P
≯ m∠X is false. Therefore m∠P > m∠X. 340 340 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Using the Hinge Theorem and Its Converse A Compare m∠PQS and m∠RQS. Compare the side lengths in △PQS and △RQS. QS = QS PQ = RQ PS > RS By the Converse of the Hinge Theorem, m∠PQS > m∠RQS. B Compare KL and MN. Compare the sides and angles in △KLN and △MNL. KN = ML LN = LN m∠LNK < m∠NLM By the Hinge Theorem, KL < MN. C Find the range of values for z. Step 1 Compare the side lengths in △TUV and △TWV. TV = TV VU = VW TU < TW By the Converse of the Hinge Theorem, m∠UVT < m∠WVT. 6z - 3 < 45 Substitute the given values. z < 8 Add 3 to both sides and divide both sides by 6. Step 2 Since ∠UVT is in a triangle, m∠UVT > 0°. 6z - 3 > 0 Substitute the given value. z > 0.5 Add 3 to both sides and divide both sides by 6. Step 3 Combine the inequalities. The range of values for z is 0.5 < z < 8. Compare the given measures. 1a. m∠EGH and m∠EGF 1b. BC and AB E X A M P L E 2 Entertainment Application The angle of the swings in a circular swing ride changes with the speed of the ride. The diagram shows the position of one swing at two different speeds. Which rider is farther from the base of the swing tower? Explain. The height of the tower and the length of the cable holding the chair are the same in both triangles. The angle formed by the swing in position A is smaller than the angle formed by the swing in position B. So rider B is farther from the base of the tower than rider A by the Hinge Theorem. 5- 6 Inequalities in Two Triangles 341 341 ��BAge07sec05L06002_A 2. When the swing ride is at full speed, the chairs are farthest from the base of the swing tower. What can you conclude about the angles of the swings at full speed versus low speed? Explain. E X A M P L E 3 Proving Triangle Relationships Write a two-column proof. ̶̶ NL Given: Prove: KM > NM ̶̶ KL ≅ Proof: ̶̶ KL ≅ ̶̶̶ LM ≅ ̶̶ NL ̶̶̶ LM 1. 2. Statements Reasons 1. Given 2. Reflex. Prop. of ≅ 3. m∠KLM = m∠NLM + m∠KLN 3. ∠ Add. Post. 4. m∠KLM > m∠NLM 4. Comparison Prop. of Inequal. 5. KM > NM 5. Hinge Thm. Write a two-column proof. 3a. Given: C is the midpoint of ̶̶ BD . m∠1 = m∠2 m∠3 > m∠4 Prove: AB > ED 3b. Given: ∠SRT ≅ ∠STR TU > RU Prove: m∠TSU > m∠RSU THINK AND DISCUSS 1. Describe a real-world object that shows the Hinge Theorem or its converse. 2. Can you make a conclusion about the triangles shown at right by applying the Hinge Theorem? Explain. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, use the given triangles to write a statement for the theorem. 342 342 Chapter 5 Properties and Attributes of Triangles �� 5-6 Exercises Exercises KEYWORD: MG7 5-6 KEYWORD: MG7 Parent GUIDED PRACTICE Compare the given measures. p. 341 1. AC and XZ 2. m∠SRT and m∠QRT 3. KL and KN Find the range of values for x. 4. 5. 6. Health A therapist can take measurements to p. 341 gauge the flexibility of a patient’s elbow joint. In which position is the angle measure at the elbow joint greater? Explain. 342 Independent Practice For See Exercises Example 9–14 15 16 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 8. Write a two-column proof. Given: ̶̶ FH is a median of △DFG. m∠DHF > m∠GHF Prove: DF > GF PRACTICE AND PROBLEM SOLVING Compare the given measures. 9. m∠DCA and m∠BCA 10. m∠GHJ and m∠KLM 11. TU and SV Find the range of values for z. 12. 13. 14. 5- 6 Inequalities in Two Triangles 343 343 �� 15. Industry The operator of a backhoe changes the distance between the cab and the bucket by changing the angle formed by the arms. In which position is the distance from the cab to the bucket greater? Explain. �� 16. Write a two-column proof. ̶̶̶ MQ , JQ > NP ̶̶ KP ≅ ̶̶̶ NM , ̶̶ JK ≅ Given: Prove: m∠K > m∠M 17. Critical Thinking ABC is an isosceles triangle with base triangle with base ̶̶ YZ . Given that ̶̶ AB ≅ ̶̶ BC . XYZ is an isosceles ̶̶ XY and m∠A = m∠X, compare BC and YZ. Compare. Write <, >, or =. 18. m∠QRP m∠SRP 19. m∠QPR m∠QRP 20. m∠PRS m∠RSP 21. m∠RSP m∠RPS 22. m∠QPR m∠RPS 23. m∠PSR m∠PQR Make a conclusion based on the Hinge Theorem or its converse. (Hint : Draw a sketch.) ̶̶ DE , ̶̶ RT . The endpoints of 25. △RST is isosceles with base 24. In △ABC and △DEF, ̶̶ BC ≅ ̶̶ AB ≅ ̶̶ EF , m∠B = 59°, and m∠E = 47°. ̶̶ SV are vertex S and a point V on ̶̶ RT . RV = 4, and TV = 5. 26. In △GHJ and △KLM, ̶̶̶ GH ≅ ̶̶ KL , and ̶̶ GJ ≅ ̶̶̶ KM . ∠G is a right angle, and ∠K is an acute angle. 27. In △XYZ, ̶̶̶ XM is the median to ̶̶ YZ , and YX > ZX. 28. Write About It The picture shows a door hinge in two different positions. Use the picture to explain why Theorem 5-6-1 is called the Hinge Theorem. 29. Write About It Compare the Hinge Theorem to the SAS Congruence Postulate. How are they alike? How are they different? 30. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The solid lines in the figure show an airline’s routes between four cities. a. A traveler wants to fly from Jackson (J) to Shelby (S), but there is no direct flight between these cities. Given that m∠NSJ < m∠HSJ, should the traveler first fly to Newton Springs (N) or to Hollis (H) if he wants to minimize the number of miles flown? Why? b. The distance from Shelby (S) to Jackson (J) is 182 mi. What is the minimum number of miles the traveler will have to fly? 344 344 Chapter 5 Properties and Attributes of Triangles �� 31. ̶̶ ML is a median of △JKL. Which inequality best describes the range of values for x? x > 2 x > 10 < 10 32. ̶̶ DC is a median of △ABC. Which of the following statements is true? BC < AC BC > AC AD = DB DC = AB 33. Short Response Two groups start hiking from the same camp. Group A hikes 6.5 miles due west and then hikes 4 miles in the direction N 35° W. Group B hikes 6.5 miles due east and then hikes 4 miles in the direction N 45° E. At this point, which group is closer to the camp? Explain. CHALLENGE AND EXTEND 34. Multi-Step In △XYZ, XZ = 5x + 15, XY = 8x - 6, and m∠XVZ > m∠XVY. Find the range of values for x. 35. Use these steps to write a paragraph proof of the Hinge Theorem. ̶̶ BC ≅ ̶̶ AB ≅ ̶̶ DE , Given: ̶̶ EF , m∠ABC > m∠DEF Prove: AC > DF a. Locate P outside △ABC so that ∠ABP ≅ ∠DEF and thus ̶̶ BP ≅ ̶̶ AP ≅ ̶̶ EF . Show that △ABP ≅ △DEF and ̶̶ DF . ̶̶ AC so that Show that △BQP ≅ △BQC and thus ̶̶ BQ bisects ∠PBC. Draw ̶̶ QP ≅ ̶̶ QC . b. Locate Q on ̶̶ QP . c. Justify the statements AQ + QP > AP, AQ + QC = AC, AQ + QC > AP, AC > AP, and AC > DF. SPIRAL REVIEW Find the range and mode, if any, of each set of data. (Previous course) 36. 2, 5, 1, 0.5, 0.75, 2 37. 95, 97, 89, 87, 85, 99 38. 5, 5, 7, 9, 4, 4, 8, 7 For the given information, show that m ǁ n. State any postulates or theorems used. (Lesson 3-3) 39. m∠2 = (3x + 21) °, m∠6 = (7x + 1) °, x = 5 40. m∠4 = (2x + 34) °, m∠7 = (15x + 27) °, x = 7 Find each measure. (Lesson 5-4) 41. DF 42. BC 43. m∠BFD 5- 6 Inequalities in Two Triangles 345 345 6.5 mi6.5 mi4 mi4 miABThird sketchge07se_c05106009aatopo mapGeometry 2007 SEHolt Rinehart WinstonKaren Minot(415)883-6560�� Simplest Radical Form Algebra When a problem involves square roots, you may be asked to give the answer in simplest radical form. Recall that the radicand is the expression under the radical sign. See Skills Bank page S55 Simplest Form of a Square-Root Expression An expression containing square roots is in simplest form when • the radicand has no perfect square factors other than 1. • the radicand has no fractions. • there are no square roots in any denominator. To simplify a radical expression, remember that the square root of a product is equal to the product of the square roots. Also, the square root of a quotient is equal to the quotient of the square roots. √  ab = √  a · √  b , when a ≥ 0 and , when a ≥ 0 and b > 0 Examples Write each expression in simplest radical form. A √  216 √  216 216 has a perfect-square factor of 36, so the expression is not in simplest radical form. √  (36) (6) Factor the radicand. √  36 · √  6 Product Property of Square Roots 6 √  6 Simplify. Try This TAKS Grades 9–11 Obj. 2 Write each expression in simplest radical form. 3. 10_ √  2 2. √3_ 1. √720 16 346 346 Chapter 5 Properties and Attributes of Triangles ) There is a square root in the denominator, so the expression is not in simplest radical form. Multiply by a form of 1 to eliminate the square root in the denominator Simplify. Divide. 4. √1_ 3 5. √45 5-7 Hands-on Proof of the Pythagorean Theorem In Lesson 1-6, you used the Pythagorean Theorem to find the distance between two points in the coordinate plane. In this activity, you will build figures and compare their areas to justify the Pythagorean Theorem. Use with Lesson 5-7 TEKS G.8.C Congruence and the geometry of size: derive, extend, and use the Pythagorean Theorem. Also G.9.B Activity 1 Draw a large scalene right triangle on graph paper. Draw three copies of the triangle. On each triangle, label the shorter leg a, the longer leg b, and the hypotenuse c. 2 Draw a square with a side length of b - a. Label each side of the square. 3 Cut out the five figures. Arrange them to make the composite figure shown at right. 4 You can think of this composite figure as being made of the two squares outlined in red. What are the side length and area of the small red square? of the large red square? 5 Use your results from Step 4 to write an algebraic expression for the area of the composite figure. 6 Now rearrange the five figures to make a single square with side length c. Write an algebraic expression f
or the area of this square. Try This 1. Since the composite figure and the square with side length c are made of the same five shapes, their areas are equal. Write and simplify an equation to represent this relationship. What conclusion can you make? 2. Draw a scalene right triangle with different side lengths. Repeat the activity. Do you reach the same conclusion? 5- 7 Geometry Lab 347 347 5-7 The Pythagorean Theorem TEKS G.8.C Congruence and the geometry of size: derive, extend, and use the Pythagorean Theorem. Also G.1.B, G.5.B, G.5.D, G.11.C Objectives Use the Pythagorean Theorem and its converse to solve problems. Use Pythagorean inequalities to classify triangles. Vocabulary Pythagorean triple Why learn this? You can use the Pythagorean Theorem to determine whether a ladder is in a safe position. (See Example 2.) The Pythagorean Theorem is probably the most famous mathematical relationship. As you learned in Lesson 1-6, it states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. For more on the Pythagorean Theorem, see the Theorem Builder on page xxvi. � � � The Pythagorean Theorem is named for the Greek mathematician Pythagoras, who lived in the sixth century B.C.E. However, this relationship was known to earlier people, such as the Babylonians, Egyptians, and Chinese. There are many different proofs of the Pythagorean Theorem. The one below uses area and algebra. PROOF PROOF Pythagorean Theorem Given: A right triangle with leg lengths a and b and hypotenuse of length c Prove: a 2 + b 2 = c 2 The area A of a square with side length s is given by the formula A = s 2 . The area A of a triangle with base b and height h is given by the formula A = 1 __ bh. 2 Proof: Arrange four copies of the triangle as shown. The sides of the triangles form two squares. The area of the outer square is (a + b) 2 . The area of the inner square is c 2 . The area of each blue triangle is 1 __ 2 ab. area of outer square = area of 4 blue triangles + area of inner square (a + b) 2 = 4 ( 1 _ 2 ab) + c 2 a 2 + 2ab + b 2 = 2ab + Substitute the areas. Simplify. Subtract 2ab from both sides. The Pythagorean Theorem gives you a way to find unknown side lengths when you know a triangle is a right triangle. 348 348 Chapter 5 Properties and Attributes of Triangles �� E X A M P L E 1 Using the Pythagorean Theorem Find the value of x. Give your answer in simplest radical form 52 = x 2 √  52 = x Pythagorean Theorem Substitute 6 for a, 4 for b, and x for c. Simplify. Find the positive square root. x = √  (4) (13) = 2 √  13 Simplify the radicalx - 1) 2 = x 2 Pythagorean Theorem Substitute 5 for a, x - 1 for b, 25 + x 2 - 2x + 1 = x 2 Multiply. and x for c. -2x + 26 = 0 Combine like terms. 26 = 2x x = 13 Add 2x to both sides. Divide both sides by 2. Find the value of x. Give your answer in simplest radical form. 1a. 1b. E X A M P L E 2 Safety Application To prevent a ladder from shifting, safety experts recommend that the ratio of a : b be 4 : 1. How far from the base of the wall should you place the foot of a 10-foot ladder? Round to the nearest inch. Let x be the distance in feet from the foot of the ladder to the base of the wall. Then 4x is the distance in feet from the top of the ladder to the base of the wall4x) 2 + x 2 = 10 2 17 x 2 = 100 x 2 = 100 _ 17  100 _ 17 x = √ ≈ 2 ft 5 in. Pythagorean Theorem Substitute. Multiply and combine like terms. Divide both sides by 17. Find the positive square root and round it. 2. What if...? According to the recommended ratio, how high will a 30-foot ladder reach when placed against a wall? Round to the nearest inch. A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2 is called a Pythagorean triple . Common Pythagorean Triples 3, 4, 5 5, 12, 13, 8, 15, 17 7, 24, 25 349 5- 7 The Pythagorean Theorem 349 ��abge07sec05l07002aAB E X A M P L E 3 Identifying Pythagorean Triples Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain 12 2 + b 2 = 15 2 b 2 = 81 b = 9 Pythagorean Theorem Substitute 12 for a and 15 for c. Multiply and subtract 144 from both sides. Find the positive square root. The side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2 , so they form a Pythagorean triple + 15 2 = c 2 306 = c 2 Pythagorean Theorem Substitute 9 for a and 15 for b. Multiply and add. c = √  306 = 3 √  34 Find the positive square root and simplify. The side lengths do not form a Pythagorean triple because 3 √  34 is not a whole number. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 3a. 3c. 3b. 3d. The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths. Theorems 5-7-1 Converse of the Pythagorean Theorem THEOREM HYPOTHESIS CONCLUSION If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. △ABC is a right triangle. a 2 + b 2 = c 2 You will prove Theorem 5-7-1 in Exercise 45. 350 350 Chapter 5 Properties and Attributes of Triangles �� You can also use side lengths to classify a triangle as acute or obtuse. Theorems 5-7-2 Pythagorean Inequalities Theorem In △ABC, c is the length of the longest side. If c 2 > a 2 + b 2 , then △ABC is an obtuse triangle. If c 2 < a 2 + b 2 , then △ABC is an acute triangle. To understand why the Pythagorean inequalities are true, consider △ABC. If c 2 = a 2 + b 2 , then △ABC is a right triangle by the Converse of the Pythagorean Theorem. So m∠C = 90°. If c 2 > a 2 + b 2 , then c has increased. By the Converse of the Hinge Theorem, m∠C has also increased. So m∠C > 90°. If c 2 < a 2 + b 2 , then c has decreased. By the Converse of the Hinge Theorem, m∠C has also decreased. So m∠C < 90°. E X A M P L E 4 Classifying Triangles By the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greater than the third side length. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. A 8, 11, 13 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 8, 11, and 13 can be the side lengths of a triangle. Step 2 Classify the triangle. c 2 ≟ a 2 + b 2 13 2 ≟ 8 2 + 11 2 169 ≟ 64 + 121 169 < 185 Compare c 2 to a 2 + b 2 . Substitute the longest side length for c. Multiply. Add and compare. Since c 2 < a 2 + b 2 , the triangle is acute. B 5.8, 9.3, 15.6 Step 1 Determine if the measures form a triangle. Since 5.8 + 9.3 = 15.1 and 15.1 ≯ 15.6, these cannot be the side lengths of a triangle. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 4a. 7, 12, 16 4b. 11, 18, 34 4c. 3.8, 4.1, 5.2 351 5- 7 The Pythagorean Theorem 351 �� THINK AND DISCUSS 1. How do you know which numbers to substitute for c, a, and b when using the Pythagorean Inequalities? 2. Explain how the figure at right demonstrates the Pythagorean Theorem. 3. List the conditions that a set of three numbers must satisfy in order to form a Pythagorean triple. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, summarize the Pythagorean relationship. 5-7 Exercises Exercises KEYWORD: MG7 5-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Do the numbers 2.7, 3.6, and 4.5 form a Pythagorean triple? Explain why or why not Find the value of x. Give your answer in simplest radical form. p. 349 2. 3. 4. 349 5. Computers The size of a computer monitor is usually given by the length of its diagonal. A monitor’s aspect ratio is the ratio of its width to its height. This monitor has a diagonal length of 19 inches and an aspect ratio of 5 : 4. What are the width and height of the monitor? Round to the nearest tenth of an inch. 350 Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 6. 7. 8. 351 Multi-Step Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 9. 7, 10, 12 , 1 3 _ , 3 1 _ 12. 1 1 _ 4 4 2 10. 9, 11, 15 13. 5.9, 6, 8.4 11. 9, 40, 41 14. 11, 13, 7 √  6 352 352 Chapter 5 Properties and Attributes of Triangles �� PRACTICE AND PROBLEM SOLVING Find the value of x. Give your answer in simplest radical form. 15. 16. 17. 18. Safety The safety rules for a playground state that the height of the slide and the distance from the base of the ladder to the front of the slide must be in a ratio of 3 : 5. If a slide is about 8 feet long, what are the height of the slide and the distance from the base of the ladder to the front of the slide? Round to the nearest inch. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 19. 20. 21. Multi-Step Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 22. 10, 12, 15 , 2, 2 1 _ 25. 1 1 _ 2 2 23. 8, 13, 23 26. 0.7, 1.1, 1.7 24. 9, 14, 17 27. 7, 12, 6 √  5 28. Surveying It is believed that surveyors in ancient Egypt laid out right angles using a rope divided into twelve sections by eleven equally spaced knots. How could the surveyors use this rope to make a right angle? 29. /////ERROR ANALYSIS///// Below are two solutions for finding x. Which is incorrect? Explain the error. Independent Practice For See Exercises Example 15–17 18 19–21 22–27 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 Surveying Ancient Egyptian surveyors were referred to as rope-stretchers. The standard surveying rope was 100 royal cubits. A cubit is 52.4 cm long. Find the value of x. Give
your answer in simplest radical form. 30. 31. 33. 34. 32. 35. 353 5- 7 The Pythagorean Theorem 353 �� 36. Space Exploration The International Space Station orbits at an altitude of about 250 miles above Earth’s surface. The radius of Earth is approximately 3963 miles. How far can an astronaut in the space station see to the horizon? Round to the nearest mile. 37. Critical Thinking In the proof of the Pythagorean Theorem on page 348, how do you know the outer figure is a square? How do you know the inner figure is a square? Multi-Step Find the perimeter and the area of each figure. Give your answer in simplest radical form. 38. 41. 39. 42. 40. 43. 44. Write About It When you apply both the Pythagorean Theorem and its converse, you use the equation a 2 + b 2 = c 2 . Explain in your own words how the two theorems are different. 45. Use this plan to write a paragraph proof of the Converse of the Pythagorean Theorem. Given: △ABC with a 2 + b 2 = c 2 Prove: △ABC is a right triangle. Plan: Draw △PQR with ∠R as the right angle, leg lengths of a and b, and a hypotenuse of length x. By the Pythagorean Theorem, a 2 + b 2 = x 2 . Use substitution to compare x and c. Show that △ABC ≅ △PQR and thus ∠C is a right angle. 46. Complete these steps to prove the Distance Formula. Given: J ( x 1 , y 1 ) and K ( x 2 , y 2 ) with x 1 ≠ x 2 and y 1 ≠ y 2 Prove: JK = √  ( ̶̶ a. Locate L so that JK is the hypotenuse of right △JKL. What are the coordinates of L? b. Find JL and LK. c. By the Pythagorean Theorem, JK 2 = JL 2 + LK 2 . Find JK. 47. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The figure shows an airline’s routes between four cities. a. A traveler wants to go from Sanak (S) to Manitou (M). To minimize the total number of miles traveled, should she first fly to King City (K) or to Rice Lake (R)? b. The airline decides to offer a direct flight from Sanak (S) to Manitou (M). Given that the length of this flight is more than 1360 mi, what can you say about m∠SRM? 354 354 Chapter 5 Properties and Attributes of Triangles �� 48. Gridded Response ̶̶ KX , ̶̶ LX , and ̶̶̶ MX are the perpendicular bisectors of △GHJ. Find GJ to the nearest tenth of a unit. 49. Which number forms a Pythagorean triple with 24 and 25? 1 7 26 49 50. The lengths of two sides of an obtuse triangle are 7 meters and 9 meters. Which could NOT be the length of the third side? 4 meters 5 meters 11 meters 12 meters 51. Extended Response The figure shows the first six triangles in a pattern of triangles. a. Find PA, PB, PC, PD, PE, and PF in simplest radical form. b. If the pattern continues, what would be the length of the hypotenuse of the ninth triangle? Explain your answer. c. Write a rule for finding the length of the hypotenuse of the nth triangle in the pattern. Explain your answer. CHALLENGE AND EXTEND 52. Algebra Find all values of k so that (-1, 2) , (-10, 5) , and (-4, k) are the vertices of a right triangle. 53. Critical Thinking Use a diagram of a right triangle to explain why a + b > √  a 2 + b 2 for any positive numbers a and b. 54. In a right triangle, the leg lengths are a and b, and the length of the altitude to the hypotenuse is h. Write an expression for h in terms of a and b. (Hint: Think of the area of the triangle.) 55. Critical Thinking Suppose the numbers a, b, and c form a Pythagorean triple. Is each of the following also a Pythagorean triple? Explain. a. a + 1, b + 1, c + 1 c. 2a, 2b, 2c d SPIRAL REVIEW Solve each equation. (Previous course) 56. (4 + x) 12 - (4x + 1) 6 = 0 57. 2x - 5 _ 3 = x 58. 4x + 3 (x + 2) = -3 (x + 3) Write a coordinate proof. (Lesson 4-7) 59. Given: ABCD is a rectangle with A (0, 0) , B (0, 2b) , C (2a, 2b) , and D (2a, 0) . M is the midpoint of ̶̶ AC . Prove: AM = MB Find the range of values for x. (Lesson 5-6) 60. 61. 355 5- 7 The Pythagorean Theorem 355 �� 5-8 Applying Special Right Triangles TEKS G.5.D Geometric patterns: identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90)... Objectives Justify and apply properties of 45°-45°-90° triangles. Justify and apply properties of 30°-60°-90° triangles. Also G.3.B, G.5.A, G.7.A Who uses this? You can use properties of special right triangles to calculate the correct size of a bandana for your dog. (See Example 2.) A diagonal of a square divides it into two congruent isosceles right triangles. Since the base angles of an isosceles triangle are congruent, the measure of each acute angle is 45°. So another name for an isosceles right triangle is a 45°-45°-90° triangle. A 45°-45°-90° triangle is one type of special right triangle. You can use the Pythagorean Theorem to find a relationship among the side lengths of a 45°-45°-90° triangle √  Pythagorean Theorem Substitute the given values. Simplify. Find the square root of both sides. Simplify. Theorem 5-8-1 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times √  2 . AC = BC = ℓ AB = Finding Side Lengths in a 45°-45°-90° Triangle Find the value of x. Give your answer in simplest radical form. A By the Triangle Sum Theorem, the measure of the third angle of the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of 7. x = 7 √  2 Hypotenuse = leg √  2 356 356 Chapter 5 Properties and Attributes of Triangles �� Find the value of x. Give your answer in simplest radical form. B The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 3. 3 = x √  2 Hypotenuse = leg √  Divide both sides by √  2 . Rationalize the denominator. Find the value of x. Give your answer in simplest radical form. 1a. 1b. E X A M P L E 2 Craft Application � � � � Tessa wants to make a bandana for her dog by folding a square of cloth into a 45°-45°-90° triangle. Her dog’s neck has a circumference of about 32 cm. The folded bandana needs to be an extra 16 cm long so Tessa can tie it around her dog’s neck. What should the side length of the square be? Round to the nearest centimeter. � � �� Tessa needs a 45°-45°-90° triangle with a hypotenuse of 48 cm. 48 = ℓ √  2 ℓ = 48 _ √  2 Divide by √  2 and round. Hypotenuse = leg √  2 ≈ 34 cm 2. What if...? Tessa’s other dog is wearing a square bandana with a side length of 42 cm. What would you expect the circumference of the other dog’s neck to be? Round to the nearest centimeter. A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths. Draw an altitude in △PQR. Since △PQS ≅ △RQS, ̶̶ PS ≅ the Pythagorean Theorem to find y. ̶̶ RS . Label the side lengths in terms of x, and use 2x ) = √  3 x 2 y = x √  3 Pythagorean Theorem Substitute x for a, y for b, and 2x for c. Multiply and combine like terms. Find the square root of both sides. Simplify. 5- 8 Applying Special Right Triangles 357 357 �� Theorem 5-8-2 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the length of the hypotenuse is is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times √  3 . AC = s AB = 2s BC = Finding Side Lengths in a 30°-60°-90° Triangle Find the values of x and y. Give your answers in simplest radical form. A B 16 = 2x Hypotenuse = 2 (shorter leg) Divide both sides by 2. Longer leg = (shorter leg) √  3 Substitute 8 for x. 11 = x √  3 Longer leg = (shorter leg) √  3 If two angles of a triangle are not congruent, the shorter side lies opposite the smaller angle. = x = x 11 _ √  3 11 √  3 _ 3 y = 2x y = 2 ( ) _ 11 √  3 3 y = 22 √  3 _ 3 Divide both sides by √  3 . Rationalize the denominator. Hypotenuse = 2 (shorter leg) Substitute 11 √  3 _ 3 for x. Simplify. Find the values of x and y. Give your answers in simplest radical form. 3a. 3c. 3b. 3d. 358 358 Chapter 5 Properties and Attributes of Triangles �� 30°-60°-90° Triangles To remember the side relationships in a 30°-60°-90° triangle, I draw a simple “1-2- √  3 ” triangle like this. 2 = 2 (1) , so hypotenuse = 2 (shorter leg1) , so longer leg = √  3 (shorter leg) . Marcus Maiello Johnson High School E X A M P L E 4 Using the 30°-60°-90° Triangle Theorem The frame of the clock shown is an equilateral triangle. The length of one side of the frame is 20 cm. Will the clock fit on a shelf that is 18 cm below the shelf above it? Step 1 Divide the equilateral triangle into two 30°-60°-90° triangles. The height of the frame is the length of the longer leg. Step 2 Find the length x of the shorter leg. Hypotenuse = 2(shorter leg) Divide both sides by 2. 20 = 2x 10 = x Step 3 Find the length h of the longer leg. h = 10 √  3 ≈ 17.3 cm Longer leg = (shorter leg) √  3 The frame is approximately 17.3 centimeters tall. So the clock will fit on the shelf. 4. What if…? A manufacturer wants to make a larger clock with a height of 30 centimeters. What is the length of each side of the frame? Round to the nearest tenth. THINK AND DISCUSS 1. Explain why an isosceles right triangle is a 45°-45°-90° triangle. 2. Describe how finding x in triangle I is different from finding x in triangle II. I. II. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, sketch the special r
ight triangle and label its side lengths in terms of s. 5- 8 Applying Special Right Triangles 359 359 �� 5-8 Exercises Exercises GUIDED PRACTICE Find the value of x. Give your answer in simplest radical form. p. 356 1. 2. 3. KEYWORD: MG7 5-8 KEYWORD: MG7 Parent . 357 4. Transportation The two arms of the railroad sign are perpendicular bisectors of each other. In Pennsylvania, the lengths marked in red must be 19.5 inches. What is the distance labeled d? Round to the nearest tenth of an inch Find the values of x and y. Give your answers in simplest radical form. p. 358 5. 6. 7. 359 8. Entertainment Regulation billiard balls are 2 1 __ 4 inches in diameter. The rack used to group 15 billiard balls is in the shape of an equilateral triangle. What is the approximate height of the triangle formed by the rack? Round to the nearest quarter of an inch. Independent Practice For See Exercises Example 9–11 12 13–15 16 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S13 Application Practice p. S32 PRACTICE AND PROBLEM SOLVING Find the value of x. Give your answer in simplest radical form. 9. 10. 11. 12. Design This tabletop is an isosceles 12. right triangle. The length of the front edge of the table is 48 inches. What is the length w of each side edge? Round to the nearest tenth of an inch. � � �� Find the value of x and y. Give your answers in simplest radical form. 13. 14. 15. 360 360 Chapter 5 Properties and Attributes of Triangles �� Pets 16. Pets A dog walk is used in dog agility 16. competitions. In this dog walk, each ramp makes an angle of 30° with the ground. a. How long is one ramp? b. b. How long is the entire dog walk, including both ramps? Multi-Step Find the perimeter and area of each figure. Give your answers in simplest radical form. 17. 17. a 45°-45°-90° triangle with hypotenuse length 12 inches 18. 18. a 30°-60°-90° triangle with hypotenuse length 28 centimeters Agility courses test the skill of both the dog and the dog’s handler. Dog agility competitions in the United States are regulated by the U.S. Dog Agility Association, headquartered in Garland, Texas. 19. 19. a square with diagonal length 18 meters 20. an equilateral triangle with side length 4 feet 21. an equilateral triangle with height 30 yards 22. Estimation The triangle loom is made from wood strips shaped into a 45°-45°-90° triangle. Pegs are placed every 1 __ 2 inch along the hypotenuse and every 1 __ 4 inch along each leg. Suppose you make a loom with an 18-inch hypotenuse. Approximately how many pegs will you need? 23. Critical Thinking The angle measures of a triangle are in the ratio 1 : 2 : 3. Are the side lengths also in the ratio 1 : 2 : 3? Explain your answer. Find the coordinates of point P under the given conditions. Give your answers in simplest radical form. 24. △PQR is a 45°-45°-90° triangle with vertices Q (4, 6) and R (-6, -4) , and m∠P = 90°. P is in Quadrant II. 25. △PST is a 45°-45°-90° triangle with vertices S (4, -3) and T (-2, 3) , and m∠S = 90°. P is in Quadrant I. 26. △PWX is a 30°-60°-90° triangle with vertices W (-1, -4) and X (4, -4) , and m∠W = 90°. P is in Quadrant II. 27. △PYZ is a 30°-60°-90° triangle with vertices Y (-7, 10) and Z (5, 10) , and m∠Z = 90°. P is in Quadrant IV. 28. Write About It Why do you think 30°-60°-90° triangles and 45°-45°-90° triangles are called special right triangles? 29. This problem will prepare you for the Multi-Step TAKS Prep on page 364. The figure shows an airline’s routes among four cities. The airline offers one frequent-flier mile for each mile flown (rounded to the nearest mile). How many frequent-flier miles do you earn for each flight? a. Nelson (N) to Belton (B) b. Idria (I) to Nelson (N) c. Belton (B) to Idria (I) 5- 8 Applying Special Right Triangles 361 361 30°30°12 ft4.5 ftge07se_c05l08007aAB�� 30. Which is a true statement? AB = BC √  2 AB = BC √  3 AC = BC √  3 AC = AB √  2 31. An 18-foot pole is broken during a storm. The top of the pole touches the ground 12 feet from the base of the pole. How tall is the part of the pole left standing? 5 feet 6 feet 13 feet 22 feet �� 32. The length of the hypotenuse of an isosceles right triangle is 24 inches. What is the length of one leg of the triangle, rounded to the nearest tenth of an inch? 13.9 inches 17.0 inches 33.9 inches 41.6 inches 33. Gridded Response Find the area of the rectangle to the nearest tenth of a square inch. CHALLENGE AND EXTEND Multi-Step Find the value of x in each figure. 34. 35. 36. Each edge of the cube has length e. a. Find the diagonal length d when e = 1, e = 2, and e = 3. Give the answers in simplest radical form. b. Write a formula for d for any positive value of e. 37. Write a paragraph proof to show that the altitude to the hypotenuse of a 30°-60°-90° triangle divides the hypotenuse into two segments, one of which is 3 times as long as the other. SPIRAL REVIEW Rewrite each function in the form y = a (x - h) 2 - k and find the axis of symmetry. (Previous course) 38. y = x 2 + 4x 39. y = x 2 - 10x -2 40. y = x 2 + 7x +15 Classify each triangle by its angle measures. (Lesson 4-1) 41. △ ADB 42. △BDC 43. △ ABC Use the diagram for Exercises 44–46. (Lesson 5-1) 44. Given that PS = SR and m∠PSQ = 65°, find m∠PQR. 45. Given that UT = TV and m∠PQS = 42°, find m∠VTS. 46. Given that ∠PQS ≅ ∠SQR, SR = 3TU, and PS = 7.5, find TV. 362 362 Chapter 5 Properties and Attributes of Triangles �� 5-8 Graph Irrational Numbers Numbers such as √  2 and √  3 are irrational. That is, they cannot be written as the ratio of two integers. In decimal form, they are infinite nonrepeating decimals. You can round the decimal form to estimate the location of these numbers on a number line, or you can use right triangles to construct their locations exactly. Use with Lesson 5-8 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.5.B Activity 1 Draw a line. Mark two points near the left side of the line and label them 0 and 1. The distance from 0 to 1 is 1 unit. 2 Set your compass to 1 unit and mark increments at 2, 3, 4, and 5 units to construct a number line. 3 Construct a perpendicular to the line through 1. 4 Using your compass, mark 1 unit up from the number line and then draw a right triangle. The legs both have length 1, so by the Pythagorean Theorem, the hypotenuse has a length of √2 . 5 Set your compass to the length of the hypotenuse. Draw an arc centered at 0 that intersects the number line at √  2 . 6 Repeat Steps 3 through 5, starting at √  2 , to construct a segment of length √  3 . Try This 1. Sketch the two right triangles from Step 6. Label the side lengths and use the Pythagorean Theorem to show why the construction is correct. 2. Construct √4 and verify that it is equal to 2. 3. Construct √5 through √9 and verify that √9 is equal to 3. 4. Set your compass to the length of the segment from 0 to √2 . Mark off another segment of length √2 to show that √8 is equal to 2 √2 . 5- 8 Geometry Lab 363 363 �� SECTION 5B Relationships in Triangles Fly Away! A commuter airline serves the four cities of Ashton, Brady, Colfax, and Dumas, located at points A, B, C, and D, respectively. The solid lines in the figure show the airline’s existing routes. The airline is building an airport at H, which will serve as a hub. This will add four new routes to their schedule: ̶̶ CH , and ̶̶ BH , ̶̶ AH , ̶̶̶ DH . 1. The airline wants to locate the airport so that the combined distance to the cities (AH + BH + CH + DH) is as small as possible. Give an indirect argument to explain why the airline should locate the airport at the ̶̶ AC and intersection of the diagonals point X inside quadrilateral ABCD results in a smaller combined distance. Then consider how AX + CX compares to AH + CH.) ̶̶ BD . (Hint: Assume that a different 2. Currently, travelers who want to go from Ashton to Colfax must first fly to Brady. Once the airport is built, they will fly from Ashton to the new airport and then to Colfax. How many miles will this save compared to the distance of the current trip? 3. Currently, travelers who want to go from Brady to Dumas must first fly to Colfax. Once the airport is built, they will fly from Brady to the new airport and then to Dumas. How many miles will this save? 4. Once the airport is built, the airline plans to serve a meal only on its longest flight. On which route should they serve the meal? How do you know that this route is the longest? 364 364 Chapter 5 Properties and Attributes of Triangles �� SECTION 5B Quiz for Lessons 5-5 Through 5-8 5-5 Indirect Proof and Inequalities in One Triangle 1. Write an indirect proof that the supplement of an acute angle cannot be an acute angle. 2. Write the angles of △KLM in order from smallest to largest. 3. Write the sides of △DEF in order from shortest to longest. Tell whether a triangle can have sides with the given lengths. Explain. 4. 8.3, 10.5, 18.8 5. 4s, s + 10, s 2 , when s = 4 6. The distance from Kara’s school to the theater is 9 km. The distance from her school to the zoo is 16 km. If the three locations form a triangle, what is the range of distances from the theater to the zoo? 5-6 Inequalities in Two Triangles 7. Compare PR and SV. 8. Compare m∠KJL and 9. Find the range of m∠MJL. values for x. 5-7 The Pythagorean Theorem 10. Find the value of x. Give the answer in simplest radical form. 11. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. 12. Tell if the measures 10, 12, and 16 can be the side lengths of a tria
ngle. If so, classify the triangle as acute, obtuse, or right. 13. A landscaper wants to place a stone walkway from one corner of the rectangular lawn to the opposite corner. What will be the length of the walkway? Round to the nearest inch. 5-8 Applying Special Right Triangles 14. A yield sign is an equilateral triangle with a side length of 36 inches. �� What is the height h of the sign? Round to the nearest inch. Find the values of the variables. Give your answers in simplest radical form. 15. 16. 17. �� Ready to Go On? 365 365 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary altitude of a triangle . . . . . . . . 316 equidistant . . . . . . . . . . . . . . . . . 300 median of a triangle . . . . . . . . 314 centroid of a triangle . . . . . . . . 314 incenter of a triangle . . . . . . . . 309 midsegment of a triangle . . . . 322 circumcenter of a triangle . . . 307 indirect proof . . . . . . . . . . . . . . . 332 orthocenter of a triangle . . . . 316 circumscribed . . . . . . . . . . . . . . 308 inscribed . . . . . . . . . . . . . . . . . . . 309 point of concurrency . . . . . . . . 307 concurrent . . . . . . . . . . . . . . . . . 307 locus . . . . . . . . . . . . . . . . . . . . . . . 300 Pythagorean triple . . . . . . . . . . 349 Complete the sentences below with vocabulary words from the list above. 1. A point that is the same distance from two or more objects is ? from the objects. ̶̶̶̶ 2. A ? is a segment that joins the midpoints of two sides of the triangle. ̶̶̶̶ 3. The point of concurrency of the angle bisectors of a triangle is the ? . ̶̶̶̶ 4. A ? is a set of points that satisfies a given condition. ̶̶̶̶ 5-1 Perpendicular and Angle Bisectors (pp. 300–306) E X A M P L E S Find each measure. ■ JL ̶̶̶ MK and ̶̶ JM ≅ ̶̶̶ ML is the Because ̶̶ ̶̶̶ ML ⊥ JK , perpendicular bisector of ̶̶ JK . EXERCISES Find each measure. 5. BD 6. YZ TEKS G.3.B, G.3.E, G.7.A, G.7.B, G.7.C, G.10.B JL = KL ⊥ Bisector Thm. JL = 7.9 Substitute 7.9 for KL. ■ m∠PQS, given that m∠PQR = 68° ̶̶ ̶̶ SP ⊥ QP , Since SP = SR, ̶̶ ̶̶  QS bisects QR , SR ⊥ and ∠PQR by the Converse of the Angle Bisector Theorem. m∠PQR m∠PQS = 1 _ 2 m∠PQS = 1 _ (68°) = 34° 2 Def. of ∠ bisector Substitute 68° for m∠PQR. 7. HT 8. m∠MNP Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints. 9. A (-4, 5) , B (6, -5) 10. X (3, 2) , Y (5, 10) Tell whether the given information allows you to conclude that P is on the bisector of ∠ABC. 11. 12. 366 366 Chapter 5 Properties and Attributes of Triangles �� 5-2 Bisectors of Triangles (pp. 307–313) TEKS G.2.A, G.2.B, G.3.B, G.7.A, G.7.B E X A M P L E S ̶̶ ̶̶ FG EG , and ■ ̶̶ DG , are the perpendicular bisectors of △ABC. Find AG. G is the circumcenter of △ABC. By the Circumcenter Theorem, G is equidistant from the vertices of △ABC. AG = CG Circumcenter Thm. AG = 5.1 Substitute 5.1 for CG. ■ ̶̶ ̶̶ QS and RS are angle bisectors of △PQR. Find the ̶̶ PR . distance from S to S is the incenter of △PQR. By the Incenter Theorem, S is equidistant from the sides of △PQR. The distance from S to ̶̶ PR is also 17. distance from S to ̶̶ PQ is 17, so the ̶̶ PY , and EXERCISES ̶̶ PX , perpendicular bisectors of △GHJ. Find each length. ̶̶ PZ are the 13. GY 15. GJ 14. GP 16. PH ̶̶ ̶̶ UA and VA are angle bisectors of △UVW. Find each measure. 17. the distance from ̶̶ UV A to 18. m∠WVA Find the circumcenter of a triangle with the given vertices. 19. M (0, 6) , N (8, 0) , O (0, 0) 20. O (0, 0) , R (0, -7) , S (-12, 0) 5-3 Medians and Altitudes of Triangles (pp. 314–320) TEKS G.2.A, G.2.B, G.3.B, G.7.A, G.7.B, G.7.C E X A M P L E S ■ In △JKL, JP = 42. Find JQ. JP JQ = 2_ 3 JQ = 2_ 3 JQ = 28 (42) Centroid Thm. Substitute 42 for JP. Multiply. ■ Find the orthocenter of △RST with vertices R (-5, 3) , S (-2, 5) , and T (-2, 0) . Since ̶̶ ST is vertical, the equation of the line containing the altitude ̶̶ ST is y = 3. from R to ̶̶ RT = slope of 3 - 0 _ -5 - (-2) = -1 ̶̶ RT is 1. The slope of the altitude to This line must pass through S (-2, 5) . y - y 1 = m (x - x 1 ) Point-slope form y - 5 = 1 (x + 2) ⎧ y = 3 ⎨ Solve the system y = x + 7 ⎩ Substitution to find that the EXERCISES In △DEF, DB = 24.6, and EZ = 11.6. Find each length. 21. DZ 22. ZB 23. ZC 24. EC Find the orthocenter of a triangle with the given vertices. 25. J (-6, 7) , K (-6, 0) , L (-11, 0) 26. A (1, 2) , B (6, 2) , C (1, -8) 27. R (2, 3) , S (7, 8) , T (8, 3) 28. X (-3, 2) , Y (5, 2) , Z (3, -4) 29. The coordinates of a triangular piece of a mobile are (0, 4) , (3, 8) , and (6, 0) . The piece will hang from a chain so that it is balanced. At what coordinates should the chain be attached? coordinates of the orthocenter are (-4, 3) . Study Guide: Review 367 367 �� TEKS G.2.A, G.2.B, G.3.B, G.5.A, G.7.B, G.9.B 5-4 The Triangle Midsegment Theorem (pp. 322–327) E X A M P L E S Find each measure. ■ NQ By the △ Midsegment Thm., NQ = 1 _ 2 KL = 45.7. ■ m∠NQM ̶̶̶ ̶̶ ML NP ǁ m∠NQM = m∠PNQ m∠NQM = 37° △ Midsegment Thm. Alt. Int.  Thm. Substitution EXERCISES Find each measure. 30. BC 31. XZ 32. XC 33. m∠BCZ 34. m∠BAX 35. m∠YXZ 36. The vertices of △GHJ are G (-4, -7) , H (2, 5) , and J (10, -3) . V is the midpoint of ̶̶ HJ . Show that W is the midpoint of and VW = 1 __ 2 GJ. ̶̶̶ GH , and ̶̶ ̶̶̶ GJ VW ǁ 5-5 Indirect Proof and Inequalities in One Triangle (pp. 332–339) TEKS G.3.B, E X A M P L E S ■ Write the angles of △RST in order from smallest to largest. EXERCISES 37. Write the sides of △ABC in order from shortest to longest. The smallest angle is opposite the shortest side. In order, the angles are ∠S, ∠R, and ∠T. 38. Write the angles of △FGH in order from smallest to largest. G.3.C, G.3.E, G.5.B ■ The lengths of two sides of a triangle are 15 inches and 12 inches. Find the range of possible lengths for the third side. Let s be the length of the third side. s + 15 > 12 s > -3 s + 12 > 15 s > 3 15 + 12 > s 27 > s By the Triangle Inequality Theorem, 3 in. < s < 27 in. 39. The lengths of two sides of a triangle are 13.5 centimeters and 4.5 centimeters. Find the range of possible lengths for the third side. Tell whether a triangle can have sides with the given lengths. Explain. 40. 6.2, 8.1, 14.2 41. z, z, 3z, when z = 5 42. Write an indirect proof that a triangle cannot have two obtuse angles. 5-6 Inequalities in Two Triangles (pp. 340–345) TEKS G.3.B, G.3.E E X A M P L E S Compare the given measures. ■ KL and ST KJ = RS, JL = RT, and m∠J > m∠R. By the Hinge Theorem, KL > ST. ■ m∠ZXY and m∠XZW XY = WZ, XZ = XZ, and YZ < XW. By the Converse of the Hinge Theorem, m∠ZXY < m∠XZW. 368 368 Chapter 5 Properties and Attributes of Triangles EXERCISES Compare the given measures. 43. PS and RS 44. m∠BCA and m∠DCA Find the range of values for n. 46. 45. �� 5-7 The Pythagorean Theorem (pp. 348–355) TEKS G.1.B, G.5.B, G.5.D, G.8.C, G.11.C E X A M P L E S EXERCISES ■ Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 62 + 32 = x2 45 = x 2 x = 3 √5 Pyth. Thm. Substitution Simplify. Find the positive square root and simplify. ■ Find the missing side length. Tell if the sides form a Pythagorean triple. Explain1.6)2 = 22 a 2 = 1.44 a = 1.2 Pyth. Thm. Substitution Solve for a 2 . Find the positive square root. The side lengths do not form a Pythagorean triple because 1.2 and 1.6 are not whole numbers. Find the value of x. Give your answer in simplest radical form. 47. 48. Find the missing side length. Tell if the sides form a Pythagorean triple. Explain. 49. 50. Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 51. 9, 12, 16 52. 11, 14, 27 53. 1.5, 3.6, 3.9 54. 2, 3.7, 4.1 5-8 Applying Special Right Triangles (pp. 356–362) TEKS G.3.B, G.5.A, G.5.D, G.7.A E X A M P L E S EXERCISES Find the values of the variables. Give your answers in simplest radical form. ■ ■ ■ This is a 45°-45°-90° triangle. x = 19 √  2 Hyp. = leg √  2 This is a 45°-45°-90° triangle. 15 = x √  2 Hyp. = leg √  2 Divide both sides by √  2 . Rationalize the denominator. This is a 30°-60°-90° triangle. 22 = 2x Hyp. = 2(shorter leg) = x 15 _ √  2 15 √  2 _ 2 = x 11 = x y = 11 √  3 Divide both sides by 2. Longer leg = (shorter leg) √  3 Find the values of the variables. Give your answers in simplest radical form. 55. 56. 57. 59. 58. 60. Find the value of each variable. Round to the nearest inch. 61. 62. Study Guide: Review 369 369 �� Find each measure. 1. KL 2. m∠WXY 3. BC 4. ̶̶̶ NQ , and ̶̶ PQ are the ̶̶̶ MQ , perpendicular bisectors of △RST. Find RS and RQ. 5. ̶̶ FG are angle ̶̶ EG and bisectors of △DEF. Find m∠GEF and the ̶̶ DF . distance from G to 6. In △XYZ, XC = 261, and ZW = 118. Find XW, BW, and BZ. 7. Find the orthocenter of △JKL with vertices J (-5, 2) , K (-5, 10) , and L (1, 4) . 8. In △GHJ at right, find PR, GJ, and m∠GRP. 9. Write an indirect proof that two obtuse angles cannot form a linear pair. 10. Write the angles of △BEH in order from smallest to largest. 11. Write the sides of △RTY in order from shortest to longest. 12. The distance from Arville to Branton is 114 miles. The distance from Branton to Camford is 247 miles. If the three towns form a triangle, what is the range of distances from Arville to Camford? 13. Compare m∠SPV and m∠ZPV. 14. Find the range of values for x. 15. Find the missing side length in the triangle. Tell
if the side lengths form a Pythagorean triple. Explain. 16. Tell if the measures 18, 20, and 27 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 17. An IMAX screen is 62 feet tall and 82 feet wide. What is the length of the screen’s diagonal? Round to the nearest inch. Find the values of the variables. Give your answers in simplest radical form. 18. 19. 20. 370 370 Chapter 5 Properties and Attributes of Triangles �� FOCUS ON SAT MATHEMATICS SUBJECT TESTS Some questions on the SAT Mathematics Subject Tests require the use of a calculator. You can take the test without one, but it is not recommended. The calculator you use must meet certain criteria. For example, calculators that make noise or have typewriter-like keypads are not allowed. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. If you have both a scientific and a graphing calculator, bring the graphing calculator to the test. Make sure you spend time getting used to a new calculator before the day of the test. 1. In △ABC, m∠C = 2m∠A, and CB = 3 units. What is AB to the nearest hundredth unit? 3. The side lengths of a right triangle are 2, 5, and c, where c > 5. What is the value of c? (A) 1.73 units (B) 4.24 units (C) 5.20 units (D) 8.49 units (E) 10.39 units 2. What is the perimeter of △ABC if D is the ̶̶ AB , E is the midpoint of ̶̶ BC , and midpoint of F is the midpoint of ̶̶ AC ? Note: Figure not drawn to scale. (A) 8 centimeters (B) 14 centimeters (C) 20 centimeters (D) 28 centimeters (E) 35 centimeters (A) √  21 (B) √  29 (C) 7 (D) 9 (E) √  145 4. In the triangle below, which of the following CANNOT be the length of the unknown side? (A) 2.2 (B) 6 (C) 12.8 (D) 17.2 (E) 18.1 5. Which of the following points is on the perpendicular bisector of the segment with endpoints (3, 4) and (9, 4) ? (A) (4, 2) (B) (4, 5) (C) (5, 4) (D) (6, -1) (E) (7, 4) College Entrance Exam Practice 371 371 �� Any Question Type: Check with a Different Method It is important to check all of your answers on a test. An effective way to do this is to use a different method to answer the question a second time. If you get the same answer with two different methods, then your answer is probably correct. Multiple Choice What are the coordinates of the centroid of △ABC with A (-2, 4) , B (4, 6) , and C (1, -1) ? (1, 5) (2.5, 2.5) (1, 3) (3, 1) Method 1: The centroid of a triangle is the point of concurrency of the medians. Write the equations of two medians and find their point of intersection. Let D be the midpoint of ̶̶ AB and let E be the midpoint of ̶̶ BC . -1,5) The median from C to D contains C (1, -1) and D (1, 5) . 6 + (-12.5, 2.5) It is vertical, so its equation is x = 1. The median from A to E contains A (-2, 4) and E (2.5, 2.5) . slope of ̶̶ AE = 4 - 2.5 _ -2 - 2.5 = 1.5 _ -4.x + 2) 3 Point-slope form Substitute 4 for y 1 , - 1 _ 3 and -2 for x 1 . for m, ⎧ x = 1 Solve the system ⎨ y - 4 = - 1 __ (x + 2) ⎩ 3 to find the point of intersection1 + 2) Substitute 1 for x. y = 3 Simplify. The coordinates of the centroid are (1, 3) . So choice H is the correct answer. Method 2: To check this answer, use a different method. By the Centroid Theorem, the centroid of a triangle is 2 __ 3 of the distance from each vertex to the midpoint of the ̶̶ CD is vertical with a length of 6 units. 2 __ (6) = 4, opposite side. 3 and the coordinates of the point that is 4 units up from C is (1, 3) . This method confirms that choice H is the correct answer. Problem Solving Strategies • Draw a Diagram • Make a Model • Guess and Test • Work Backward • Find a Pattern • Make a Table • Solve a Simpler Problem • Use Logical Reasoning • Use a Venn Diagram • Make an Organized List 372 372 Chapter 5 Properties and Attributes of Triangles �� If you can’t think of a different method to use to check your answer, circle the question and come back to it later. Item C Gridded Response Find the area of the square in square centimeters. Read each test item and answer the questions that follow. Item A Multiple Choice Given that ℓ is the perpendicular bisector of and BC = 6n - 11, what is the value of n? ̶̶ AB , AC = 3n + 1, 5. How can you use special right triangles to answer this question? 6. Explain how you can check your answer by using the Pythagorean Theorem. -4 3 _ 4 4 _ 3 4 1. How can you use the given answer choices to solve this problem? 2. Describe how to solve this problem directly. Item B Multiple Choice Which number forms a Pythagorean triple with 15 and 17? 5 7 8 10 3. How can you use the given answer choices to find the answer? 4. Describe a different method you can use to check your answer. Item D Multiple Choice Which coordinates for point Z form a right triangle with the points X (-8, 4) and Y (0, -2) ? Z (4, 4) Z (4, 6) Z (3, 2) Z (8, 4) 7. Explain how to use slope to determine if △XYZ is a right triangle. 8. How can you use the Converse of the Pythagorean Theorem to check your answer? Item E Multiple Choice What are the coordinates of the orthocenter of △RST? (0, 2) (0, 1) (-1, 3) (1, 2) 9. Describe how you would solve this problem directly. 10. How can you use the third altitude of the triangle to confirm that your answer is correct? TAKS Tackler 373 373 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–5 Multiple Choice 1. ̶̶ GJ is a midsegment of △DEF, and midsegment of △GFJ. What is the length of ̶̶ HK is a ̶̶ HK ? 2.25 centimeters 4 centimeters 7.5 centimeters 9 centimeters 2. In △RST, SR < ST, and RT > ST. If m∠R = (2x + 10) ° and m∠T = (3x - 25) °, which is a possible value of x? 25 30 35 40 3. The vertex angle of an isosceles triangle measures (7a - 2) °, and one of the base angles measures (4a + 1) °. Which term best describes this triangle? Acute Equiangular Right Obtuse 4. The lengths of two sides of an acute triangle are 8 inches and 10 inches. Which of the following could be the length of the third side? 5 inches 6 inches 12 inches 13 inches 5. For the coordinates M (-1, 0) , N (-2, 2) , P (10, y) , ̶̶̶ MN ǁ ̶̶ PQ . What is the value of y? and Q (4, 6) , -18 -6 6 18 6. What is the area of an equilateral triangle that has a perimeter of 18 centimeters? 9 square centimeters 9 √  3 square centimeters 18 square centimeters 18 √  3 square centimeters 7. In △ABC and △DEF, ̶̶ AC ≅ ̶̶ DE , and ∠A ≅ ∠E. Which of the following would allow you to conclude by SAS that these triangles are congruent? ̶̶ AB ≅ ̶̶ AC ≅ ̶̶ BA ≅ ̶̶ CB ≅ ̶̶ DF ̶̶ EF ̶̶ FE ̶̶ DF 8. For the segment below, AB = 1 __ AC, and CD = 2BC. 2 Which expression is equal to the length of ̶̶̶ AD ? 2AB + BC 2AC + AB 3AB 4BC 9. In △DEF, m∠D = 2 (m∠E + m∠F) . Which term best describes △DEF? Acute Equiangular Right Obtuse 10. Which point of concurrency is always located inside the triangle? The centroid of an obtuse triangle The circumcenter of an obtuse triangle The circumcenter of a right triangle The orthocenter of a right triangle 374 374 Chapter 5 Properties and Attributes of Triangles �� If a diagram is not provided, draw your own. Use the given information to label the diagram. 11. The length of one leg of a right triangle is 3 times the length of the other, and the length of the hypotenuse is 10. What is the length of the longest leg? 3 3 √  10 √  10 12 √  5 12. Which statement is true by the Transitive Property of Congruence? If ∠A ≅ ∠T, then ∠T ≅ ∠A. If m∠L = m∠S, then ∠L ≅ ∠S. 5QR + 10 = 5 (QR + 2) ̶̶ ̶̶ ̶̶ EF , then DE ≅ DE and If ̶̶ BD ≅ ̶̶ BD ≅ ̶̶ EF . Gridded Response 13. P is the incenter of △JKL. The distance from P ̶̶ KL is 2y - 9. What is the distance from P to to ̶̶ JK ? STANDARDIZED TEST PREP Short Response 17. In △RST, S is on the perpendicular bisector of m∠S = (4n + 16) °, and m∠R = (3n - 18) °. Find m∠R. Show your work and explain how you determined your answer. ̶̶ RT , 18. Given that ̶̶ BD ǁ ̶̶ AC and ̶̶ AB ≅ ̶̶ BD , explain why AC < DC. 19. Write an indirect proof that an acute triangle cannot contain a pair of complementary angles. Given: △XYZ is an acute triangle. Prove: △XYZ does not contain a pair of complementary angles. 20. Find the coordinates of the orthocenter of △JKL. Show your work and explain how you found your answer. 14. In a plane, r ǁ s, and s ⊥ t. How many right angles are formed by the lines r, s, and t? 15. What is the measure, in degrees, of ∠H? 16. The point T is in the interior of ∠XYZ. If m∠XYZ = (25x + 10) °, m∠XYT = 90°, and m∠TYZ = (9x) °, what is the value of x? Extended Response 21. Consider the statement “If a triangle is equiangular, then it is acute.” a. Write the converse, inverse, and contrapositive of this conditional statement. b. Write a biconditional statement from the conditional statement. c. Determine the truth value of the biconditional statement. If it is false, give a counterexample. d. Determine the truth value of each statement below. Give an example or counterexample to justify your reasoning. “For any conditional, if the inverse and contrapositive are true, then the biconditional is true.” “For any conditional, if the inverse and converse are true, then the biconditional is true.” Cumulative Assessment, Chapters 1–5 375 375 �� Polygons and Quadrilaterals 6A Polygons and Parallelograms Lab Construct Regular Polygons 6-1 Properties and Attributes of Polygons Lab Explore Properties of Parallelograms 6-2 Properties of Parallelograms 6-3 Conditions for Parallelograms 6B Other Special Quadrilaterals 6-4 Properties of Special Parallelograms Lab Predict Conditions for Special Parallelograms 6-5 Conditions for Special Parallelograms Lab Explore Isosceles Tra
pezoids 6-6 Properties of Kites and Trapezoids KEYWORD: MG7 ChProj This tile mosaic showing the Alamo and surrounding buildings is on the Riverwalk in San Antonio. 376 376 Chapter 6 Vocabulary Match each term on the left with a definition on the right. 1. exterior angle A. lines that intersect to form right angles 2. parallel lines B. lines in the same plane that do not intersect 3. perpendicular lines C. two angles of a polygon that share a side 4. polygon 5. quadrilateral D. a closed plane figure formed by three or more segments that intersect only at their endpoints E. a four-sided polygon F. an angle formed by one side of a polygon and the extension of a consecutive side Triangle Sum Theorem Find the value of x. 6. 7. 8. 9. Parallel Lines and Transversals Find the measure of each numbered angle. 10. 11. 12. Special Right Triangles Find the value of x. Give the answer in simplest radical form. 13. 15. 14. 16. Conditional Statements Tell whether the given statement is true or false. Write the converse. Tell whether the converse is true or false. 17. If two angles form a linear pair, then they are supplementary. 18. If two angles are congruent, then they are right angles. 19. If a triangle is a scalene triangle, then it is an acute triangle. Polygons and Quadrilaterals 377 377 �� Key Vocabulary/Vocabulario concave diagonal cóncavo diagonal isosceles trapezoid trapecio isósceles kite cometa parallelogram paralelogramo rectangle rectángulo regular polygon polígono regular rhombus square trapezoid rombo cuadrado trapecio Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word concave is made up of two parts: con and cave. Sketch a polygon that looks like it caves in. 2. If a triangle is isosceles, then it has two congruent legs. What do you think is a special property of an isosceles trapezoid ? 3. A parallelogram has four sides. What do you think is a special property of the sides of a parallelogram? Geometry TEKS G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships 6-1 Geo. Lab ★ 6-2 Geo. Lab Les. 6-1 Les. 6-2 Les. 6-3 Les. 6-4 6-5 Tech. Lab 6-6 Tech. Lab Les. 6-5 Les. 6-6 ★ ★ ★ ★ ★ ★ G.2.B Geometric structure* make conjectures about ... ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ polygons ... and determine the validity of the conjectures, choosing from a variety of approaches ... G.3.B Geometric structure* construct and justify statements ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ about geometric figures and their properties G.3.E Geometric structure* use deductive reasoning to prove ★ ★ ★ ★ ★ a statement G.5.B Geometric patterns* use numeric and geometric ★ ★ patterns to make generalizations about geometric properties, including properties of polygons,... and angle relationships in polygons ... G.7.A Dimensionality and the geometry of location* use one- and two-dimensional coordinate systems to represent ... figures G.7.B Dimensionality and the geometry of location* use slopes and equations of lines to investigate geometric relationships ... G.7.C Dimensionality and the geometry of location* ... use formulas involving length, slope, and midpoint ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ G.9.B Congruence and the geometry of size* formulate and ★ ★ ★ ★ test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models * Knowledge and skills are written out completely on pages TX28–TX35. 378 378 Chapter 6 Writing Strategy: Write a Convincing Argument Throughout this book, the exercises that require you to write an explanation or argument to support an idea. Your response to a Write About It exercise shows that you have a solid understanding of the mathematical concept. icon identifies To be effective, a written argument should contain • a clear statement of your mathematical claim. • evidence or reasoning that supports your claim. From Lesson 5-4 36. Write About It An isosceles triangle has two congruent sides. Does it also have two congruent midsegments? Explain. Step 1 Make a statement of your mathematical claim. Make a statement of your mathematical claim. Draw a sketch to investigate the properties of the midsegments of an isosceles triangle. You will find that the midsegments parallel to the legs of the isosceles triangle are congruent. Claim: The midsegments parallel to the legs of an isosceles triangle are congruent. Step 2 Give evidence to support your claim. Identify any properties or theorems that support your claim. In this case, the Triangle Midsegment Theorem states that the length of a midsegment of a triangle is 1 __ 2 the length of the parallel side. To clarify your argument, label your diagram and use it in your response. Step 3 Write a complete response. Write a complete response. Yes, the two midsegments parallel to the legs of an isosceles triangle ̶̶ ̶̶ are congruent. Suppose △ABC is isosceles with YZ XZ and are midsegments of △ABC. By the Triangle Midsegment Theorem, ̶̶ ̶̶ AC , AB = AC. So 1 __ 2 AB = 1 __ 2 AC XZ = 1 __ 2 AC and YZ = 1 __ 2 AB. Since AB ≅ by the Multiplication Property of Equality. By substitution, XZ = YZ, so ̶̶ AB ≅ ̶̶ XZ ≅ ̶̶ AC . ̶̶ YZ . Try This Write a convincing argument. 1. Compare the circumcenter and the incenter of a triangle. 2. If you know the side lengths of a triangle, how do you determine which angle is the largest? Polygons and Quadrilaterals 379 379 6-1 Use with Lesson 6-1 Activity 1 Construct Regular Polygons In Chapter 4, you learned that an equilateral triangle is a triangle with three congruent sides. You also learned that an equilateral triangle is equiangular, meaning that all its angles are congruent. In this lab, you will construct polygons that are both equilateral and equiangular by inscribing them in circles. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures .... Also G.2.B, G.3.B, G.3.D, G.5.B, G.9.B 1 Construct circle P. Draw a diameter ̶̶ AC . 2 Construct the perpendicular bisector of of the bisector and the circle as B and D. ̶̶ AC . Label the intersections 3 Draw ̶̶ AB , ̶̶ BC , ̶̶ CD , and ̶̶ DA . The polygon ABCD is a regular quadrilateral. This means it is a four-sided polygon that has four congruent sides and four congruent angles. Try This 1. Describe a different method for constructing a regular quadrilateral. 2. The regular quadrilateral in Activity 1 is inscribed in the circle. What is the relationship between the circle and the regular quadrilateral? 3. A regular octagon is an eight-sided polygon that has eight congruent sides and eight congruent angles. Use angle bisectors to construct a regular octagon from a regular quadrilateral. Activity 2 1 Construct circle P. Draw a point A on the circle. 2 Use the same compass setting. Starting at A, draw arcs to mark off equal parts along the circle. Label the other points where the arcs intersect the circle as B, C, D, E, and F. 3 Draw ̶̶ BC , ̶̶ CD , ̶̶ AB , ̶̶ FA . The polygon ABCDEF is a regular hexagon. This means it is a six-sided polygon that has six congruent sides and six congruent angles. ̶̶ EF , and ̶̶ DE , Try This 4. Justify the conclusion that ABCDEF is a regular hexagon. (Hint: Draw ̶̶ AD , ̶̶ BE , and ̶̶ CF . What types of triangles are formed?) diameters 5. A regular dodecagon is a 12-sided polygon that has 12 congruent sides and 12 congruent angles. Use the construction of a regular hexagon to construct a regular dodecagon. Explain your method. 380 380 Chapter 6 Polygons and Quadrilaterals �� Activity 3 1 Construct circle P. Draw a diameter ̶̶ AB . 2 Construct the perpendicular bisector of ̶̶ AB . Label one point where the bisector intersects the circle as point E. 3 Construct the midpoint of radius ̶̶ PB . Label it as point C. 4 Set your compass to the length CE. Place the compass point at C and draw an arc that intersects ̶̶ AB . Label the point of intersection D. 5 Set the compass to the length ED. Starting at E, draw arcs to mark off equal parts along the circle. Label the other points where the arcs intersect the circle as F, G, H, and J. 6 Draw ̶̶ EF , ̶̶ FG , ̶̶̶ GH , ̶̶ JE . The polygon EFGHJ is a regular pentagon. This means it is a five-sided polygon that has five congruent sides and five congruent angles. ̶̶ HJ , and Steps 1–3 Step 4 Step 5 Step 6 Try This 6. A regular decagon is a ten-sided polygon that has ten congruent sides and ten congruent angles. Use the construction of a regular pentagon to construct a regular decagon. Explain your method. 7. Measure each angle of the regular polygons in Activities 1–3 and complete the following table. REGULAR POLYGONS Number of Sides Measure of Each Angle Sum of Angle Measures 3 60° 180° 4 5 6 8. Make a Conjecture What is a general rule for finding the sum of the angle measures in a regular polygon with n sides? 9. Make a Conjecture What is a general rule for finding the measure of each angle in a regular polygon with n sides? 6-1 Geometry Lab 381 381 �� 6-1 Properties and Attributes of Polygons TEKS G.5.B Geometric patterns: use … patterns to make generalizations about ... properties of ... and angle relationships in polygons .... Objectives Classify polygons based on their sides and angles. Find and use the measures of interior and exterior angles of polygons. Why learn this? The opening that lets light into a camera lens is created by an aperture, a set of blades whose edges may form a polygon. (See Example 5.) Vocabulary side of a polygon vertex of a polygon diagonal regular polygon concave convex Also G.2.B, G.3.B, G.4.A, G.5.A, G.7.A In Lesson 2-4, you learned the definition of a polygon. Now you will learn about the parts of a polygon and about way
s to classify polygons. Each segment that forms a polygon is a side of the polygon . The common endpoint of two sides is a vertex of the polygon . A segment that connects any two nonconsecutive vertices is a diagonal . You can name a polygon by the number of its sides. The table shows the names of some common polygons. Polygon ABCDE is a pentagon. Number of Sides Name of Polygon 3 4 5 6 7 8 9 10 12 n Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon E X A M P L E 1 Identifying Polygons Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. A B C polygon, pentagon not a polygon polygon, octagon Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1a. 1b. 1c. All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. 382 382 Chapter 6 Polygons and Quadrilaterals �� A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex . A regular polygon is always convex. E X A M P L E 2 Classifying Polygons Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. A B C irregular, convex regular, convex irregular, concave Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. 2a. 2b. To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon. By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Polygon Number of Sides Number of Triangles Triangle Quadrilateral Pentagon Hexagon n-gon Sum of Interior Angle Measures (1) 180° = 180° (2) 180° = 360° (3) 180° = 540° (4) 180° = 720° (n - 2) 180° In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles is (n - 2) 180°. Theorem 6-1-1 Polygon Angle Sum Theorem The sum of the interior angle measures of a convex polygon with n sides is (n - 2) 180°. 6- 1 Properties and Attributes of Polygons 383 383 �� E X A M P L E 3 Finding Interior Angle Measures and Sums in Polygons A Find the sum of the interior angle measures of a convex octagon. (n - 2) 180° (8 - 2) 180° 1080° Polygon ∠ Sum Thm. An octagon has 8 sides, so substitute 8 for n. Simplify. B Find the measure of each interior angle of a regular nonagon. Step 1 Find the sum of the interior angle measures. (n - 2) 180° (9 - 2) 180° = 1260° Polygon ∠ Sum Thm. Substitute 9 for n and simplify. Step 2 Find the measure of one interior angle. 1260° _ 9 = 140° The int.  are ≅, so divide by 9. C Find the measure of each interior angle of quadrilateral PQRS. (4 - 2) 180° = 360° Polygon ∠ Sum Thm. Polygon ∠ Sum Thm. m∠P + m∠Q + m∠R + m∠S = 360° c + 3c + c + 3c = 360 8c = 360 c = 45 Substitute. Combine like terms. Divide both sides by 8. m∠P = m∠R = 45° m∠Q = m∠S = 3 (45°) = 135° 3a. Find the sum of the interior angle measures of a convex 15-gon. 3b. Find the measure of each interior angle of a regular decagon. In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°. An exterior angle is formed by one side of a polygon and the extension of a consecutive side. Theorem 6-1-2 Polygon Exterior Angle Sum Theorem The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360°. E X A M P L E 4 Finding Exterior Angle Measures in Polygons A Find the measure of each exterior angle of a regular hexagon. A hexagon has 6 sides and 6 vertices. sum of ext.  = 360° measure of one ext. ∠ = 360° _ = 60° 6 Polygon Ext. ∠ Sum Thm. A regular hexagon has 6 ≅ ext. , so divide the sum by 6. The measure of each exterior angle of a regular hexagon is 60°. 384 384 Chapter 6 Polygons and Quadrilaterals �� B Find the value of a in polygon RSTUV. 7a° + 2a° + 3a° + 6a° + 2a° = 360° Polygon Ext. ∠ Sum Thm. 20a = 360 a = 18 Combine like terms. Divide both sides by 20. 4a. Find the measure of each exterior angle of a regular dodecagon. 4b. Find the value of r in polygon JKLM. E X A M P L E 5 Photography Application The aperture of the camera is formed by ten blades. The blades overlap to form a regular decagon. What is the measure of ∠CBD? � � � � ∠CBD is an exterior angle of a regular decagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angle measures is 360°. m∠CBD = 360° _ = 36° 10 A regular decagon has 10 ≅ ext.  , so divide the sum by 10. 5. What if…? Suppose the shutter were formed by 8 blades. What would the measure of each exterior angle be? THINK AND DISCUSS 1. Draw a concave pentagon and a convex pentagon. Explain the difference between the two figures. 2. Explain why you cannot use the expression 360° ____ n to find the measure of an exterior angle of an irregular n-gon. 3. GET ORGANIZED Copy and complete the graphic organizer. In each cell, write the formula for finding the indicated value for a regular convex polygon with n sides. 6- 1 Properties and Attributes of Polygons 385 385 �� 6-1 Exercises Exercises KEYWORD: MG7 6-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain why an equilateral polygon is not necessarily a regular polygon. 382 Tell whether each outlined shape is a polygon. If it is a polygon, name it by the number of its sides. 2. 2. 3. 3. 4. 5. 5 Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. p. 383 6. 7. 8. Find the measure of each interior angle of pentagon ABCDE. p. 384 10. Find the measure of each interior angle of a regular dodecagon. 11. Find the sum of the interior angle measures of a convex 20-gon 12. Find the value of y in polygon JKLM. �� p. 384 13. Find the measure of each exterior angle of a regular pentagon. �� . 385 Safety Use the photograph of the traffic sign for Exercises 14 and 15. 14. Name the polygon by the number of its sides. 15. In the polygon, ∠P, ∠R, and ∠T are right angles, and ∠Q ≅ ∠S. What are m∠Q and m∠S? � � � � PRACTICE AND PROBLEM SOLVING Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 16. 17. 18. Independent Practice For See Exercises Example 16–18 19–21 22–24 25–26 27–28 1 2 3 4 5 Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. TEKS TEKS TAKS TAKS 19. Skills Practice p. S14 Application Practice p. S33 20. 21. 386 386 Chapter 6 Polygons and Quadrilaterals �� 22. Find the measure of each interior angle of quadrilateral RSTV. � 23. Find the measure of each interior angle of a regular 18-gon. 24. Find the sum of the interior angle measures of a convex heptagon. 25. Find the measure of each exterior angle of a regular nonagon. �� 26. A pentagon has exterior angle measures of 5a°, 4a°, 10a°, 3a°, and 8a°. Find the value of a. Crafts The folds on the lid of the gift box form a regular hexagon. Find each measure. 27. m∠JKM 28. m∠MKL � � � � Algebra Find the value of x in each figure. 29. 30. 31. Find the number of sides a regular polygon must have to meet each condition. 32. Each interior angle measure equals each exterior angle measure. 33. Each interior angle measure is four times the measure of each exterior angle. 34. Each exterior angle measure is one eighth the measure of each interior angle. Name the convex polygon whose interior angle measures have each given sum. 35. 540° 36. 900° 37. 1800° 38. 2520° Multi-Step An exterior angle measure of a regular polygon is given. Find the number of its sides and the measure of each interior angle. 39. 120° 40. 72° 41. 36° 42. 24° 43. /////ERROR ANALYSIS///// Which conclusion is incorrect? Explain the error. 44. Estimation Graph the polygon formed by the points A (-2, -6) , B (-4, -1) , C (-1, 2) , D (4, 0) , and E (3, -5) . Estimate the measure of each interior angle. Make a conjecture about whether the polygon is equiangular. Now measure each interior angle with a protractor. Was your conjecture correct? 45. This problem will prepare you for the Multi-Step TAKS Prep on page 406. In this quartz crystal, m∠A = 95°, m∠B = 125°, m∠E = m∠D = 130°, and ∠C ≅ ∠F ≅ ∠G. a. Name polygon ABCDEFG by the number of sides. b. What is the sum of the interior angle measures � � of ABCDEFG? c. Find m∠F. � � � � � 6- 1 Properties and Attributes of Polygons 387 387 �� 46. The perimeter of a regular polygon is 45 inches. The length of one side is 7.5 inches. Name the polygon by the number of its sides. Draw an example of each figure. 47. a regular quadrilateral 48. an irregular concave heptagon 49. an irregular convex pentagon 50. an equilateral polygon that is not equiangular 51. Write About It Use the terms from the lesson to describe the figure as specifically as possible. 52. Critical Thinking What geometric figure does a regular polygon begin to resemble as the number of sides increases? 53. Which terms describe the figure shown? I. quadrilateral II. concave III. regular I only II only I and II I and III 54. Which statement is NOT true about a regular 16-gon? It is a convex polygon. It has 16 congruent sides. The sum of the interior angle measures is 2880°. The sum of the ex
terior angles, one at each vertex, is 360°. 55. In polygon ABCD, m∠A = 49°, m∠B = 107°, and m∠C = 2m∠D. What is m∠C? 24° 68° 102° 136° CHALLENGE AND EXTEND 56. The interior angle measures of a convex pentagon are consecutive multiples of 4. Find the measure of each interior angle. 57. Polygon PQRST is a regular pentagon. Find the values of x, y, and z. 58. Multi-Step Polygon ABCDEFGHJK is a regular decagon. ̶̶ DE are extended so that they meet at point L Sides in the exterior of the polygon. Find m∠BLD. ̶̶ AB and 59. Critical Thinking Does the Polygon Angle Sum Theorem work for concave polygons? Draw a sketch to support your answer. SPIRAL REVIEW Solve by factoring. (Previous course) 60. x 2 + 3x - 10 = 0 61. x 2 - x - 12 = 0 62. x 2 - 12x = -35 The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side. (Lesson 5-5) 63. 4, 4 64. 6, 12 65. 3, 7 Find each side length for a 30°-60°-90° triangle. (Lesson 5-8) 66. the length of the hypotenuse when the length of the shorter leg is 6 67. the length of the longer leg when the length of the hypotenuse is 10 388 388 Chapter 6 Polygons and Quadrilaterals �� Relations and Functions Algebra Many numeric relationships in geometry can be represented by algebraic relations. These relations may or may not be functions, depending on their domain and range. A relation is a set of ordered pairs. All the first coordinates in the set of ordered pairs are the domain of the relation. All the second coordinates are the range of the relation. A function is a type of relation that pairs each element in the domain with exactly one element in the range. See Skills Bank page S61 Example Give the domain and range of the relation y = 6_ x - 6 . Tell whether the relation is a function. Step 1 Make a table of values for the relation. x y -6 -0.5 0 -1 5 -6 6 Undefined 7 6 12 1 Step 2 Plot the points and connect them with smooth curves. Step 3 Identify the domain and range. Since y is undefined at x = 6, the domain of the relation is the set of all real numbers except 6. Since there is no x-value such that y = 0, the range of the relation is the set of all real numbers except 0. Step 4 Determine whether the relation is a function. From the graph, you can see that only one y-value exists for each x-value, so the relation is a function. Try This TAKS Grades 9–11 Obj. 2 Give the domain and range of each relation. Tell whether the relation is a function. 1. y = (x - 2) 180 2. y = 360 4. y = 360_ x 7. x = -2 5. x = 3y - 10 8. y = x 2 + 4 3. y = (x - 2) 180_ x 6. x 2 + y 2 = 9 9. -x + 8y = 5 On Track for TAKS 389 389 �� 6-2 Use with Lesson 6-2 Activity Explore Properties of Parallelograms In this lab you will investigate the relationships among the angles and sides of a special type of quadrilateral called a parallelogram. You will need to apply the Transitive Property of Congruence. That is, if figure A ≅ figure B and figure B ≅ figure C, then figure A ≅ figure C. TEKS G.9.B Congruence and the geometry of size: formulate and test conjectures about ... polygons ... based on explorations and concrete models. Also G.2.B, G.3.B, G.10.A 1 Use opposite sides of an index card to draw a set of parallel lines on a piece of patty paper. Then use opposite sides of a ruler to draw a second set of parallel lines that intersects the first. Label the points of intersection A, B, C, and D, in that order. Quadrilateral ABCD has two pairs of parallel sides. It is a parallelogram. 2 Place a second piece of patty paper over the first and trace ABCD. Label the points that correspond to A, B, C, and D as Q, R, S, and T, in that order. The parallelograms ABCD and QRST are congruent. Name all the pairs of congruent corresponding sides and angles. 3 Lay ABCD over QRST so that ̶̶ AB overlays ̶̶ ST . What do ̶̶ AB and you notice about their lengths? What does this tell ̶̶ you about DA ̶̶ RS . What do you notice about their lengths? overlays ̶̶ DA and What does this tell you about ̶̶ CD ? Now move ABCD so that ̶̶ BC ? 4 Lay ABCD over QRST so that ∠A overlays ∠S. What do you notice about their measures? What does this tell you about ∠A and ∠C? Now move ABCD so that ∠B overlays ∠T. What do you notice about their measures? What does this tell you about ∠B and ∠D? 5 Arrange the pieces of patty paper so that ̶̶ QR and ̶̶ RS overlays ̶̶ AB ? What does this ̶̶ AD . What do you notice about tell you about ∠A and ∠R? What can you conclude about ∠A and ∠B? 6 Draw diagonals ̶̶ AC and ̶̶ BD . Fold ABCD so that A matches C, making a crease. Unfold the paper and fold it again so that B matches D, making another crease. What do you notice about the creases? What can you conclude about the diagonals? Try This 1. Repeat the above steps with a different parallelogram. Do you get the same results? 2. Make a Conjecture How do you think the sides of a parallelogram are related to each other? the angles? the diagonals? Write your conjectures as conditional statements. 390 390 Chapter 6 Polygons and Quadrilaterals 6-2 Properties of Parallelograms TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.B, G.3.E, G.7.A, G.7.B, G.7.C, G.10.B Objectives Prove and apply properties of parallelograms. Use properties of parallelograms to solve problems. Vocabulary parallelogram Who uses this? Race car designers can use a parallelogram-shaped linkage to keep the wheels of the car vertical on uneven surfaces. (See Example 1.) Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names. Opposite sides of a quadrilateral do not share a vertex. Opposite angles do not share a side. A quadrilateral with two pairs of parallel sides is a parallelogram . To write the name of a parallelogram, you use the symbol . Parallelogram ABCD ABCD ̶̶ AB ǁ ̶̶ CD, ̶̶ BC ǁ ̶̶ DA Theorem 6-2-1 Properties of Parallelograms THEOREM HYPOTHESIS CONCLUSION If a quadrilateral is a parallelogram, then its opposite sides are congruent. ( → opp. sides ≅) ̶̶ AB ≅ ̶̶ BC ≅ ̶̶ CD ̶̶ DA PROOF PROOF Theorem 6-2-1 Given: JKLM is a parallelogram. Prove: ̶̶ KL ≅ ̶̶ JK ≅ ̶̶̶ LM , ̶̶ MJ Proof: Statements Reasons 1. JKLM is a parallelogram. ̶̶ KL ǁ ̶̶ JK ǁ ̶̶̶ LM , ̶̶ MJ 2. 3. ∠1 ≅ ∠2, ∠3 ≅ ∠4 ̶̶ JL ≅ ̶̶ JL 4. 5. △JKL ≅ △LMJ ̶̶ JK ≅ ̶̶̶ LM , ̶̶ KL ≅ ̶̶ MJ 6. 1. Given 2. Def. of  3. Alt. Int.  Thm. 4. Reflex. Prop. of ≅ 5. ASA Steps 3, 4 6. CPCTC 6- 2 Properties of Parallelograms 391 391 �� Theorems Properties of Parallelograms THEOREM HYPOTHESIS CONCLUSION 6-2-2 6-2-3 6-2-4 If a quadrilateral is a parallelogram, then its opposite angles are congruent. ( → opp.  ≅) If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. ( → cons.  supp.) If a quadrilateral is a parallelogram, then its diagonals bisect each other. ( → diags. bisect each other) ∠A ≅ ∠C ∠B ≅ ∠D m∠A + m∠B = 180° m∠B + m∠C = 180° m∠C + m∠D = 180° m∠D + m∠A = 180° ̶̶ AZ ≅ ̶̶ BZ ≅ ̶̶ CZ ̶̶ DZ You will prove Theorems 6-2-3 and 6-2-4 in Exercises 45 and 44. E X A M P L E 1 Racing Application Racing The diagram shows the parallelogram-shaped linkage that joins the frame of a race car to one wheel of the car. In PQRS, QR = 48 cm, RT = 30 cm, and m∠QPS = 73°. Find each measure. ̶̶ QR A PS ̶̶ PS ≅ PS = QR PS = 48 cm B m∠PQR  → opp. sides ≅ Def. of ≅ segs. Substitute 48 for QR. Texas Motor Speedway, located in Fort Worth, is home to both NASCAR and Indy car racing events. With seating for 150,061 spectators, it is the second-largest sporting facility in the country. m∠PQR + m∠QPS = 180°  → cons.  supp. m∠PQR + 73 = 180 m∠PQR = 107° Substitute 73 for m∠QPS. Subtract 73 from both sides. ̶̶ RT C PT ̶̶ PT ≅ PT = RT PT = 30 cm  → diags. bisect each other Def. of ≅ segs. Substitute 30 for RT. In KLMN, LM = 28 in., LN = 26 in., and m∠LKN = 74°. Find each measure. 1a. KN 1b. m∠NML 1c. LO � � � � � 392 392 Chapter 6 Polygons and Quadrilaterals ��PQRST E X A M P L E 2 Using Properties of Parallelograms to Find Measures ABCD is a parallelogram. Find each measure. A AD ̶̶ ̶̶ AD ≅ BC AD = BC 7x = 5x + 19 2x = 19 x = 9.5 AD = 7x = 7 (9.5) = 66.5  → opp. sides ≅ Def. of ≅ segs. Substitute the given values. Subtract 5x from both sides. Divide both sides by 2. B m∠B m∠A + m∠B = 180°  → cons.  supp. (10y - 1) + (6y + 5) = 180 16y + 4 = 180 16y = 176 y = 11 ⎤ ⎦ ⎡ ⎣ m∠B = (6y + 5) ° = 6 (11) + 5 ° = 71° Substitute the given values. Combine like terms. Subtract 4 from both sides. Divide both sides by 16. EFGH is a parallelogram. Find each measure. 2a. JG 2b. FH E X A M P L E 3 Parallelograms in the Coordinate Plane Three vertices of ABCD are A (1, -2) , B (-2, 3) , and D (5, -1) . Find the coordinates of vertex C. Since ABCD is a parallelogram, both pairs of opposite sides must be parallel. Step 1 Graph the given points. ̶̶ AB by counting Step 2 Find the slope of the units from A to B. The rise from -2 to 3 is 5. The run from 1 to -2 is -3. Step 3 Start at D and count the same number of units. A rise of 5 from -1 is 4. A run of -3 from 5 is 2. Label (2, 4) as vertex C. When you are drawing a figure in the coordinate plane, the name ABCD gives the order of the vertices. Step 4 Use the slope formula to verify that ̶̶ BC ǁ ̶̶ AD . slope of ̶̶ BC = 4 - 3 _ = 1 _ 4 2 - (-2) -1 - (-2) _ 5 - 1 = 1 _ 4 slope of ̶̶ AD = The coordinates of vertex C are (2, 4) . 3. Three vertices of PQRS are P (-3, -2) , Q (-1, 4) , and S (5, 0) . Find the coordinates of vertex R. 6- 2 Properties of Parallelograms 393 393 �� E X A M P L E 4 Using Properties of Parallelograms in a Proof Write a two-column proof. A Theorem 6-2-2 Given: ABCD is a parallelogram. Prove: ∠BAD ≅ ∠DCB, ∠ABC ≅ ∠CDA Proof: Statements Reasons 1. ABCD is a parallelogram. ̶̶̶ DA ≅ ̶̶ BC 2. ̶̶ AB ≅ ̶̶ BD ≅ ̶̶ CD , ̶̶ BD 3. 4. △BAD ≅ △DCB 5. ∠BAD ≅ ∠DCB ̶̶ AC ≅ ̶̶
AC 6. 7. △ABC ≅ △CDA 8. ∠ABC ≅ ∠CDA 1. Given 2.  → opp. sides ≅ 3. Reflex. Prop. of ≅ 4. SSS Steps 2, 3 5. CPCTC 6. Reflex. Prop. of ≅ 7. SSS Steps 2, 6 8. CPCTC B Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear. Prove: ∠G ≅ ∠L Proof: Statements Reasons 1. GHJN and JKLM are parallelograms. 1. Given 2. ∠HJN ≅ ∠G, ∠MJK ≅ ∠L 3. ∠HJN ≅ ∠MJK 4. ∠G ≅ ∠L 2.  → opp.  ≅ 3. Vert.  Thm. 4. Trans. Prop. of ≅ 4. Use the figure in Example 4B to write a two-column proof. Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear. Prove: ∠N ≅ ∠K THINK AND DISCUSS 1. The measure of one angle of a parallelogram is 71°. What are the measures of the other angles? 2. In VWXY, VW = 21, and WY = 36. Find as many other measures as you can. Justify your answers. 3. GET ORGANIZED Copy and complete the graphic organizer. In each cell, draw a figure with markings that represents the given property. 394 394 Chapter 6 Polygons and Quadrilaterals �� 6-2 Exercises Exercises KEYWORD: MG7 6-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Explain why the figure at right is NOT a parallelogram. 2. Draw PQRS. Name the opposite sides and opposite angles. 392 Safety The handrail is made from congruent parallelograms. In ABCD, AB = 17.5, DE = 18, and m∠BCD = 110°. Find each measure. 3. BD 5. BE 7. m∠ADC 4. CD 6. m∠ABC 8. m∠DAB � � � � � JKLM is a parallelogram. Find each measure. p. 393 9. JK 11. m∠L 10. LM 12. m∠ 13. Multi-Step Three vertices of DFGH are D (-9, 4) , F (-1, 5) , and G (2, 0) . p. 393 Find the coordinates of vertex H 14. Write a two-column proof. p. 394 Given: PSTV is a parallelogram. Prove: ∠STV ≅ ∠R ̶̶ PQ ≅ ̶̶ RQ Independent Practice Shipping Cranes can be used to load PRACTICE AND PROBLEM SOLVING � For See Exercises Example 15–20 21–24 25 26 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S14 Application Practice p. S33 cargo onto ships. In JKLM, JL = 165.8, JK = 110, and m∠JML = 50°. Find the measure of each part of the crane. 15. JN 17. LN 16. LM 18. m∠JKL 19. m∠KLM 20. m∠MJK � � � � WXYZ is a parallelogram. Find each measure. 21. WV 23. XZ 22. YW 24. ZV 25. Multi-Step Three vertices of PRTV are P (-4, -4) , R (-10, 0) , and V (5, -1) . Find the coordinates of vertex T. 26. Write a two-column proof. Given: ABCD and AFGH are parallelograms. Prove: ∠C ≅ ∠G 6- 2 Properties of Parallelograms 395 395 �� Algebra The perimeter of PQRS is 84. Find the length of each side of PQRS under the given conditions. 28. QR = 3 (RS) 27. PQ = QR 31. Cars To repair a large truck, a mechanic might use a parallelogram lift. In the lift, ̶̶ ̶̶ ̶̶ HJ . KJ , and FG ≅ a. Which angles are congruent to ∠1? ̶̶̶ GH ≅ ̶̶ GK ≅ ̶̶ LK ≅ ̶̶ FL ≅ 29. RS = SP - 7 Justify your answer. b. What is the relationship between ∠1 and each of the remaining labeled angles? Justify your answer. � � � � 30. SP = RS 2 � � � � � � � � � � Complete each statement about KMPR. Justify your answer. 32. ∠MPR ≅ ̶̶ PR ≅ 35. ? ̶̶̶̶ ? 33. ∠PRK ≅ ̶̶̶̶ ? 34. ̶̶̶̶ 36. ̶̶̶ MP ǁ ? ̶̶̶̶ ̶̶̶ MT ≅ ̶̶̶ MK ǁ ? ̶̶̶̶ ? ̶̶̶̶ 37. 38. ∠MPK ≅ ? 39. ∠MTK ≅ ̶̶̶̶ ? 40. m∠MKR + m∠PRK = ̶̶̶̶ ? ̶̶̶̶ Find the values of x, y, and z in each parallelogram. 41. 42. 43. 44. Complete the paragraph proof of Theorem 6-2-4 by filling in the blanks. Given: ABCD is a parallelogram. ̶̶ BD bisect each other at E. ̶̶ AC and Prove: Proof: It is given that ABCD is a parallelogram. By the definition of a parallelogram, ̶̶ AB ǁ a. ∠3 ≅ c. by e. bisect each other at E by the definition of g. ? . By the Alternate Interior Angles Theorem, ∠1 ≅ b. ̶̶̶̶ ? . ̶̶̶̶ ? , and ̶̶̶̶ ̶̶ CD because d. ? . This means that △ABE ≅ △CDE ̶̶̶̶ ̶̶ ̶̶ DE . Therefore CE , and ? , ̶̶̶̶ ̶̶ AB ≅ ? . So by f. ̶̶̶̶ ̶̶ AC and ̶̶ BE ≅ ̶̶ AE ≅ ̶̶ BD ? . ̶̶̶̶ 45. Write a two-column proof of Theorem 6-2-3: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Algebra Find the values of x and y in each parallelogram. 46. 47. 48. This problem will prepare you for the Multi-Step TAKS Prep on page 406. In this calcite crystal, the face ABCD is a parallelogram. a. In ABCD, m∠B = (6x + 12) °, and m∠D = (9x - 33) °. Find m∠B. b. Find m∠A and m∠C. Which theorem or theorems did you use to find these angle measures? � � � � 396 396 Chapter 6 Polygons and Quadrilaterals �� 49. Critical Thinking Draw any parallelogram. Draw a second parallelogram whose corresponding sides are congruent to the sides of the first parallelogram but whose corresponding angles are not congruent to the angles of the first. a. Is there an SSSS congruence postulate for parallelograms? Explain. b. Remember the meaning of triangle rigidity. Is a parallelogram rigid? Explain. 50. Write About It Explain why every parallelogram is a quadrilateral but every quadrilateral is not necessarily a parallelogram. 51. What is the value of x in PQRS? 15 20 30 70 52. The diagonals of JKLM intersect at Z. Which statement is true? JL = 1 _ 2 JL = 1 _ 2 JL = KM KM JZ JL = 2JZ 53. Gridded Response In ABCD, BC = 8.2, and CD = 5. What is the perimeter of ABCD? CHALLENGE AND EXTEND The coordinates of three vertices of a parallelogram are given. Give the coordinates for all possible locations of the fourth vertex. 54. (0, 5) , (4, 0) , (8, 5) 55. (-2, 1) , (3, -1) , (-1, -4) 56. The feathers on an arrow form two congruent parallelograms that share a common side. Each parallelogram is the reflection of the other across the line they share. Show that y = 2x. 57. Prove that the bisectors of two consecutive angles of a parallelogram are perpendicular. SPIRAL REVIEW Describe the correlation shown in each scatter plot as positive, negative, or no correlation. (Previous course) 58. 59. Classify each angle pair. (Lesson 3-1) 60. ∠2 and ∠7 61. ∠5 and ∠4 62. ∠6 and ∠7 63. ∠1 and ∠3 An interior angle measure of a regular polygon is given. Find the number of sides and the measure of each exterior angle. (Lesson 6-1) 64. 120° 65. 135° 66. 156° 6- 2 Properties of Parallelograms 397 397 �� 6-3 Conditions for Parallelograms TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.3.E, G.7.A, G.7.B, G.7.C Objective Prove that a given quadrilateral is a parallelogram. Who uses this? A bird watcher can use a parallelogram mount to adjust the height of a pair of binoculars without changing the viewing angle. (See Example 4.) You have learned to identify the properties of a parallelogram. Now you will be given the properties of a quadrilateral and will have to tell if the quadrilateral is a parallelogram. To do this, you can use the definition of a parallelogram or the conditions below. Theorems Conditions for Parallelograms THEOREM EXAMPLE In the converse of a theorem, the hypothesis and conclusion are exchanged. 6-3-1 6-3-2 6-3-3 If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. (quad. with pair of opp. sides ǁ and ≅ → ) If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. (quad. with opp. sides ≅ → ) If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. (quad. with opp.  ≅ → ) You will prove Theorems 6-3-2 and 6-3-3 in Exercises 26 and 29. PROOF PROOF Theorem 6-3-1 ̶̶ KL ǁ ̶̶ MJ , ̶̶ KL ≅ ̶̶ MJ Given: Prove: JKLM is a parallelogram. Proof: ̶̶ KL ≅ ̶̶ MJ . Since ̶̶ KL ǁ ̶̶ MJ , ∠1 ≅ ∠2 by the It is given that Alternate Interior Angles Theorem. By the Reflexive Property ̶̶ ̶̶ JL ≅ JL . So △JKL ≅ △LMJ by SAS. By CPCTC, of Congruence, ̶̶̶ ̶̶ ∠3 ≅ ∠4, and LM by the Converse of the Alternate JK ǁ Interior Angles Theorem. Since the opposite sides of JKLM are parallel, JKLM is a parallelogram by definition. 398 398 Chapter 6 Polygons and Quadrilaterals �� The two theorems below can also be used to show that a given quadrilateral is a parallelogram. Theorems Conditions for Parallelograms THEOREM EXAMPLE 6-3-4 6-3-5 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. (quad. with ∠ supp. to cons.  → ) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. (quad. with diags. bisecting each other → ) You will prove Theorems 6-3-4 and 6-3-5 in Exercises 27 and 30. E X A M P L E 1 Verifying Figures are Parallelograms A Show that ABCD is a parallelogram for x = 7 and y = 4. Step 1 Find BC and DA. BC = x + 14 BC = 7 + 14 = 21 Step 2 Find AB and CD. AB = 5y - 4 AB = 5 (4) - 4 = 16 Given Substitute and simplify. DA = 3x DA = 3x = 3 (7) = 21 Given Substitute and simplify. CD = 2y + 8 CD = 2 (4) + 8 = 16 Since BC = DA and AB = CD, ABCD is a parallelogram by Theorem 6-3-2. B Show that EFGH is a parallelogram for z = 11 and w = 4.5. m∠F = (9z + 19) ° ⎤ ⎦ ⎡ ⎣ 9 (11) + 19 ° = 118° m∠F = m∠H = (11z - 3) ° ⎤ ⎦ ⎡ ⎣ 11 (11) - 3 ° = 118° m∠H = m∠G = (14w - 1) ° ⎤ ⎦ ⎡ ⎣ 14 (4.5) - 1 ° = 62° m∠G = Given Substitute 11 for z and simplify. Given Substitute 11 for z and simplify. Given Substitute 4.5 for w and simplify. Since 118° + 62° = 180°, ∠G is supplementary to both ∠F and ∠H. EFGH is a parallelogram by Theorem 6-3-4. 1. Show that PQRS is a parallelogram for a = 2.4 and b = 9. 6- 3 Conditions for Parallelograms 399 399 �� E X A M P L E 2 Applying Conditions for Parallelograms Determine if each quadrilateral must be a parallelogram. Justify your answer. A B No. One pair of opposite sides are
parallel. A different pair of opposite sides are congruent. The conditions for a parallelogram are not met. Yes. The diagonals bisect each other. By Theorem 6-3-5, the quadrilateral is a parallelogram. Determine if each quadrilateral must be a parallelogram. Justify your answer. 2a. 2b. E X A M P L E 3 Proving Parallelograms in the Coordinate Plane Show that quadrilateral ABCD is a parallelogram by using the given definition or theorem. A A (-3, 2) , B (-2, 7) , C (2, 4) , D (1, -1) ; definition of parallelogram Find the slopes of both pairs of opposite sides. To say that a quadrilateral is a parallelogram by definition, you must show that both pairs of opposite sides are parallel. slope of ̶̶ AB = 7 - 2 _ -2 - (-3) slope of slope of ̶̶ CD = -1 - 4 _ 1 - 2 ̶̶ BC = 5 _ = 5 -1 _ = - 3 = -4 2 - (-2) 2 - (-1) _ -3 - 1 slope of ̶̶ DA = Since both pairs of opposite sides are parallel, ABCD is a parallelogram by definition. B F (-4, -2) , G (-2, 2) , H (4, 3) , J (2, -1) ; Theorem 6-3-1 Find the slopes and lengths of one pair of opposite sides. = 1 _ 6 ̶̶ GH = 3 - 2 _ slope of slope of ̶̶ JF = 4 - (-2) -2 - (-1) _ -3 - 2) 2 = √  37 4 - (-2) = -1 _ -6 = 1 _ 6 GH = √  ⎤ ⎦ ⎡ ⎣ 2 = √  37 (-4 - 2) 2 + -2 - (-1) JF = √  ̶̶̶ GH and Since GH = JF, FGHJ is a parallelogram. ̶̶ JF have the same slope, so ̶̶ JF . So by Theorem 6-3-1, ̶̶̶ GH ≅ ̶̶̶ GH ǁ ̶̶ JF . 400 400 Chapter 6 Polygons and Quadrilaterals �� 3. Use the definition of a parallelogram to show that the quadrilateral with vertices K (-3, 0) , L (-5, 7) , M (3, 5) , and N (5, -2) is a parallelogram. You have learned several ways to determine whether a quadrilateral is a parallelogram. You can use the given information about a figure to decide which condition is best to apply. Conditions for Parallelograms Both pairs of opposite sides are parallel. (definition) One pair of opposite sides are parallel and congruent. (Theorem 6-3-1) Both pairs of opposite sides are congruent. (Theorem 6-3-2) Both pairs of opposite angles are congruent. (Theorem 6-3-3) One angle is supplementary to both of its consecutive angles. (Theorem 6-3-4) The diagonals bisect each other. (Theorem 6-3-5) To show that a quadrilateral is a parallelogram, you only have to show that it satisfies one of these sets of conditions. E X A M P L E 4 Bird-Watching Application Bird-Watching The westernmost bald eagle nest in Texas is 9 miles north of Llano, where a family of bald eagles can be seen from the side of the highway during their winter nesting season. In the parallelogram mount, there are bolts at P, Q, R, and S such that PQ = RS and QR = SP. The frame PQRS moves when you raise or lower the binoculars. Why is PQRS always a parallelogram? When you move the binoculars, the angle measures change, but PQ, QR, RS, and SP stay the same. So it is always true that PQ = RS and QR = SP. Since both pairs of opposite sides of the quadrilateral are congruent, PQRS is always a parallelogram. � � � � � � 4. The frame is attached to the tripod at points A and B such that AB = RS and BR = SA. So ABRS is also a parallelogram. How does this ensure that the angle of the binoculars stays the same? THINK AND DISCUSS 1. What do all the theorems in this lesson have in common? 2. How are the theorems in this lesson different from the theorems in Lesson 6-2? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write one of the six conditions for a parallelogram. Then sketch a parallelogram and label it to show how it meets the condition. 6- 3 Conditions for Parallelograms 401 401 �� 6-3 Exercises Exercises KEYWORD: MG7 6-3 KEYWORD: MG7 Parent GUIDED PRACTICE . Show that EFGH is a parallelogram 2. Show that KLPQ is a parallelogram p. 399 for s = 5 and t = 6. for m = 14 and n = 12.5 Determine if each quadrilateral must be a parallelogram. Justify your answer. p. 400 3. 4. 5. 400 Show that the quadrilateral with the given vertices is a parallelogram. 6. W (-5, -2) , X (-3, 3) , Y (3, 5) , Z (1, 0) 7. R (-1, -5) , S (-2, -1) , T (4, -1) , U (5, -5. Navigation A parallel rule can be used p. 401 Independent Practice For See Exercises Example 9–10 11–13 14–15 16 1 2 3 4 TEKS TEKS TAKS TAKS ̶̶ BC . You place the edge of one ruler on to plot a course on a navigation chart. The tool is made of two rulers connected ̶̶ AD at hinges to two congruent crossbars and your desired course and then move the second ruler over the compass rose on the chart to read the bearing for your course. ̶̶ AB always parallel to If ̶̶ BC , why is ̶̶ AD ǁ ̶̶ CD ? � � � � PRACTICE AND PROBLEM SOLVING 9. Show that BCGH is a parallelogram 10. Show that TUVW is a parallelogram for for x = 3.2 and y = 7. for a = 19.5 and b = 22. Determine if each quadrilateral must be a parallelogram. Justify your answer. Skills Practice p. S14 Application Practice p. S33 11. 12. 13. Show that the quadrilateral with the given vertices is a parallelogram. 14. J (-1, 0) , K (-3, 7) , L (2, 6) , M (4, -1) 15. P (-8, -4) , Q (-5, 1) , R (1, -5) , S (-2, -10) 402 402 Chapter 6 Polygons and Quadrilaterals �� 16. Design The toolbox has cantilever trays � that pull away from the box so that you can reach the items beneath them. Two congruent brackets connect each tray to the box. Given that AD = BC, ̶̶ CD keep how do the brackets the tray horizontal? ̶̶ AB and � � � Determine if each quadrilateral must be a parallelogram. Justify your answer. 17. 18. 19. Algebra Find the values of a and b that would make the quadrilateral a parallelogram. 20. 22. 21. 23. 24. Critical Thinking Draw a quadrilateral that has congruent diagonals but is not a parallelogram. What can you conclude about using congruent diagonals as a condition for a parallelogram? 25. Social Studies The angles at the corners of the flag of the Republic of the Congo are right angles. The red and green triangles are congruent isosceles right triangles. Why is the shape of the yellow stripe a parallelogram? 26. Complete the two-column proof of Theorem 6-3-2 by filling in the blanks. Given: ̶̶ CD , ̶̶ DA Prove: ABCD is a parallelogram. ̶̶ AB ≅ ̶̶ BC ≅ Proof: Statements Reasons ̶̶ AB ≅ ̶̶ BD ≅ ̶̶ CD , ̶̶ BD 1. 2. ̶̶ BC ≅ ̶̶̶ DA 3. △DAB ≅ b. ? ̶̶̶̶̶ 4. ∠1 ≅ d. ̶̶ CD , ̶̶ AB ǁ ? , ∠4 ≅ e. ̶̶̶̶̶ ̶̶ ̶̶̶ DA BC ǁ 5. ? ̶̶̶̶̶ 6. ABCD is a parallelogram. 1. Given 2. a. 3. c. ? ̶̶̶̶̶ ? ̶̶̶̶̶ 4. CPCTC 5. f. 6. g. ? ̶̶̶̶̶ ? ̶̶̶̶̶ 6- 3 Conditions for Parallelograms 403 403 �� Measurement Ancient balance scales had one beam that moved on a single hinge. The stress on the hinge often made the scale imprecise. 27. Complete the paragraph proof of Theorem 6-3-4 by filling in the blanks. Given: ∠P is supplementary to ∠Q. ∠P is supplementary to ∠S. Prove: PQRS is a parallelogram. Proof: It is given that ∠P is supplementary to a. ? . ̶̶̶̶ By the Converse of the Same-Side Interior Angles Theorem, ? and b. ̶̶̶̶ ̶̶ QR ǁ c. ? and ̶̶̶̶ by the definition of e. ̶̶ PQ ǁ d. ? . ̶̶̶̶ ? . So PQRS is a parallelogram ̶̶̶̶ 28. Measurement In the eighteenth century, Gilles Personne de Roberval designed a scale with two beams and two hinges. In ABCD, ̶̶ AB , and F is the midpoint E is the midpoint of ̶̶ CD . Write a paragraph proof that AEFD and of EBCF are parallelograms. Prove each theorem. 29. Theorem 6-3-3 Given: ∠E ≅ ∠G, ∠F ≅ ∠H Prove: EFGH is a parallelogram. Plan: Show that the sum of the interior angles of EFGH is 360°. Then apply properties of equality to show that m∠E + m∠F = 180° and m∠E + m∠H = 180°. ̶̶ FG ǁ Then you can conclude that ̶̶̶ GH and ̶̶ EF ǁ ̶̶ HE . � � � � � � 30. Theorem 6-3-5 ̶̶ JL and Given: Prove: JKLM is a parallelogram. ̶̶̶ KM bisect each other. Plan: Show that △JNK ≅ △LNM and △KNL ≅ △MNJ. Then use the fact that the corresponding angles are congruent to show ̶̶ JK ǁ ̶̶̶ LM and ̶̶ KL ǁ ̶̶ MJ . 31. Prove that the figure formed by two midsegments of a triangle and their corresponding bases is a parallelogram. 32. Write About It Use the theorems from Lessons 6-2 and 6-3 to write three biconditional statements about parallelograms. 33. Construction Explain how you can construct a parallelogram based on the conditions of Theorem 6-3-1. Use your method to construct a parallelogram. 34. This problem will prepare you for the Multi-Step TAKS Prep on page 406. A geologist made the following observations while examining this amethyst crystal. Tell whether each set of observations allows the geologist to conclude that PQRS is a parallelogram. If so, explain why. ̶̶ PQ ≅ ̶̶ SR , and ̶̶ PS ǁ ̶̶ QR . a. b. ∠S and ∠R are supplementary, and ̶̶ PQ ǁ c. ∠S ≅ ∠Q, and ̶̶ SR . ̶̶ PS ≅ ̶̶ QR . � � � � 404 404 Chapter 6 Polygons and Quadrilaterals �� 35. What additional information would allow you to conclude that WXYZ is a parallelogram? ̶̶ XY ≅ ̶̶̶ WX ≅ ̶̶̶ ZW ̶̶ YZ ̶̶̶ WY ≅ ̶̶̶ WZ ∠XWY ≅ ∠ZYW 36. Which could be the coordinates of the fourth vertex of ABCD with A (-1, -1) , B (1, 3) , and C (6, 1) ? D (8, 5) D (4, -3) D (13, 3) D (3, 7) 37. Short Response The vertices of quadrilateral RSTV are R (-5, 0) , S (-1, 3) , T (5, 1) , and V (2, -2) . Is RSTV a parallelogram? Justify your answer. CHALLENGE AND EXTEND 38. Write About It As the upper platform of the movable staircase is raised and lowered, the height of each step changes. How does the upper platform remain parallel to the ground? 39. Multi-Step The diagonals of a parallelogram intersect at (-2, 1.5) . Two vertices are located at (-7, 2) and (2, 6.5) . Find the coordinates of the other two vertices. 40. Given: D is the midpoint of ̶̶ AC , and E is the midpoint of ̶̶ BC . Prove: ̶̶ DE ǁ ̶̶ AB , DE = 1 _ 2 AB (Hint: Extend ̶̶ ̶̶ EF ≅ DE to form Then show that DFBA is a parallelogram.) ̶̶ DF so that ̶̶ DE . SPIRAL REVIEW   . -5, -2, 0, 0.5 ⎬ ⎨ Compl
ete a table of values for each function. Use the domain   (Previous course) 41. f (x) = 7x - 3 42. f (x) = x + 2 _ 2 43. f (x) = 3x 2 + 2 Use SAS to explain why each pair of triangles are congruent. (Lesson 4-4) 44. △ABD ≅ △CDB 45. △TUW ≅ △VUW For JKLM, find each measure. (Lesson 6-2) 46. NM 48. JL 47. LM 49. JK 6- 3 Conditions for Parallelograms 405 405 �� SECTION 6A Polygons and Parallelograms Crystal Clear A crystal is a mineral formation that has polygonal faces. Geologists classify crystals based on the types of polygons that the faces form. 1. What type of polygon is ABCDE in the fluorite crystal? Given that m∠B = 120°, m∠E = 65°, and ∠C ≅ ∠D, find m∠A. ̶̶ AE ǁ ̶̶ CD , � � � � � � � � � 2. The pink crystals are called rhodochrosite. The face FGHJ is a parallelogram. Given that m∠F = (9x - 13) ° and m∠J = (7x + 1) °, find m∠G. Explain how you found this angle measure. 3. While studying the amazonite crystal, a geologist ̶̶̶ MN ≅ ̶̶ QP and ∠NQP ≅ ∠QNM. Can the found that geologist conclude that MNPQ is a parallelogram? Why or why not? Justify your answer. � � � � 406 406 Chapter 6 Polygons and Quadrilaterals SECTION 6A Quiz for Lessons 6-1 Through 6-3 6-1 Properties and Attributes of Polygons Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1. 2. 3. 4. 5. Find the sum of the interior angle measures of a convex 16-gon. 6. The surface of a trampoline is in the shape of a regular hexagon. Find the measure of each interior angle of the trampoline. 7. A park in the shape of quadrilateral PQRS is bordered by four sidewalks. Find the measure of each exterior angle of the park. 8. Find the measure of each exterior angle of a regular decagon. 6-2 Properties of Parallelograms A pantograph is used to copy drawings. Its legs form a parallelogram. In JKLM, LM = 17 cm, KN = 13.5 cm, and m∠KJM = 102°. Find each measure. 9. KM 10. KJ 11. MN 13. m∠JML 12. m∠JKL 15. Three vertices of ABCD are A (-3, 1) , B (5, 7) , and C (6, 2) . Find the coordinates of vertex D. 14. m∠KLM WXYZ is a parallelogram. Find each measure. 16. WX 18. m∠X 17. YZ 19. m∠W � � � � � 6-3 Conditions for Parallelograms 20. Show that RSTV is a parallelogram 21. Show that GHJK is a parallelogram for x = 6 and y = 4.5. for m = 12 and n = 9.5. Determine if each quadrilateral must be a parallelogram. Justify your answer. 22. 23. 24. 25. Show that a quadrilateral with vertices C (-9, 4) , D (-4, 8) , E (2, 6) , and F (-3, 2) is a parallelogram. Ready to Go On? 407 407 �� 6-4 Properties of Special Parallelograms TEKS G.3.B Geometric structure: construct and justify statements about geometric figures .... Also G.2.A, G.2.B, G.3.E, G.7.A, G.7.B, G.7.C Objectives Prove and apply properties of rectangles, rhombuses, and squares. Use properties of rectangles, rhombuses, and squares to solve problems. Vocabulary rectangle rhombus square Who uses this? Artists who work with stained glass can use properties of rectangles to cut materials to the correct sizes. A second type of special quadrilateral is a rectangle. A rectangle is a quadrilateral with four right angles. Theorems Properties of Rectangles THEOREM HYPOTHESIS CONCLUSION 6-4-1 If a quadrilateral is a rectangle, then it is a parallelogram. (rect. → ) 6-4-2 If a parallelogram is a rectangle, then its diagonals are congruent. (rect. → diags. ≅) ABCD is a parallelogram. ̶̶ AC ≅ ̶̶ BD You will prove Theorems 6-4-1 and 6-4-2 in Exercises 38 and 35. Since a rectangle is a parallelogram by Theorem 6-4-1, a rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2. E X A M P L E 1 Craft Application An artist connects stained glass pieces with lead strips. In this rectangular window, the strips are cut so that FG = 14 in. and FH = 20 in. Find JG. ̶̶ ̶̶ EG ≅ FH EG = FH = 20 JG = 1 _ 2 JG = 1 _ (20) = 10 in. 2 EG Rect. → diags. ≅ Def. of ≅ segs.  → diags. bisect each other Substitute and simplify. � � � � � Carpentry The rectangular gate has diagonal braces. Find each length. 1a. HJ 1b. HK 408 408 Chapter 6 Polygons and Quadrilaterals ��HGLJK48 in.30.8 in.ge07se_c06l04003aAB A rhombus is another special quadrilateral. A rhombus is a quadrilateral with four congruent sides. Theorems Properties of Rhombuses THEOREM HYPOTHESIS CONCLUSION 6-4-3 If a quadrilateral is a rhombus, then it is a parallelogram. (rhombus → ) 6-4-4 If a parallelogram is a rhombus, then its diagonals are perpendicular. (rhombus → diags. ⊥) 6-4-5 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. (rhombus → each diag. bisects opp. ) ABCD is a parallelogram. ̶̶ AC ⊥ ̶̶ BD ∠1 ≅ ∠2 ∠3 ≅ ∠4 ∠5 ≅ ∠6 ∠7 ≅ ∠8 You will prove Theorems 6-4-3 and 6-4-4 in Exercises 34 and 37. PROOF PROOF Theorem 6-4-5 Given: JKLM is a rhombus. Prove: ̶̶ JL bisects ∠KJM and ∠KLM. ̶̶̶ KM bisects ∠JKL and ∠JML. Proof: Since JKLM is a rhombus, ̶̶ JK ≅ ̶̶ JM , and ̶̶ KL ≅ ̶̶̶ ML by the definition of a rhombus. By the Reflexive Property of Congruence, Thus △JKL ≅ △JML by SSS. Then ∠1 ≅ ∠2, and ∠3 ≅ ∠4 by CPCTC. ̶̶ JL bisects ∠KJM and ∠KLM by the definition of an angle bisector. So ̶̶̶ KM bisects ∠JKL and ∠JML. By similar reasoning, ̶̶ JL ≅ ̶̶ JL . Like a rectangle, a rhombus is a parallelogram. So you can apply the properties of parallelograms to rhombuses. E X A M P L E 2 Using Properties of Rhombuses to Find Measures RSTV is a rhombus. Find each measure. A VT ST = SR 4x + 7 = 9x - 11 18 = 5x Def. of rhombus Substitute the given values. Subtract 4x from both sides and add 11 to both sides. Divide both sides by 5. Def. of rhombus 3.6 = x VT = ST VT = 4x + 7 VT = 4 (3.6) + 7 = 21.4 Substitute 4x + 7 for ST. Substitute 3.6 for x and simplify. 6- 4 Properties of Special Parallelograms 409 409 �� RSTV is a rhombus. Find each measure. B m∠WSR m∠SWT = 90° 2y + 10 = 90 y = 40 Rhombus → diags. ⊥ Substitute 2y + 10 for m∠SWT. Subtract 10 from both sides and divide both sides by 2. m∠WSR = m∠TSW m∠WSR = (y + 2) ° m∠WSR = (40 + 2) ° = 42° Rhombus → each diag. bisects opp.  Substitute y + 2 for m∠TSW. Substitute 40 for y and simplify. CDFG is a rhombus. Find each measure. 2a. CD 2b. m∠GCH if m∠GCD = (b + 3) ° and m∠CDF = (6b - 40) ° Rectangles, rhombuses, and squares are sometimes referred to as special parallelograms. A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three. E X A M P L E 3 Verifying Properties of Squares Show that the diagonals of square ABCD are congruent perpendicular bisectors of each other. Step 1 Show that ̶̶ AC and ̶̶ BD are congruent. AC = √  ⎤ ⎦ ⎡ ⎣ 2 + (7 - 0) 2 = √  58 2 - (-1) BD = √  ⎤ ⎦ ⎡ ⎣ 2 + (2 - 5) 2 = √  58 4 - (-3) ̶̶ AC ≅ ̶̶ AC and ̶̶ BD . ̶̶ BD are perpendicular. Since AC = BD, Step 2 Show that slope of 2 - (-1) _ ̶̶ 7 AC = 3 _ 7 7 ̶̶ BD = 2 - 5 _ slope of _ _ ) = -1, ) (- 3 Since ( 7 7 3 4 - (-3) ̶̶ AC ⊥ ̶̶ BD . Step 3 Show that ̶̶ AC and ̶̶ BD bisect each other. mdpt. of mdpt. of ̶̶ AC : ( BD : ( ̶̶ 0 + 7 - Since ̶̶ AC and ̶̶ BD have the same midpoint, they bisect each other. The diagonals are congruent perpendicular bisectors of each other. 3. The vertices of square STVW are S (-5, -4) , T (0, 2) , V (6, -3) , and W (1, -9) . Show that the diagonals of square STVW are congruent perpendicular bisectors of each other. 410 410 Chapter 6 Polygons and Quadrilaterals �� Special Parallelograms To remember the properties of rectangles, rhombuses, and squares, I start with a square, which has all the properties of the others. To get a rectangle that is not a square, I stretch the square in one direction. Its diagonals are still congruent, but they are no longer perpendicular. Taylor Gallinghouse Central High School To get a rhombus that is not a square, I go back to the square and slide the top in one direction. Its diagonals are still perpendicular and bisect the opposite angles, but they aren’t congruent. E X A M P L E 4 Using Properties of Special Parallelograms in Proofs Given: EFGH is a rectangle. J is the midpoint of Prove: △FJG is isosceles. Proof: ̶̶ EH . Statements Reasons 1. EFGH is a rectangle. J is the midpoint of ̶̶ EH . 2. ∠E and ∠H are right angles. 3. ∠E ≅ ∠H 4. EFGH is a parallelogram. ̶̶ EF ≅ ̶̶ EJ ≅ ̶̶̶ HG ̶̶ HJ 5. 6. 7. △FJE ≅ △GJH ̶̶ GJ ̶̶ FJ ≅ 8. 9. △FJG is isosceles. 1. Given 2. Def. of rect. 3. Rt. ∠ ≅ Thm. 4. Rect. →  5.  → opp. sides ≅ 6. Def. of mdpt. 7. SAS Steps 3, 5, 6 8. CPCTC 9. Def. of isosc. △ 4. Given: PQTS is a rhombus with diagonal ̶̶ PR . Prove: ̶̶ RQ ≅ ̶̶ RS THINK AND DISCUSS 1. Which theorem means “The diagonals of a rectangle are congruent”? Why do you think the theorem is written as a conditional? 2. What properties of a rhombus are the same as the properties of all parallelograms? What special properties does a rhombus have? 3. GET ORGANIZED Copy and complete the graphic organizer. Write the missing terms in the three unlabeled sections. Then write a definition of each term. 6- 4 Properties of Special Parallelograms 411 411 �� 6-4 Exercises Exercises KEYWORD: MG7 6-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary What is another name for an equilateral quadrilateral? an equiangular quadrilateral? a regular quadrilateral. 408 Engineering The braces of the bridge support lie along the diagonals of rectangle PQRS. RS = 160 ft, and QS = 380 ft. Find each length. 2. TQ 4. ST 3. PQ 5. PR ABCD is a rhombus. Find each measure. p. 409 6. AB 7. m∠ABC . Multi-Step The vertices of square JKLM p. 410 are J (-3, -5) , K (-4, 1) , L (2, 2) , and M (3, -4) . Show
that the diagonals of square JKLM are congruent perpendicular bisectors of each other. 411 Independent Practice For See Exercises Example 10–13 14–15 16 17 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S15 Application Practice p. S33 9. Given: RECT is a rectangle. Prove: △REY ≅ △TCX ̶̶ RX ≅ ̶̶ TY PRACTICE AND PROBLEM SOLVING Carpentry A carpenter measures the diagonals of a piece of wood. In rectangle JKLM, JM = 25 in., and JP = 14 1 __ in. Find each length. 2 10. JL 11. KL 12. KM 13. MP VWXY is a rhombus. Find each measure. 14. VW 15. m∠VWX and m∠WYX if m∠WVY = (4b + 10) ° and m∠XZW = (10b - 5) ° 16. Multi-Step The vertices of square PQRS are P (-4, 0) , Q (4, 3) , R (7, -5) , and S (-1, -8) . Show that the diagonals of square PQRS are congruent perpendicular bisectors of each other. 17. Given: RHMB is a rhombus with diagonal ̶̶ HB . Prove: ∠HMX ≅ ∠HRX Find the measures of the numbered angles in each rectangle. 18. 19. 20. 412 412 Chapter 6 Polygons and Quadrilaterals �� Find the measures of the numbered angles in each rhombus. 21. 22. 23. Tell whether each statement is sometimes, always, or never true. (Hint: Refer to your graphic organizer for this lesson.) 24. A rectangle is a parallelogram. 25. A rhombus is a square. 26. A parallelogram is a rhombus. 27. A rhombus is a rectangle. 28. A square is a rhombus. 29. A rectangle is a quadrilateral. 30. A square is a rectangle. 31. A rectangle is a square. 32. Critical Thinking A triangle is equilateral if and only if the triangle is equiangular. Can you make a similar statement about a quadrilateral? Explain your answer. 33. History There are five shapes of clay tiles in this tile mosaic from the ruins of Pompeii. a. Make a sketch of each shape of tile and tell whether the shape is a polygon. b. Name each polygon by its number of sides. Does each shape appear to be regular or irregular? c. Do any of the shapes appear to be special parallelograms? If so, identify them by name. d. Find the measure of each interior angle of the center polygon. 34. /////ERROR ANALYSIS///// Find and correct the error in this proof of Theorem 6-4-3. Given: JKLM is a rhombus. Prove: JKLM is a parallelogram. Proof: History Pompeii was located in what is today southern Italy. In C.E. 79, Mount Vesuvius erupted and buried Pompeii in volcanic ash. The ruins have been excavated and provide a glimpse into life in ancient Rome. It is given that JKLM is a rhombus. So by the definition of a rhombus, ̶̶ ̶̶ JK ≅ MJ . Theorem 6-2-1 states that if a quadrilateral is a parallelogram, then its opposite sides are congruent. So JKLM is a ̶̶̶ LM , and ̶̶ KL ≅ parallelogram by Theorem 6-2-1. 35. Complete the two-column proof of Theorem 6-4-2 by filling in the blanks. Given: EFGH is a rectangle. ̶̶ GE Prove: ̶̶ FH ≅ Proof: Statements Reasons 1. EFGH is a rectangle. 1. Given 2. EFGH is a parallelogram. ̶̶ EF ≅ b. ̶̶ ̶̶ EH EH ≅ 3. 4. ? ̶̶̶̶̶ 5. ∠FEH and ∠GHE are right angles. 2. a. ? ̶̶̶̶̶ 3.  → opp. sides ≅ 4. c. 5. d. ? ̶̶̶̶̶ ? ̶̶̶̶̶ 6. ∠FEH ≅ e. ? ̶̶̶̶̶ 6. Rt. ∠ ≅ Thm. 7. △FEH ≅ △GHE ̶̶ GE ̶̶ FH ≅ 8. 7. f. 8. g. ? ̶̶̶̶̶ ? ̶̶̶̶̶ 6- 4 Properties of Special Parallelograms 413 413 �� 36. This problem will prepare you for the Multi-Step TAKS Prep on page 436. The organizers of a fair plan to fence off a plot of land given by the coordinates A (2, 4) , B (4, 2) , C (-1, -3) , and D (-3, -1) . a. Find the slope of each side of quadrilateral ABCD. b. What type of quadrilateral is formed by the fences? Justify your answer. c. The organizers plan to build a straight path connecting A and C and another path connecting B and D. Explain why these two paths will have the same length. 37. Use this plan to write a proof of Theorem 6-4-4. Given: VWXY is a rhombus. ̶̶̶ WY ̶̶ VX ⊥ Prove: Plan: Use the definition of a rhombus and the properties of parallelograms to show that △WZX ≅ △YZX. Then use CPCTC to show that ∠WZX and ∠YZX are right angles. 38. Write a paragraph proof of Theorem 6-4-1. Given: ABCD is a rectangle. Prove: ABCD is a parallelogram. 39. Write a two-column proof. Given: ABCD is a rhombus. E, F, G, and H are the midpoints of the sides. Prove: EFGH is a parallelogram. Multi-Step Find the perimeter and area of each figure. Round to the nearest hundredth, if necessary. 40. 41. 42. 43. Write About It Explain why each of these conditional statements is true. a. If a quadrilateral is a square, then it is a parallelogram. b. If a quadrilateral is a square, then it is a rectangle. c. If a quadrilateral is a square, then it is a rhombus. 44. Write About It List the properties that a square “inherits” because it is (1) a parallelogram, (2) a rectangle, and (3) a rhombus. 45. Which expression represents the measure of ∠J in rhombus JKLM? x° 2x° (180 - x) ° (180 - 2x) ° 46. Short Response The diagonals of rectangle QRST intersect at point P. If QR = 1.8 cm, QP = 1.5 cm, and QT = 2.4 cm, find the perimeter of △RST. Explain how you found your answer. 414 414 Chapter 6 Polygons and Quadrilaterals �� 47. Which statement is NOT true of a rectangle? Both pairs of opposite sides are congruent and parallel. Both pairs of opposite angles are congruent and supplementary. All pairs of consecutive sides are congruent and perpendicular. All pairs of consecutive angles are congruent and supplementary. CHALLENGE AND EXTEND 48. Algebra Find the value of x in the rhombus. 49. Prove that the segment joining the midpoints of two consecutive sides of a rhombus is perpendicular to one diagonal and parallel to the other. 50. Extend the definition of a triangle midsegment to write a definition for the midsegment of a rectangle. Prove that a midsegment of a rectangle divides the rectangle into two congruent rectangles. 51. The figure is formed by joining eleven congruent squares. How many rectangles are in the figure? SPIRAL REVIEW 52. The cost c of a taxi ride is given by c = 2 + 1.8 (m - 1) , where m is the length of the trip in miles. Mr. Hatch takes a 6-mile taxi ride. How much change should he get if he pays with a $20 bill and leaves a 10% tip? (Previous course) Determine if each conditional is true. If false, give a counterexample. (Lesson 2-2) 53. If a number is divisible by -3, then it is divisible by 3. 54. If the diameter of a circle is doubled, then the area of the circle will double. Determine if each quadrilateral must be a parallelogram. Justify your answer. (Lesson 6-3) 55. 56. Construction Rhombus     ̶̶ PS . Set the compass ̶̶ PS . Place Draw to the length of the compass point at P and ̶̶ draw an arc above PS . Label a point Q on the arc. Place the compass point at Q and draw an arc to the right of Q. Place the compass point at S and draw an arc that intersects the arc drawn from Q. Label the point of intersection R. Draw ̶̶ PQ , ̶̶ QR , and ̶̶ RS . 6- 4 Properties of Special Parallelograms 415 415 �� 6-5 Predict Conditions for Special Parallelograms In this lab, you will use geometry software to predict the conditions that are sufficient to prove that a parallelogram is a rectangle, rhombus, or square. Use with Lesson 6-5 Activity 1 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.2.B, G.3.B, G.9.B KEYWORD: MG7 Lab6 1 Construct ̶̶ AB and ̶̶ AD with a common endpoint A. Construct a line through D parallel to Construct a line through B parallel to ̶̶ AB . ̶̶ AD . 2 Construct point C at the intersection of the ̶̶ BC two lines. Hide the lines and construct ̶̶ CD to complete the parallelogram. and 3 Measure the four sides and angles of the parallelogram. 4 Move A so that m∠ABC = 90°. What type of special parallelogram results? 5 Move A so that m∠ABC ≠ 90°. 6 Construct ̶̶ AC and ̶̶ BD and measure their lengths. Move A so that AC = BD. What type of special parallelogram results? Try This 1. How does the method of constructing ABCD in Steps 1 and 2 guarantee that the quadrilateral is a parallelogram? 2. Make a Conjecture What are two conditions for a rectangle? Write your conjectures as conditional statements. 416 416 Chapter 6 Polygons and Quadrilaterals Activity 2 1 Use the parallelogram you constructed in Activity 1. Move A so that AB = BC. What type of special parallelogram results? 2 Move A so that AB ≠ BC. 3 Label the intersection of the diagonals as E. Measure ∠AEB. 4 Move A so that m∠AEB = 90°. What type of special parallelogram results? 5 Move A so that m∠AEB ≠ 90°. 6 Measure ∠ABD and ∠CBD. Move A so that m∠ABD = m∠CBD. What type of special parallelogram results? Try This 3. Make a Conjecture What are three conditions for a rhombus? Write your conjectures as conditional statements. 4. Make a Conjecture A square is both a rectangle and a rhombus. What conditions do you think must hold for a parallelogram to be a square? 6- 5 Technology Lab 417 417 6-5 Conditions for Special Parallelograms TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.3.E, G.7.A, G.7.B, G.7.C Objective Prove that a given quadrilateral is a rectangle, rhombus, or square. Who uses this? Building contractors and carpenters can use the conditions for rectangles to make sure the frame for a house has the correct shape. When you are given a parallelogram with certain properties, you can use the theorems below to determine whether the parallelogram is a rectangle. Theorems Conditions for Rectangles THEOREM EXAMPLE 6-5-1 6-5-2 If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. ( with one rt. ∠ → rect.) If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. ( with diags. ≅ → rect.) ̶̶ AC ≅ ̶̶ BD You will prove Theorems 6-5-1 and 6-5-2 in Exercises 31 and 28. E X A M P L E 1 Carpentry Application ̶̶
WZ and A contractor built a wood frame for the side of a house so that ̶̶ ̶̶ XY ≅ XW ≅ tape measure, the contractor found that XZ = WY. Why must the frame be a rectangle? ̶̶ YZ . Using a Both pairs of opposite sides of WXYZ are congruent, so WXYZ is a parallelogram. Since XZ = WY, the diagonals of WXYZ are congruent. Therefore the frame is a rectangle by Theorem 6-5-2. 418 418 Chapter 6 Polygons and Quadrilaterals �� 1. A carpenter’s square can be used to test that an angle is a right angle. How could the contractor use a carpenter’s square to check that the frame is a rectangle? Below are some conditions you can use to determine whether a parallelogram is a rhombus. Theorems Conditions for Rhombuses THEOREM EXAMPLE In order to apply Theorems 6-5-1 through 6-5-5, the quadrilateral must be a parallelogram. 6-5-3 6-5-4 6-5-5 If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. ( with one pair cons. sides ≅ → rhombus) If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. ( with diags. ⊥ → rhombus) If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. ( with diag. bisecting opp.  → rhombus) You will prove Theorems 6-5-3 and 6-5-4 in Exercises 32 and 30. PROOF PROOF Theorem 6-5-5 Given: JKLM is a parallelogram. ̶̶ JL bisects ∠KJM and ∠KLM. Prove: JKLM is a rhombus. Proof: Statements Reasons 1. JKLM is a parallelogram. ̶̶ JL bisects ∠KJM and ∠KLM. 1. Given 2. ∠1 ≅ ∠2, ∠3 ≅ ∠4 ̶̶ JL ≅ ̶̶ JL 3. 4. △JKL ≅ △JML ̶̶ JK ≅ ̶̶ JM 5. 6. JKLM is a rhombus. 2. Def. of ∠ bisector 3. Reflex. Prop. of ≅ 4. ASA Steps 2, 3 5. CPCTC 6.  with one pair cons. sides ≅ → rhombus To prove that a given quadrilateral is a square, it is sufficient to show that the figure is both a rectangle and a rhombus. You will explain why this is true in Exercise 43. 6- 5 Conditions for Special Parallelograms 419 419 �� E X A M P L E 2 Applying Conditions for Special Parallelograms Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. A Given: ̶̶ AB ≅ ̶̶ AD ⊥ ̶̶ BC ≅ ̶̶ AC ⊥ ̶̶ CD , ̶̶ DC , Conclusion: ABCD is a square. Step 1 Determine if ABCD is a parallelogram. ̶̶ AD , ̶̶ BD ̶̶ AB ≅ ̶̶ CD , ̶̶ BC ≅ ̶̶ AD Given ABCD is a parallelogram. Quad. with opp. sides ≅ →  Step 2 Determine if ABCD is a rectangle. ̶̶ AD ⊥ ̶̶ DC , so ∠ADC is a right angle. Def. of ⊥ ABCD is a rectangle.  with one rt. ∠ → rect. Step 3 Determine if ABCD is a rhombus. ̶̶ AC ⊥ ̶̶ BD ABCD is a rhombus. Given  with diags. ⊥ → rhombus Step 4 Determine if ABCD is a square. Since ABCD is a rectangle and a rhombus, it has four right angles and four congruent sides. So ABCD is a square by definition. The conclusion is valid. ̶̶ BC ̶̶ AB ≅ B Given: You can also prove that a given quadrilateral is a rectangle, rhombus, or square by using the definitions of the special quadrilaterals. Conclusion: ABCD is a rhombus. The conclusion is not valid. By Theorem 6-5-3, if one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. To apply this theorem, you must first know that ABCD is a parallelogram. 2. Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: ∠ABC is a right angle. Conclusion: ABCD is a rectangle. E X A M P L E 3 Identifying Special Parallelograms in the Coordinate Plane Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. A A (0, 2) , B (3, 6) , C (8, 6) , D (5, 2) Step 1 Graph ABCD. Step 2 Determine if ABCD is a rectangle. √  (8 - 0) 2 + (6 - 2) 2 AC = = √  80 = 4 √  5 BD = √  (5 - 3) 2 + (2 - 6) 2 = √  20 = 2 √  5 Since 4 √  5 ≠ 2 √  5 , ABCD is not a rectangle. Thus ABCD is not a square. 420 420 Chapter 6 Polygons and Quadrilaterals �� Step 3 Determine if ABCD is a rhombus. slope of ̶̶ AC = 6 - 2 _ ) (-2) = -1, B E (-4, -1) , F (-3, 2) , G (3, 0) , H (2, -3) Since ( 1 _ = 1 _ 2 ̶̶ AC ⊥ 8 - 0 2 slope of ̶̶ BD = 2 - 6 _ = -2 5 - 3 ̶̶ BD . ABCD is a rhombus. Step 1 Graph EFGH. Step 2 Determine if EFGH is a rectangle. EG = √  ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ 2 2 + 0 - (-1) 3 - (-4) = √  50 = 5 √  2 FH = √  ⎤ ⎦ ⎡ ⎣ 2 + (-3 - 2) 2 2 - (-3) = √  50 = 5 √  2 Since 5 √  2 = 5 √  2 , the diagonals are congruent. EFGH is a rectangle. Step 3 Determine if EFGH is a rhombus. slope of ̶̶ EG = 0 - (-1) _ 3 - (-4) slope of Since ( 1_ ̶̶ FH = -3 - 2 _ 7)(-1) ≠ -1, 2 - (-3) ̶̶ EG ⊥/ = 1 _ 7 = -5 _ 5 ̶̶ FH . = -1 So EFGH is a not a rhombus and cannot be a square. Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 3a. K (-5, -1) , L (-2, 4) , M (3, 1) , N (0, -4) 3b. P (-4, 6) , Q (2, 5) , R (3, -1) , S (-3, 0) THINK AND DISCUSS 1. What special parallelogram is formed when the diagonals of a parallelogram are congruent? when the diagonals are perpendicular? when the diagonals are both congruent and perpendicular? 2. Draw a figure that shows why this statement is not necessarily true: If one angle of a quadrilateral is a right angle, then the quadrilateral is a rectangle. 3. A rectangle can also be defined as a parallelogram with a right angle. Explain why this definition is accurate. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, write at least three conditions for the given parallelogram. 6- 5 Conditions for Special Parallelograms 421 421 �� 6-5 Exercises Exercises GUIDED PRACTICE . Gardening A city garden club is planting a KEYWORD: MG7 6-5 KEYWORD: MG7 Parent p. 418 . 420 square garden. They drive pegs into the ground at each corner and tie strings between each pair. ̶̶̶ ZW . The pegs are spaced so that How can the garden club use the diagonal strings to verify that the garden is a square? ̶̶̶ WX ≅ ̶̶ XY ≅ ̶̶ YZ ≅ Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. 2. Given: ̶̶ AC ≅ ̶̶ BD Conclusion: ABCD is a rectangle. 3. Given: ̶̶ AB ǁ ̶̶ AB ⊥ Conclusion: ABCD is a rectangle. ̶̶ AB ≅ ̶̶ CD , ̶̶ CD , ̶̶ BC . 420 Multi-Step Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 4. P (-5, 2) , Q (4, 5) , R (6, -1) , S (-3, -4) 5. W (-6, 0) , X (1, 4) , Y (2, -4) , Z (-5, -8) Independent Practice For See Exercises Example 6 7–8 9–10 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S15 Application Practice p. S33 PRACTICE AND PROBLEM SOLVING 6. Crafts A framer uses a clamp to hold together the pieces of a picture frame. ̶̶ RS and The pieces are cut so that ̶̶ ̶̶ QR ≅ SP . The clamp is adjusted so that PZ, QZ, RZ, and SZ are all equal. Why must the frame be a rectangle? ̶̶ PQ ≅ � � � � � Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. ̶̶ FH ̶̶ FH bisect each other. ̶̶ EG and 7. Given: ̶̶ EG ⊥ Conclusion: EFGH is a rhombus. 8. Given: ̶̶ FH bisects ∠EFG and ∠EHG. Conclusion: EFGH is a rhombus. Multi-Step Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 9. A (-10, 4) , B (-2, 10) , C (4, 2) , D (-4, -4) 10. J (-9, -7) , K (-4, -2) , L (3, -3) , M (-2, -8) Tell whether each quadrilateral is a parallelogram, rectangle, rhombus, or square. Give all the names that apply. 11. 12. 13. 422 422 Chapter 6 Polygons and Quadrilaterals ge07sec06l05004aABeckmannXWYZV�� Tell whether each quadrilateral is a parallelogram, rectangle, rhombus, or square. Give all the names that apply. 14. 15. 16. 17. /////ERROR ANALYSIS///// In ABCD, ̶̶ AC ≅ ̶̶ BD . Which conclusion is incorrect? Explain the error. Give one characteristic of the diagonals of each figure that would make the conclusion valid. 18. Conclusion: JKLM is a rhombus. 19. Conclusion: PQRS is a square. The coordinates of three vertices of ABCD are given. Find the coordinates of D so that the given type of figure is formed. 20. A (4, -2) , B (-5, -2) , C (4, 4) ; rectangle 21. A (-5, 5) , B (0, 0) , C (7, 1) ; rhombus 22. A (0, 2) , B (4, -2) , C (0, -6) ; square 23. A (2, 1) , B (-1, 5) , C (-5, 2) ; square Find the value of x that makes each parallelogram the given type. 24. rectangle 25. rhombus 26. square 27. Critical Thinking The diagonals of a quadrilateral are perpendicular bisectors of each other. What is the best name for this quadrilateral? Explain your answer. 28. Complete the two-column proof of Theorem 6-5-2 by filling in the blanks. Given: EFGH is a parallelogram. ̶̶ EG ≅ ̶̶ HF Prove: EFGH is a rectangle. Proof: Statements Reasons 1. EFGH is a parallelogram. ̶̶ EG ≅ ̶̶ HF ̶̶ EF ≅ ̶̶̶ HG 2. 3. b. ? ̶̶̶̶̶ 4. △EFH ≅ △HGE 5. ∠FEH ≅ d. ? ̶̶̶̶̶ 6. ∠FEH and ∠GHE are supplementary. 7. g. ? ̶̶̶̶̶ 8. EFGH is a rectangle. 1. Given 2. a. ? ̶̶̶̶̶ 3. Reflex. Prop. of ≅ 4. c. 5. e. 6. f. ? ̶̶̶̶̶ ? ̶̶̶̶̶ ? ̶̶̶̶̶ 7. ≅  supp. → rt.  8. h. ? ̶̶̶̶̶ 6- 5 Conditions for Special Parallelograms 423 423 �� 29. This problem will prepare you for the Multi-Step TAKS Prep on page 436. A state fair takes place on a plot of land given by the coordinates A (-2, 3) , B (1, 2) , C (2, -1) , and D (-1, 0) . a. Show that the opposite sides of quadrilateral ABCD are parallel. b. A straight path connects A and C, and another path connects B and D. Use slopes to prove that these two paths are perpendicular. c. What can you conclude about ABCD? Explain your answer. 30. Complete the paragraph proof of Theorem 6-5-4 by filling in the blanks. Given: PQRS is a parallelogram. Prov
e: PQRS is a rhombus. ̶̶ PR ⊥ ̶̶ QS Proof: that ̶̶ PR ⊥ It is given that PQRS is a parallelogram. The diagonals of a ? . By the ̶̶̶̶ ? . It is given ̶̶̶̶ parallelogram bisect each other, so Reflexive Property of Congruence, ̶̶ PT ≅ a. ̶̶ QT ≅ b. ̶̶ QS , so ∠QTP and ∠QTR are right angles by the ? . Then ∠QTP ≅ ∠QTR by the d. ̶̶̶̶ definition of c. So △QTP ≅ △QTR by e. By Theorem 6-5-3, if one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a ? , and ̶̶̶̶ ̶̶ QP ≅ f. ? . ̶̶̶̶ ? , by CPCTC. ̶̶̶̶ g. ? . Therefore PQRS is rhombus. ̶̶̶̶ 31. Write a two-column proof of Theorem 6-5-1. Given: ABCD is a parallelogram. ∠A is a right angle. Prove: ABCD is a rectangle. 32. Write a paragraph proof of Theorem 6-5-3. Given: JKLM is a parallelogram. Prove: JKLM is a rhombus. ̶̶ JK ≅ ̶̶ KL 33. Algebra Four lines are represented by the equations below. m: y = -x + 7 ℓ: y = -x + 1 a. Graph the four lines in the coordinate plane. b. Classify the quadrilateral formed by the lines. c. What if…? Suppose the slopes of lines n and p change to 1. n: y = 2x + 1 p: y = 2x + 7 Reclassify the quadrilateral. 34. Write a two-column proof. Given: FHJN and GLMF are parallelograms. ̶̶ FG ≅ ̶̶ FN Prove: FGKN is a rhombus. 35. Write About It Use Theorems 6-4-2 and 6-5-2 to write a biconditional statement about rectangles. Use Theorems 6-4-4 and 6-5-4 to write a biconditional statement about rhombuses. Can you combine Theorems 6-4-5 and 6-5-5 to write a biconditional statement? Explain your answer. Construction Use the diagonals to construct each figure. Then use the theorems from this lesson to explain why your method works. 36. rectangle 37. rhombus 38. square 424 424 Chapter 6 Polygons and Quadrilaterals �� 39. In PQRS, ̶̶ PR and ̶̶ QS intersect at T. What additional information is needed to conclude that PQRS is a rectangle? ̶̶ PT ≅ ̶̶ PT ≅ ̶̶ QT ̶̶ RT ̶̶ ̶̶ PT ⊥ QT ̶̶ PT bisects ∠QPS. 40. Which of the following is the best name for figure WXYZ with vertices W (-3, 1) , X (1, 5) , Y (8, -2) , and Z (4, -6) ? Parallelogram Rectangle Rhombus Square 41. Extended Response a. Write and solve an equation to find the value of x. b. Is JKLM a parallelogram? Explain. c. Is JKLM a rectangle? Explain. d. Is JKLM a rhombus? Explain. CHALLENGE AND EXTEND ̶̶ ̶̶ 42. Given: DF , BC , ̶̶ EF , ̶̶ AB ≅ ̶̶ BC ǁ ̶̶ AC ≅ ̶̶ BE ⊥ ̶̶ DE , ̶̶ EF ̶̶ AB ⊥ ̶̶ DE ⊥ ̶̶ EF , Prove: EBCF is a rectangle. 43. Critical Thinking Consider the following statement: If a quadrilateral is a rectangle and a rhombus, then it is a square. a. Explain why the statement is true. b. If a quadrilateral is a rectangle, is it necessary to show that all four sides are congruent in order to conclude that it is a square? Explain. c. If a quadrilateral is a rhombus, is it necessary to show that all four angles are right angles in order to conclude that it is a square? Explain. 44. Cars As you turn the crank of a car jack, the platform that supports the car raises. Use the diagonals of the parallelogram to explain whether the jack forms a rectangle, rhombus, or square. SPIRAL REVIEW Sketch the graph of each function. State whether the function is linear or nonlinear. (Previous course) 45. y = -3x + 1 46. y = x 2 - 4 47. y = 3 Find the perimeter of each figure. Round to the nearest tenth. (Lesson 5-7) 48. 49. Find the value of each variable that would make the quadrilateral a parallelogram. (Lesson 6-3) 50. x 51. y 52. z 6- 5 Conditions for Special Parallelograms 425 425 �� 6-6 Explore Isosceles Trapezoids In this lab you will investigate the properties and conditions of an isosceles trapezoid. A trapezoid is a quadrilateral with one pair of parallel sides, called bases. The sides that are not parallel are called legs. In an isosceles trapezoid, the legs are congruent. Use with Lesson 6-6 Activity 1 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.2.B, G.3.B, G.9.B KEYWORD: MG7 Lab6 1 Draw ̶̶ AB and a point C not on ̶̶ AB . Construct a parallel line ℓ through C. 2 Draw point D on line ℓ. Construct ̶̶ AC and ̶̶ BD . 3 Measure AC, BD, ∠CAB, ∠ABD, ∠ACD, and ∠CDB. 4 Move D until AC = BD. What do you notice about m∠CAB and m∠ABD? What do you notice about m∠ACD and m∠CDB? 5 Move D so that AC ≠ BD. Now move D so that m∠CAB = m∠ABD. What do you notice about AC and BD? Try This 1. Make a Conjecture What is true about the base angles of an isosceles trapezoid? Write your conjecture as a conditional statement. 2. Make a Conjecture How can the base angles of a trapezoid be used to determine if the trapezoid is isosceles? Write your conjecture as a conditional statement. Activity 2 1 Construct ̶̶ AD and ̶̶ CB . 2 Measure AD and CB. 3 Move D until AC = BD. What do you notice about AD and CB? 4 Move D so that AC ≠ BD. Now move D so that AD = BC. What do you notice about AC and BD? Try This 3. Make a Conjecture What is true about the diagonals of an isosceles trapezoid? Write your conjecture as a conditional statement. 4. Make a Conjecture How can the diagonals of a trapezoid be used to determine if the trapezoid is isosceles? Write your conjecture as a conditional statement. 426 426 Chapter 6 Polygons and Quadrilaterals 6-6 Properties of Kites and Trapezoids TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.3.E, G.7.A, G.7.B, G.7.C Why learn this? The design of a simple kite flown at the beach shares the properties of the geometric figure called a kite. A kite is a quadrilateral with exactly two pairs of congruent consecutive sides. Objectives Use properties of kites to solve problems. Use properties of trapezoids to solve problems. Vocabulary kite trapezoid base of a trapezoid leg of a trapezoid base angle of a trapezoid isosceles trapezoid midsegment of a trapezoid Theorems Properties of Kites THEOREM HYPOTHESIS CONCLUSION 6-6-1 6-6-2 If a quadrilateral is a kite, then its diagonals are perpendicular. (kite → diags. ⊥) If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. (kite → one pair opp.  ≅) ̶̶ AC ⊥ ̶̶ BD ∠B ≅ ∠D ∠A ≇ ∠C You will prove Theorem 6-6-1 in Exercise 39. PROOF PROOF Theorem 6-6-2 ̶̶ JK ≅ Given: JKLM is a kite with Prove: ∠K ≅ ∠M, ∠KJM ≇ ∠KLM ̶̶ JM and ̶̶ KL ≅ ̶̶̶ ML . Proof: Step 1 Prove ∠K ≅ ∠M. ̶̶ JK ≅ ̶̶ JL ≅ It is given that of Congruence, So ∠K ≅ ∠M by CPCTC. ̶̶ KL ≅ ̶̶ ̶̶̶ ML . By the Reflexive Property JM and ̶̶ JL . This means that △JKL ≅ △JML by SSS. Step 2 Prove ∠KJM ≇ ∠KLM. If ∠KJM ≅ ∠KLM, then both pairs of opposite angles of JKLM are congruent. This would mean that JKLM is a parallelogram. But this contradicts the given fact that JKLM is a kite. Therefore ∠KJM ≇ ∠KLM. 6- 6 Properties of Kites and Trapezoids 427 427 �� E X A M P L E 1 Problem-Solving Application Alicia is using a pattern to make a kite. She has made the frame of the kite by placing wooden sticks along the diagonals. She also has cut four triangular pieces of fabric and has attached them to the frame. To finish the kite, Alicia must cover the outer edges with a cloth binding. There are 2 yards of binding in one package. What is the total amount of binding needed to cover the edges of the kite? How many packages of binding must Alicia buy? � � � �� Understand the Problem The answer has two parts. • the total length of binding Alicia needs • the number of packages of binding Alicia must buy Make a Plan The diagonals of a kite are perpendicular, so the four triangles are right triangles. Use the Pythagorean Theorem and the properties of kites to find the unknown side lengths. Add these lengths to find the perimeter of the kite. Solve PQ = = √  √  16 2 + 13 2 425 = 5 √  17 in. RQ = PQ = 5 √  17 in. √  16 2 + 22 2 740 = 2 √  PS = = √  185 in. Pyth. Thm. ̶̶ PQ ≅ ̶̶ RQ Pyth. Thm. RS = PS = 2 √  185 in. ̶̶ RS ≅ ̶̶ PS perimeter of PQRS = 5 √  17 + 5 √  17 + 2 √  185 + 2 √  185 ≈ 95.6 in. Kites The Zilker Kite Festival, held in Austin, Texas, has been an annual event since 1929. It is the longest continuously running kite festival in the country. Participants compete in categories such as highest kite, steadiest kite, strongest kite, largest kite, and most unusual kite. Alicia needs approximately 95.6 inches of binding. One package of binding contains 2 yards, or 72 inches. 95.6 _ 72 ≈ 1.3 packages of binding In order to have enough, Alicia must buy 2 packages of binding. Look Back To estimate the perimeter, change the side lengths into decimals and round. 5 √  17 ≈ 21, and 2 √  2 (21) + 2 (27) = 96. So 95.6 is a reasonable answer. 185 ≈ 27. The perimeter of the kite is approximately 1. What if...? Daryl is going to make a kite by doubling all the measures in the kite above. What is the total amount of binding needed to cover the edges of his kite? How many packages of binding must Daryl buy? 428 428 Chapter 6 Polygons and Quadrilaterals 1234 E X A M P L E 2 Using Properties of Kites In kite EFGH, m∠FEJ = 25°, and m∠FGJ = 57°. Find each measure. A m∠GFJ m∠FJG = 90° Kite → diags. ⊥ m∠GFJ + m∠FGJ = 90 m∠GFJ + 57 = 90 m∠GFJ = 33° Acute  of rt. △ are comp. Substitute 57 for m∠FGJ. Subtract 57 from both sides. B m∠JFE △FJE is also a right triangle, so m∠JFE + m∠FEJ = 90°. By substituting 25° for m∠FEJ, you find that m∠JFE = 65°. C m∠GHE ∠GHE ≅ ∠GFE m∠GHE = m∠GFE m∠GFE = m∠GFJ + m∠JFE m∠GHE = 33° + 65° = 98° Kite → one pair opp.  ≅ Def. of ≅  ∠ Add. Post. Substitute. In kite PQRS, m∠PQR = 78°, and m∠TRS = 59°. Find each measure. 2a. m∠QRT 2b. m∠QPS 2c. m∠PSR A trapezoid is a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a base . The nonparallel sides are called legs . Base angles of a trapezoid are two consecutive angles whose common side
is a base. If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid . The following theorems state the properties of an isosceles trapezoid. Theorems Isosceles Trapezoids THEOREM DIAGRAM EXAMPLE 6-6-3 6-6-4 If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. (isosc. trap. → base  ≅) If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. (trap. with pair base  ≅ → isosc. trap.) 6-6-5 A trapezoid is isosceles if and only if its diagonals are congruent. (isosc. trap. ↔ diags. ≅) Theorem 6-6-5 is a biconditional statement. So it is true both “forward” and “backward.” ∠A ≅ ∠D ∠B ≅ ∠C ABCD is isosceles. ̶̶ AC ≅ ̶̶ DB ↔ ABCD is isosceles. 6- 6 Properties of Kites and Trapezoids 429 429 �� E X A M P L E 3 Using Properties of Isosceles Trapezoids A Find m∠Y. m∠W + m∠X = 180° 117 + m∠X = 180 m∠X = 63° ∠Y ≅ ∠X m∠Y = m∠X m∠Y = 63° Same-Side Int.  Thm. Substitute 117 for m∠W. Subtract 117 from both sides. Isosc. trap. → base  ≅ Def. of ≅  Substitute 63 for m∠X. B RT = 24.1, and QP = 9.6. Find PS. ̶̶ RT ̶̶ QS ≅ QS = RT QS = 24.1 QP + PS = QS 9.6 + PS = 24.1 PS = 14.5 Isosc. trap. → diags. ≅ Def. of ≅ segs. Substitute 24.1 for RT. Seg. Add. Post. Substitute 9.6 for QP and 24.1 for QS. Subtract 9.6 from both sides. 3a. Find m∠F. 3b. JN = 10.6, and NL = 14.8. Find KM. E X A M P L E 4 Applying Conditions for Isosceles Trapezoids A Find the value of y so that EFGH is isosceles. ∠E ≅ ∠H Trap. with pair base  ≅ m∠E = m∠H 2y 2 - 25 = y 2 + 24 → isosc. trap. Def. of ≅  Substitute 2 y 2 - 25 for m∠E and y 2 + 24 for m∠H. y 2 = 49 Subtract y 2 from both sides and add 25 to both sides. y = 7 or y = -7 Find the square root of both sides. B JL = 5z + 3, and KM = 9z - 12. Find the value of z so that JKLM is isosceles. ̶̶̶ KM ̶̶ JL ≅ JL = KM 5z + 3 = 9z - 12 Diags. ≅ → isosc. trap. Def. of ≅ segs. Substitute 5z + 3 for JL and 9z - 12 for KM. 15 = 4z Subtract 5z from both sides and add 12 to both sides. 3.75 = z Divide both sides by 4. 4. Find the value of x so that PQST is isosceles. 430 430 Chapter 6 Polygons and Quadrilaterals �� The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. In Lesson 5-1, you studied the Triangle Midsegment Theorem. The Trapezoid Midsegment Theorem is similar to it. Theorem 6-6-6 Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. ̶̶ XY ǁ XY = 1 _ 2 ̶̶ BC , ̶̶ XY ǁ ̶̶ AD (BC + AD ) You will prove the Trapezoid Midsegment Theorem in Exercise 46. E X A M P L E 5 Finding Lengths Using Midsegments Find ST. MN = 1 _ (ST + RU) 2 31 = 1 _ (ST + 38 ) 2 62 = ST + 38 24 = ST Trap. Midsegment Thm. Substitute the given values. Multiply both sides by 2. Subtract 38 from both sides. 5. Find EH. THINK AND DISCUSS 1. Is it possible for the legs of a trapezoid to be parallel? Explain. 2. How is the midsegment of a trapezoid similar to a midsegment of a triangle? How is it different? 3. GET ORGANIZED Copy and complete the graphic organizer. Write the missing terms in the unlabeled sections. Then write a definition of each term. (Hint: This completes the Venn diagram you started in Lesson 6-4.) 6- 6 Properties of Kites and Trapezoids 431 431 �� 6-6 Exercises Exercises KEYWORD: MG7 6-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 428 1. In trapezoid PRSV, name the bases, the legs, and the midsegment. 2. Both a parallelogram and a kite have two pairs of congruent sides. How are the congruent sides of a kite different from the congruent sides of a parallelogram? 3. Crafts The edges of the kite-shaped glass in the sun catcher are sealed with lead strips. JH, KH, and LH are 2.75 inches, and MH is 5.5 inches. How much lead is needed to seal the edges of the sun catcher? If the craftsperson has two 3-foot lengths of lead, how many sun catchers can be sealed. 429 In kite WXYZ, m∠WXY = 104°, and m∠VYZ = 49°. Find each measure. 4. m∠VZY 5. m∠VXW 6. m∠XWZ . Find m∠A. p. 430 8. RW = 17.7, and SV = 23.3. Find TW. 430 9. Find the value of z so that EFGH is isosceles. 10. MQ = 7y - 6, and LP = 4y + 11. Find the value of y so that LMPQ is isosceles 11. Find QR. 12. Find AZ. p. 431 432 432 Chapter 6 Polygons and Quadrilaterals �� PRACTICE AND PROBLEM SOLVING 13. Design Each square section in the iron railing contains four small kites. The figure shows the dimensions of one kite. What length of iron is needed to outline one small kite? How much iron is needed to outline one complete section, including the square? In kite ABCD, m∠DAX = 32°, and m∠XDC = 64°. Find each measure. 14. m∠XDA 15. m∠ABC 16. m∠BCD Independent Practice For See Exercises Example 13 14–16 17–18 19–20 21–22 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S15 Application Practice p. S33 17. Find m∠Q. 18. SZ = 62.6, and KZ = 34. Find RJ. 19. Algebra Find the value of a so that XYZW is isosceles. Give your answer as a simplified radical. 20. Algebra GJ = 4x - 1, and FH = 9x - 15. Find the value of x so that FGHJ is isosceles. 21. Find PQ. 22. Find KR. Tell whether each statement is sometimes, always, or never true. 23. The opposite angles of a trapezoid are supplementary. 24. The opposite angles of a kite are supplementary. 25. A pair of consecutive angles in a kite are supplementary. 26. Estimation Hal is building a trapezoid-shaped frame for a flower bed. The lumber costs $1.29 per foot. Based on Hal’s sketch, estimate the cost of the lumber. (Hint: Find the angle measures in the triangle formed by the dashed line.) Find the measure of each numbered angle. 27. 30. 28. 31. 29. 32. 6- 6 Properties of Kites and Trapezoids 433 433 ��6 ft60°20 ft6 ftge07se_c06l06005aAB�� 33. This problem will prepare you for the Multi-Step TAKS Prep on page 436. The boundary of a fairground is a quadrilateral with vertices at E (-1, 3) , F (3, 4) , G (2, 0) , and H (-3, -2) . a. Use the Distance Formula to show that EFGH is a kite. b. The organizers need to know the angle measure at each vertex. Given that m∠H = 46° and m∠F = 62°, find m∠E and m∠G. Algebra Find the length of the midsegment of each trapezoid. 34. 35. 36. Mechanics The Peaucellier cell, invented in 1864, converts circular motion into linear motion. This type of linkage was supposedly used in the fans that ventilated the Houses of Parliament in London prior to the invention of electric fans. 37. Mechanics A Peaucellier cell is made of seven rods ̶̶ OA ≅ connected by joints at the labeled points. AQBP is a ̶̶ OB . As P moves along a circular rhombus, and path, Q moves along a linear path. In the position shown, m∠AQB = 72°, and m∠AOB = 28°. What are m∠PAQ, m∠OAQ, and m∠OBP? 38. Prove that one diagonal of a kite bisects a pair of opposite angles and the other diagonal. 39. Prove Theorem 6-6-1: If a quadrilateral is a kite, then its diagonals are perpendicular. Multi-Step Give the best name for a quadrilateral with the given vertices. 40. (-4, -1) , (-4, 6) , (2, 6) , (2, -4) 41. (-5, 2) , (-5, 6) , (-1, 6) , (2, -1) 42. (-2, -2) , (1, 7) , (4, 4) , (1, -5) 43. (-4, -3) , (0, 3) , (4, 3) , (8, -3) 44. Carpentry The window frame is a regular octagon. It is made from eight pieces of wood shaped like congruent isosceles trapezoids. What are m∠A, m∠B, m∠C, and m∠D? 45. Write About It Compare an isosceles trapezoid to a trapezoid that is not isosceles. What properties do the figures have in common? What properties does one have that the other does not? � � � � 46. Use coordinates to verify the Trapezoid Midsegment Theorem. a. M is the midpoint of b. N is the midpoint of ̶̶ c. Find the slopes of QR , you conclude? ̶̶ QP . What are its coordinates? ̶̶ RS . What are its coordinates? ̶̶ PS , and ̶̶̶ MN . What can d. Find QR, PS, and MN. Show that MN = 1 __ 2 (PS + QR) . 47. In trapezoid PQRS, what could be the lengths of ̶̶ QR and ̶̶ PS ? 6 and 10 6 and 26 8 and 32 10 and 24 434 434 Chapter 6 Polygons and Quadrilaterals �� 48. Which statement is never true for a kite? The diagonals are perpendicular. One pair of opposite angles are congruent. One pair of opposite sides are parallel. Two pairs of consecutive sides are congruent. 49. Gridded Response What is the length of the midsegment of trapezoid ADEB in inches? CHALLENGE AND EXTEND 50. Write a two-column proof. (Hint: If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. Use this fact to draw ̶̶̶ WZ .) auxiliary lines ̶̶ VY so that ̶̶ VY ⊥ ̶̶ UX and ̶̶ UX ⊥ ̶̶ XZ ≅ Given: WXYZ is a trapezoid with Prove: WXYZ is an isosceles trapezoid. ̶̶̶ WZ and ̶̶̶ YW . 51. The perimeter of isosceles trapezoid ABCD is 27.4 inches. If BC = 2 (AB) , find AD, AB, BC, and CD. SPIRAL REVIEW 52. An empty pool is being filled with water. After 10 hours, 20% of the pool is full. If the pool is filled at a constant rate, what fraction of the pool will be full after 25 hours? (Previous course) Write and solve an inequality for x. (Lesson 3-4) 53. 54. Tell whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. (Lesson 6-5) 55. (-3, 1) , (-1, 3) , (1, 1) , and (-1, -1) 55. 56. (1, 1) , (4, 5) , (4, 0) , and (1, -4) Construction Kite     Draw a segment ̶̶ AC . Construct line ℓ as the perpendicular bisector ̶̶ of AC . Label the intersection a
s X. Draw a point B on ℓ ̶̶ AB above ̶̶ CB . and ̶̶ AC . Draw ̶̶ AC so that Draw a point D on ℓ below DX ≠ BX. Draw and ̶̶ AD ̶̶ CD . 1. Critical Thinking How would you modify the construction above so that ABCD is a concave kite? 6- 6 Properties of Kites and Trapezoids 435 435 �� SECTION 6B Other Special Quadrilaterals A Fair Arrangement The organizers of a county fair are using a coordinate plane to plan the layout of the fairground. The fence that surrounds the fairground will have vertices at A (-1, 4) , B (7, 8) , C (3, 0) , and D (-5, -4) . 1. The organizers consider creating two straight paths through the fairground: one from point A to point C and another from point B to point D. Use a theorem from Lesson 6-4 to prove that these paths would be perpendicular. 2. The organizers instead decide to put an entry gate at the midpoint of each side of the fence, as shown. They plan to create straight paths that connect the gates. Show that the paths ̶̶ PQ , ̶̶ SP form a parallelogram. ̶̶ QR , ̶̶ RS , and ̶̶ PR and 3. Use the paths ̶̶ SQ to tell whether PQRS is a rhombus, rectangle, or square. 4. One section of the fair will contain all the rides and games. The organizers will fence off this area within the fairground by using the existing fences along ̶̶ CE , where E has coordinates (-1, 0) . What type of along quadrilateral will be formed by these four fences? ̶̶ BC and adding fences ̶̶ AB and ̶̶ AE and 5. To construct the fences, the organizers need to know the angle measures at each vertex. Given that m∠B = 37°, find the measures of the other angles in quadrilateral ABCE. 436 436 Chapter 6 Polygons and Quadrilaterals �� SECTION 6B Quiz for Lessons 6-4 Through 6-6 6-4 Properties of Special Parallelograms The flag of Jamaica is a rectangle with stripes along the diagonals. In rectangle QRST, QS = 80.5, and RS = 36. Find each length. 1. SP 2. QT 3. TR 4. TP GHJK is a rhombus. Find each measure. 5. HJ 6. m∠HJG and m∠GHJ if m∠JLH = (4b - 6) ° and m∠JKH = (2b + 11) ° 7. Given: QSTV is a rhombus. Prove: ̶̶ PQ ≅ ̶̶ RQ ̶̶ PT ≅ ̶̶ RT 6-5 Conditions for Special Parallelograms Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. ̶̶ AC ⊥ 8. Given: ̶̶ BD Conclusion: ABCD is a rhombus. 9. Given: ̶̶ AB ≅ ̶̶ AB ǁ Conclusion: ABCD is a rectangle. ̶̶ AC ≅ ̶̶ CD , ̶̶ BD , ̶̶ CD Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 10. W (-2, 2) , X (1, 5) , Y (7, -1) , Z (4, -4) ̶̶ ZX are midsegments of △TWY. 11. M (-4, 5) , N (1, 7) , P (3, 2) , Q (-2, 0) ̶̶ VX and 12. Given: ̶̶̶ TW ≅ ̶̶ TY Prove: TVXZ is a rhombus. 6-6 Properties of Kites and Trapezoids In kite EFGH, m∠FHG = 68°, and m∠FEH = 62°. Find each measure. 13. m∠FEJ 15. m∠FGJ 14. m∠EHJ 16. m∠EHG 17. Find m∠R. 18. YZ = 34.2, and VX = 53.4. Find WZ. 19. A dulcimer is a trapezoid-shaped stringed instrument. The bases are 43 in. and 23 in. long. If a string is attached at the midpoint of each leg of the trapezoid, how long is the string? Ready to Go On? 437 437 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary base of a trapezoid . . . . . . . . . . 429 kite . . . . . . . . . . . . . . . . . . . . . . . . 427 rhombus . . . . . . . . . . . . . . . . . . . 409 base angle of a trapezoid . . . . 429 leg of a trapezoid . . . . . . . . . . . 429 side of a polygon . . . . . . . . . . . . 382 concave . . . . . . . . . . . . . . . . . . . . 383 midsegment of a trapezoid . . 431 square . . . . . . . . . . . . . . . . . . . . . 410 convex . . . . . . . . . . . . . . . . . . . . . 383 parallelogram . . . . . . . . . . . . . . 391 trapezoid . . . . . . . . . . . . . . . . . . . 429 diagonal . . . . . . . . . . . . . . . . . . . 382 rectangle . . . . . . . . . . . . . . . . . . . 408 vertex of a polygon . . . . . . . . . . 382 isosceles trapezoid . . . . . . . . . . 429 regular polygon . . . . . . . . . . . . . 382 Complete the sentences below with vocabulary words from the list above. 1. The common endpoint of two sides of a polygon is a(n) ? . ̶̶̶̶ 2. A polygon is ? if no diagonal contains points in the exterior. ̶̶̶̶ 3. A(n) ? is a quadrilateral with four congruent sides. ̶̶̶̶ 4. Each of the parallel sides of a trapezoid is called a(n) ? . ̶̶̶̶ 6-1 Properties and Attributes of Polygons (pp. 382–388) E X A M P L E S EXERCISES TEKS G.2.B, G.3.B, G.4.A, G.5.A, G.5.B, G.7.A ■ Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides. The figure is a closed plane figure made of segments that intersect only at their endpoints, so it is a polygon. It has six sides, so it is a hexagon. ■ Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. The polygon is equilateral, but it is not equiangular. So it is not regular. No diagonal contains points in the exterior, so it is convex. Find each measure. ■ the sum of the interior angle measures of a convex 11-gon (n - 2) 180° (11 - 2) 180° = 1620° Polygon ∠ Sum Thm. Substitute 11 for n. ■ the measure of each exterior angle of a regular pentagon sum of ext.  = 360° Polygon Ext. ∠ SumThm. measure of one ext. ∠ = 360° _ = 72° 5 Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 5. 6. 7. Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. 8. 10. 9. Find each measure. 11. the sum of the interior angle measures of a convex dodecagon 12. the measure of each interior angle of a regular 20-gon 13. the measure of each exterior angle of a regular quadrilateral 14. the measure of each interior angle of hexagon ABCDEF 438 438 Chapter 6 Polygons and Quadrilaterals �� 6-2 Properties of Parallelograms (pp. 391–397) TEKS G.2.B, G.3.B, G.3.E, G.7.A, G.7.B, G.7.C, G.10.B E X A M P L E S ■ In PQRS, m∠RSP = 99°, PQ = 19.8, and RT = 12.3. Find PT. ̶̶ RT ̶̶ PT ≅ PT = RT PT = 12.3  → diags. bisect each other Def. of ≅ segs. Substitute 12.3 for RT. JKLM is a parallelogram. Find each measure. ■ LK ̶̶ ̶̶ LK JM ≅ JM = LK 2y - 9 = y + 7 y = 16  → opp. sides ≅ Def. of ≅ segs. Substitute the given values. Solve for y. LK = 16 + 7 = 23 ■ m∠M m∠J + m∠M = 180°  → cons.  supp. Substitute the given (x + 4) + 3x = 180 x = 44 m∠M = 3 (44) = 132° values. Solve for x. EXERCISES In ABCD, m∠ABC = 79°, BC = 62.4, and BD = 75. Find each measure. 15. BE 16. AD 17. ED 18. m∠CDA 19. m∠BCD 20. m∠DAB WXYZ is a parallelogram. Find each measure. 21. WX 22. YZ 23. m∠W 25. m∠Y 24. m∠X 26. m∠Z 27. Three vertices of RSTV are R (-8, 1) , S (2, 3) , and V (-4, -7) . Find the coordinates of vertex T. 28. Write a two-column proof. Given: GHLM is a parallelogram. ∠L ≅ ∠JMG Prove: △GJM is isosceles. 6-3 Conditions for Parallelograms (pp. 398–405) E X A M P L E S ■ Show that MNPQ is a parallelogram for a = 6 and b = 1.6. MN = 2a + 5 MN = 2 (6) + 5 = 17 QP = 4 (6) - 7 = 17 MQ = 7b MQ = 7 (1.6) = 11.2 NP = 2b + 8 NP = 2 (1.6) + 8 = 11.2 QP = 4a - 7 Since its opposite sides are congruent, MNPQ is a parallelogram. ■ Determine if the quadrilateral must be a parallelogram. Justify your answer. No. One pair of opposite angles are congruent, and one pair of consecutive sides are congruent. None of the conditions for a parallelogram are met. TEKS G.2.A, G.2.B, G.3.B, G.3.E, G.7.A, G.7.B, G.7.C EXERCISES Show that the quadrilateral is a parallelogram for the given values of the variables. 29. m = 13, n = 27 30. x = 25, y = 7 Determine if the quadrilateral must be a parallelogram. Justify your answer. 31. 32. 33. Show that the quadrilateral with vertices B (-4, 3) , D (6, 5) , F (7, -1) , and H (-3, -3) is a parallelogram. Study Guide: Review 439 439 �� 6-4 Properties of Special Parallelograms (pp. 408–415) E X A M P L E S In rectangle JKLM, KM = 52.8, and JM = 45.6. Find each length. ■ KL JKLM is a . KL = JM = 45.6 Rect. →   → opp. sides ≅ ■ NL JL = KM = 52.8 NL = 1_ 2 JL = 26.4 Rect. → diags. ≅  → diags. bisect each other ■ PQRS is a rhombus. Find m∠QPR, given that m∠QTR = (6y + 6) ° and m∠SPR = 3y°. m∠QTR = 90° 6y + 6 = 90 y = 14 m∠QPR = m∠SPR m∠QPR = 3 (14) ° = 42° Rhombus → diags. ⊥ Substitute the given value. Solve for y. Rhombus → each diag. bisects opp.  ■ The vertices of square ABCD are A (5, 0) , B (2, 4) , C (-2, 1) , and D (1, -3) . Show that the diagonals of square ABCD are congruent perpendicular bisectors of each other. AC = BD = 5 √  2 ̶̶ AC = - 1_ slope of 7 ̶̶ BD = 7 ̶̶ AC Product of slopes is -1, so diags. are ⊥. Diags. are ≅. slope of mdpt. of = mdpt. of ̶̶ BD = ( 3 _ ) , 1 _ 2 2 Diags. bisect each other. TEKS G.2.A, G.2.B, G.3.B, G.3.E, G.7.A, G.7.B, G.7.C EXERCISES In rectangle ABCD, CD = 18, and CE = 19.8. Find each length. 34. AB 35. AC 36. BD 37. BE In rhombus WXYZ, WX = 7a + 1, WZ = 9a - 6, and VZ = 3a. Find each measure. 38. WZ 39. XV 40. XY 41. XZ In rhombus RSTV, m∠TZV = (8n + 18) °, and m∠SRV = (9n + 1) °. Find each measure. 42. m∠TRS 43. m∠RSV 44. m∠STV 45. m∠TVR Find the measures of the numbered angles in each figure. 46. rectangle MNPQ 47. rhombus CDGH Show that the diagonals of the square with the given vertices are congruent perpendicular bisectors of each other. 48. R (-5, 0) , S (-1, -2) , T (-3, -6) , and U (-7, -4) 49. E (2, 1) , F (5, 1) , G (5, -2) , and H (2, -2) 6-5 Conditions for Special Parallelograms (pp. 418–425) E X A M P L E S EXERCISES TEKS G.2.A, G.2.B, G.3.B, G.3.E, G.7.A, G.7.B, G.7.C ■ Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. ̶̶ LP ⊥ ̶̶ KN Given: Conclusion: KLNP is a rhombus. Determine if the conclusion is valid. If not, tell wh
at additional information is needed to make it valid. 50. Given: ̶̶ FS , Conclusion: EFRS is a square. ̶̶ ER ⊥ ̶̶ ER ≅ ̶̶ FS The conclusion is not valid. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. To apply this theorem, you must first know that KLNP is a parallelogram. 440 440 Chapter 6 Polygons and Quadrilaterals 51. Given: ̶̶ FS bisect each other. ̶̶ ER and ̶̶ ̶̶ FS ER ≅ Conclusion: EFRS is a rectangle. ̶̶ ES ̶̶ EF ≅ ̶̶ FR ǁ ̶̶ EF ǁ ̶̶ RS , ̶̶ ES , 52. Given: Conclusion: EFRS is a rhombus. �� ■ Use the diagonals to tell whether a parallelogram with vertices P (-5, 3) , Q (0, 1) , R (2, -4) , and S (-3, -2) is a rectangle, rhombus, or square. Give all the names that apply. PR = √  98 = 7 √  2 QS = √  18 = 3 √  2 Distance Formula Distance Formula Use the diagonals to tell whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 53. B (-3, 0) , F (-2, 7) , J (5, 8) , N (4, 1) 54. D (-4, -3) , H (5, 6) , L (8, 3) , P (-1, -6) 55. Q (-8, -2) , T (-6, 8) , W (4, 6) , Z (2, -4) Since PR ≠ QS, PQRS is not a rectangle and not a square. slope of = -1 ̶̶ PR = 7 _ -7 ̶̶ QS = 3 _ 3 = 1 slope of Slope Formula Slope Formula Since the product of the slopes is -1, the diagonals are perpendicular. PQRS is a rhombus. 6-6 Properties of Kites and Trapezoids (pp. 427–435) E X A M P L E S EXERCISES TEKS G.2.A, G.2.B, G.3.B, G.3.E, G.7.A, G.7.B, G.7.C ■ In kite PQRS, m∠SRT = 24°, and m∠TSP = 53°. Find m∠SPT. △PTS is a right triangle. Kite → diags. ⊥ Acute  of rt. △ are comp. m∠SPT + m∠TSP = 90° m∠SPT + 53 = 90 Substitute 53 for m∠TSP. m∠SPT = 37° Subtract 53 from both sides. ■ Find m∠D. m∠C + m∠D = 180° Same-Side Int.  Thm. Substitute 51 for m∠C. Subtract. 51 + m∠D = 180 m∠D = 129° ■ In trapezoid HJLN, In kite WXYZ, m∠VXY = 58°, and m∠ZWX = 50°. Find each measure. 56. m∠XYZ 57. m∠ZWV 58. m∠VZW 59. m∠WZY Find each measure. 60. m∠R and m∠S 61. BZ if ZH = 70 and EK = 121.6 62. MN 63. EQ JP = 32.5, and HL = 50. Find PN. ̶̶ ̶̶ JN ≅ HL JN = HL = 50 JP + PN = JN 32.5 + PN = 50 PN = 17.5 ■ Find WZ. Isosc. trap. → diags. ≅ Def. of ≅ segs. Seg. Add. Post. Substitute. Subtract 32.5 from both sides. AB = 1 _ (XY + WZ) Trap. Midsegment Thm. 2 73.5 = 1 _ (42 + WZ) 2 147 = 42 + WZ 105 = WZ Multiply both sides by 2. Solve for WZ. Substitute. 64. Find the value of n so that PQXY is isosceles. Give the best name for a quadrilateral whose vertices have the given coordinates. 65. (-4, 5) , (-1, 8) , (5, 5) , (-1, 2) 66. (1, 4) , (5, 4) , (5, -4) , (1, -1) 67. (-6, -1) , (-4, 2) , (0, 2) , (2, -1) Study Guide: Review 441 441 �� Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1. 2. 3. The base of a fountain is in the shape of a quadrilateral, as shown. Find the measure of each interior angle of the fountain. 4. Find the sum of the interior angle measures of a convex nonagon. 5. Find the measure of each exterior angle of a regular 15-gon. � �� 6. In EFGH, EH = 28, HZ = 9, and m∠EHG = 145°. Find FH and m∠FEH. 7. JKLM is a parallelogram. Find KL and m∠L. 8. Three vertices of PQRS are P (-2, -3) , R (7, 5) , and S (6, 1) . Find the coordinates of Q. 9. Show that WXYZ is a parallelogram for a = 4 and b = 3. 10. Determine if CDGH must be a parallelogram. Justify your answer. 11. Show that a quadrilateral with vertices K (-7, -3) , L (2, 0) , S (5, -4) , and T (-4, -7) is a parallelogram. 12. In rectangle PLCM, LC = 19, and LM = 23. Find PT and PM. 13. In rhombus EHKN, m∠NQK = (7z + 6) °, and m∠ENQ = (5z + 1) °. Find m∠HEQ and m∠EHK. Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. ̶̶ NP ≅ 14. Given: 15. Given: ̶̶̶ MQ , Conclusion: MNPQ is a rectangle. ̶̶̶ NM ≅ ̶̶ NP ≅ ̶̶ PQ , ̶̶̶ NQ ≅ ̶̶ ̶̶̶ MN PQ ≅ Conclusion: MNPQ is a square. ̶̶̶ QM ≅ ̶̶̶ MP Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 16. A (-5, 7) , C (3, 6) , E (7, -1) , G (-1, 0) 17. P (4, 1) , Q (3, 4) , R (-3, 2) , S (-2, -1) 18. m∠JFR = 43°, and m∠JNB = 68°. Find m∠FBN. 20. Find HR. � � � 442 442 Chapter 6 Polygons and Quadrilaterals �� 19. PV = 61.1, and YS = 24.7. Find MY. � � � �� FOCUS ON SAT The scores for each SAT section range from 200 to 800. Your score is calculated by subtracting a fraction for each incorrect multiple-choice answer from the total number of correct answers. No points are deducted for incorrect grid-in answers or items you left blank. If you have time, go back through each section of the test and check as many of your answers as possible. Try to use a different method of solving the problem than you used the first time. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. 1. Given the quadrilateral below, what value of x 3. Which of the following terms best describes the would allow you to conclude that the figure is a parallelogram? (A) -2 (B) 0 (C) 1 (D) 2 (E) 3 2. In the figure below, if ABCD is a rectangle, what type of triangle must △ABE be? (A) Equilateral (B) Right (C) Equiangular (D) Isosceles (E) Scalene figure below? (A) Rhombus (B) Trapezoid (C) Quadrilateral (D) Square (E) Parallelogram 4. Three vertices of MNPQ are M (3, 1) , N (0, 6) , and P (4, 7) . Which of the following could be the coordinates of vertex Q? (A) (7, 0) (B) (–1, 1) (C) (7, 2) (D) (11, 3) (E) (9, 4) 5. If ABCDE is a regular pentagon, what is the measure of ∠C? (A) 45° (B) 60° (C) 90° (D) 108° (E) 120° College Entrance Exam Practice 443 443 �� Multiple Choice: Eliminate Answer Choices For some multiple-choice test items, you can eliminate one or more of the answer choices without having to do many calculations. Use estimation or logic to help you decide which answer choices can be eliminated. What is the value of x in the figure? 3° 63° 83° 153° The sum of the exterior angle measures of a convex polygon is 360°. By rounding, you can estimate the sum of the given angle measures. 100° + 30° + 140° + 30° = 300° If x = 153°, the sum of the angle measures would be far greater than 360°. So eliminate D. If x = 3°, the sum would be far less than 360°. So eliminate A. From your estimate, it seems likely that the correct choice is B, 63°. Confirm that this is correct by doing the actual calculation. 98° + 32° + 63° + 135° + 32° = 360° The correct answer is B, 63°. What is m∠B in the isosceles trapezoid? 216° 108° 72° 58° Base angles of an isosceles trapezoid are congruent. Since ∠D and ∠B are not a pair of base angles, their measures are not equal. Eliminate G, 108°. ∠D and ∠C are base angles, so m∠C = 108°. ∠B and ∠C are same-side interior angles formed by parallel lines. So they are supplementary angles. Therefore the measure of angle B cannot be greater than 180°. You can eliminate F. m∠B = 180° - 108° = 72° The correct answer is H, 72°. 444 444 Chapter 6 Polygons and Quadrilaterals �� Try to eliminate unreasonable answer choices. Some choices may be too large or too small or may have incorrect units. Item C In isosecles trapezoid ABCD, AC = 18.2, and DG = 6.3. What is GB? Read each test item and answer the questions that follow. Item A The diagonals of rectangle MNPQ intersect at S. If MN = 4.1 meters, MS = 2.35 meters, and MQ = 2.3 meters, what is the area of △MPQ to the nearest tenth? 24.5 11.9 6.3 2.9 6. Will the measure of ̶̶ than, or equal to the measure of AC ? What answer choices can you eliminate and why? ̶̶ GB be more than, less 4.7 square meters 5.4 meters 9.4 square meters 12.8 meters 1. Are there any answer choices you can eliminate immediately? If so, which choices and why? 2. Describe how to use estimation to eliminate at least one more answer choice. Item B What is the sum of the interior angles of a convex hexagon? 7. Explain how to use estimation to answer this problem. Item D In trapezoid LMNP, XY = 25 feet. What are two ̶̶ LM and possible lengths for ̶̶ PN ? 18 feet and 32 feet 49 feet and 2 feet 10 feet and 15 feet 7 inches and 43 inches 180° 500° 720° 1080° 3. Can any of the answer choices be eliminated immediately? If so, which choices and why? 4. How can you use the fact that 500 is not a multiple of 180 to eliminate choice G? 5. A student answered this problem with J. Explain the mistake the student made. 8. Which answer choice can you eliminate immediately? Why? 9. A student used logic to eliminate choice H. Do you agree with the student’s decision? Explain. 10. A student used estimation and answered this problem with G. Explain the mistake the student made. TAKS Tackler 445 445 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–6 Multiple Choice Use the figure below for Items 6 and 7. 1. The exterior angles of a triangle have measures of (x + 10) °, (2x + 20) °, and 3x°. What is the measure of the smallest interior angle of the triangle? 15° 35° 55° 65° 2. If a plant is a monocot, then its leaves have parallel veins. If a plant is an orchid, then it is a monocot. A Mexican vanilla plant is an orchid. Based on this information, which conclusion is NOT valid? The leaves of a Mexican vanilla plant have parallel veins. A Mexican vanilla plant is a monocot. All orchids have leaves with parallel veins. All monocots are orchids. 3. If △ABC ≅ △PQR and △RPQ ≅ △XYZ, which of the following angles is congruent to ∠CAB? ∠QRP ∠XZY ∠YXZ ∠XYZ 4. Which line coincides with the line 2y + 3x = 4? x + 2 3y + 2x = 4 y = 2 _ 3 a line through (-1, 1) and (2, 3) a line through (0, 2) and (4, -4) 5. What is the value of x in polygon ABCDEF? 12 18 24 36 446 446 Chapter 6
Polygons and Quadrilaterals 6. If ̶̶ JK ǁ ̶̶̶ ML , what additional information do you need to prove that quadrilateral JKLM is a parallelogram? ̶̶ JM ≅ ̶̶̶ MN ≅ ̶̶ KL ̶̶ LN ∠MLK and ∠LKJ are right angles. ∠JML and ∠KLM are supplementary. 7. Given that JKLM is a parallelogram and that m∠KLN = 25°, m∠JMN = 65°, and m∠JML = 130°, which term best describes quadrilateral JKLM? Rectangle Rhombus Square Trapezoid 8. For two lines and a transversal, ∠1 and ∠2 are same-side interior angles, ∠2 and ∠3 are vertical angles, and ∠3 and ∠4 are alternate exterior angles. Which classification best describes the angle pair ∠2 and ∠4? Adjacent angles Alternate interior angles Corresponding angles Vertical angles 9. For △ABC and △DEF, ∠A ≅ ∠F, and ̶̶ AC ≅ ̶̶ EF . Which of the following would allow you to conclude that these triangles are congruent by AAS? ∠ABC ≅ ∠EDF ∠ACB ≅ ∠EDF ∠BAC ≅ ∠FDE ∠CBA ≅ ∠FED �� 10. The vertices of ABCD are A (1, 4) , B (4, y) , C (3, -2) , and D (0, -3) . What is the value of y? 3 4 5 6 STANDARDIZED TEST PREP Short Response 17. In △ABC, AE = 9x - 11.25, and AF = x + 4. 11. Quadrilateral RSTU is a kite. What is the length ̶̶ RV ? of 4 inches 5 inches 6 inches 13 inches 12. What is the measure of each interior angle in a regular dodecagon? 30° 144° 150° 162° 13. The coordinates of the vertices of quadrilateral RSTU are R (1, 3) , S (2, 7) , T (10, 5) , and U (9, 1) . Which term best describes quadrilateral RSTU? Parallelogram Rectangle Rhombus Trapezoid �� Mixed numbers cannot be entered into the grid for gridded-response questions. For example, if you get an answer of 7 1 __ 7.25 or 29 __ . 4 , you must grid either 4 Gridded Response 14. If quadrilateral MNPQ is a parallelogram, what is the value of x? 15. What is the greatest number of line segments determined by six coplanar points when no three are collinear? 16. Quadrilateral RSTU is a rectangle with diagonals SU = 6a - 25, what is the value of a? ̶̶ SU . If RT = 4a + 2 and ̶̶ RT and a. Find the value of x. Show your work and explain how you found your answer. b. If ̶̶ DF ≅ ̶̶ EF , show that △AFD ≅ △CFE. State any theorems or postulates used. 18. Consider quadrilateral ABCD. a. Show that ABCD is a trapezoid. Justify your answer. b. What are the coordinates for the endpoints of the midsegment of trapezoid ABCD? 19. Suppose that ∠M is complementary to ∠N and ∠N is complementary to ∠P. Explain why the measurements of these three angles cannot be the angle measurements of a triangle. Extended Response 20. Given △ABC and △XYZ, suppose that ̶̶ AB ≅ ̶̶ XY and ̶̶ BC ≅ ̶̶ YZ . a. If AB = 5, BC = 6, AC = 8, and m∠B < m∠Y, explain why △XYZ is obtuse. Justify your reasoning and state any theorems or postulates used. b. If AB = 3, BC = 5, AC = 5, and m∠B > m∠Y, ̶̶ XZ so that △XYZ is a right find the length of triangle. Justify your reasoning and state any theorems or postulates used. c. If AB = 8 and BC = 4, find the range of possible ̶̶ AC . Justify your answer. values for the length of Cumulative Assessment, Chapters 1–6 447 447 �� T E X A S TAKS Grades 9–11 Obj. 10 Arlington Georgetown Southwestern University Southwestern University, a nationally recognized liberal arts university in Georgetown, Texas, is Texas’s oldest university. Southwestern was officially chartered in 1875 ge07ts_c0 6psl001aa but was formed from the resources of four existing schools—Rutersville College 2nd pass (chartered in 1840), Wesleyan College (chartered in 1844), McKenzie College (chartered in 1848), and Soule University (chartered in 1856). 6/20/5 cmurphy Choose one or more strategies to solve each problem. 1. The trusses that line the ceiling of the McCombs Campus Center are made of triangular shapes. The center shape ̶̶ AC of the resembles an equilateral triangle. If the side triangle is 42 inches long, about how tall is the center truss ̶̶ CD ? Round to the nearest inch. 2. The floor of the Rockwell Rotunda in the McCombs Campus Center is in the shape of a regular octagon. What is the measure of each interior angle of the rotunda floor? 3. Each section of the stained-glass window is made of five polygonal shapes. Name each polygon by the number of its sides. Tell whether each polygon appears to be regular or irregular, and concave or convex. Identify which of the five polygons appear to be special quadrilaterals. 4. Square ABEG at the center of the marble fireplace is composed of three shapes— a smaller square DCFG and two congruent quadrilaterals ABCD and EBCF. Find the angle measures in ABCD, and explain why ABCD must be a trapezoid. 448 448 Chapter 6 Polygons and Quadrilaterals �� Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Titan When it opened in April 2001 at Six Flags Over Texas in Arlington, Titan became the tallest roller coaster in Texas and the third tallest in the world. Titan trains travel up to 85 mi/h and cover over 5312 feet of track. The first hill features a 255-foot drop at a 65° angle into a dark, 120-foot-long tunnel! Choose one or more strategies to solve each problem. 1. If a Titan train takes 3 minutes and 30 seconds to travel the entire track, what is the roller coaster’s average speed in miles per hour? 2. If a Titan train travels through the tunnel at its maximum speed, about how long does it take the train to pass through the tunnel? Round to the nearest hundredth of a second. 3. Titan has three trains, each of which holds 30 passengers. If the roller coaster can accommodate 1600 passengers per hour, about how many trains run each hour? The figure below shows the structure of the first hill of Titan. For 4–6, use the figure. 4. Titan reaches its maximum height at the top of the first hill. The ascent covers a horizontal distance AE of about 350 feet. What is the ̶̶ AB to the nearest foot? length of the ascent 5. The length of the descent ̶̶ CD is about 270 feet. What is FD to the nearest foot? 6. Event organizers plan to hang a banner across the first hill of Titan from X to Y, ̶̶ AB and Y is where X is the midpoint of ̶̶ CD . What is the width the midpoint of of the banner to the nearest foot? 449449 �� Similarity 7A Similarity Relationships 7-1 Ratio and Proportion Lab Explore the Golden Ratio 7-2 Ratios in Similar Polygons Lab Predict Triangle Similarity Relationships 7-3 Triangle Similarity: AA, SSS, and SAS 7B Applying Similarity Lab Investigate Angle Bisectors of a Triangle 7-4 Applying Properties of Similar Triangles 7-5 Using Proportional Relationships 7-6 Dilations and Similarity in the Coordinate Plane KEYWORD: MG7 ChProj The Lighthouse Rock is located in Palo Duro Canyon or “the Grand Canyon of Texas.” 450 450 Chapter 7 Vocabulary Match each term on the left with a definition on the right. 1. side of a polygon A. two nonadjacent angles formed by two intersecting lines 2. denominator 3. numerator 4. vertex of a polygon 5. vertical angles B. the top number of a fraction, which tells how many parts of a whole are being considered C. a point that corresponds to one and only one number D. the intersection of two sides of a polygon E. one of the segments that form a polygon F. the bottom number of a fraction, which tells how many equal parts are in the whole Simplify Fractions Write each fraction in simplest form. 6. 16_ 20 7. 14_ 21 8. 33_ 121 9. 56_ 80 Ratios Use the table to write each ratio in simplest form. 10. jazz CDs to country CDs 11. hip-hop CDs to jazz CDs 12. rock CDs to total CDs 13. total CDs to country CDs Identify Polygons Ryan’s CD Collection Rock Jazz Hip-hop Country 36 18 34 24 Determine whether each figure is a polygon. If so, name it by the number of sides. 14. 15. 16. 17. Find Perimeter Find the perimeter of each figure. 18. rectangle PQRS 20. rhombus JKLM 19. regular hexagon ABCDEF 21. regular pentagon UVWXY Similarity 451 451 �� Key Vocabulary/Vocabulario dilation dilatación proportion proporción ratio scale razón escala scale drawing dibujo a escala Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. When an eye doctor dilates your eyes, the pupils become enlarged. What might it mean for one geometric figure to be a dilation of another figure? scale factor factor de escala 2. A blueprint is a scale drawing of a building. similar semejante similar polygons polígonos semejantes similarity ratio razón de semejanza What do you think is the definition of a scale drawing ? 3. What does the word similar mean in everyday language? What do you think the term similar polygons means? 4. Bike riders often talk about gear ratios. Give examples of situations where the word ratio is used. What do these examples have in common? Geometry TEKS G.1.B Geometric structure* recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships G.3.B Geometric structure* construct and justify statements about geometric figures and their properties G.5.B Geometric patterns* use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures ... 7-2 Tech. Lab Les. 7-1 7-3 Tech. Lab Les. 7-2 7-4 Tech. Lab Les. 7-3 Les. 7-4 Les. 7-5 Les. 7-.9.B Congruence and the geometry of size* formulate and test ★ ★ ★ ★ ★ conjectures about the properties and attributes of polygons ... based on explorations and concrete models G.11.A Similarity and the geometry of shape* use and extend similarity
properties and transformations to explore and justify conjectures about geometric figures. ★ ★ ★ ★ ★ ★ G.11.B Similarity and the geometry of shape* use ratios to solve ★ ★ ★ ★ ★ problems involving similar figures G.11.D Similarity and the geometry of shape* describe the effect on perimeter, area ... when one or more dimensions of a figure are changed and apply this idea in solving problems ★ * Knowledge and skills are written out completely on pages TX28–TX35. 452 452 Chapter 7 Reading Strategy: Read and Understand the Problem Many of the concepts you are learning are used in real-world situations. Throughout the text, there are examples and exercises that are real-world word problems. Listed below are strategies for solving word problems. Problem Solving Strategies • Read slowly and carefully. Determine what information is given and what you are asked to find. • If a diagram is provided, read the labels and make sure that you understand the information. If you do not, resketch and relabel the diagram so it makes sense to you. If a diagram is not provided, make a quick sketch and label it. • Use the given information to set up and solve the problem. • Decide whether your answer makes sense. From Lesson 6-1: Look at how the Polygon Exterior Angle Theorem is used in photography. Photography Application The aperture of the camera shown is formed by ten blades. The blades overlap to form a regular decagon. What is the measure of ∠CBD? � � � � Step Understand the Problem Procedure Result • List the important information. • The answer will be the measure of ∠CBD. ∠CBD is one of the exterior angles of the regular decagon formed by the apeture. Make a Plan • A diagram is provided, and it is labeled accurately. Solve • You can use the Polygon Exterior Angle Theorem. Then divide to find the measure of one of the exterior angles. m∠CBD = 360° _ = 36° 10 Look Back • The answer is reasonable since a decagon has 10 exterior angles. 10 (36°) = 360° Try This Use the problem-solving strategies for the following problem. 1. A painter’s scaffold is constructed so that the braces lie along the diagonals of rectangle PQRS. Given RS = 28 and QS = 85, find QT. Similarity 453 453 �� 7-1 Ratio and Proportion TEKS G.11.B Similarity and the geometry of shape: use ratios to solve problems involving similar figures. Also G.5.B, G.7.B, G.7.C Objectives Write and simplify ratios. Use proportions to solve problems. Who uses this? Filmmakers use ratios and proportions when creating special effects. (See Example 5.) Vocabulary ratio proportion extremes means cross products The Lord of the Rings movies transport viewers to the fantasy world of Middle Earth. Many scenes feature vast fortresses, sprawling cities, and bottomless mines. To film these images, the moviemakers used ratios to help them build highly detailed miniature models. A ratio compares two numbers by division. The ratio of two numbers a and b can be written as a to b, a : b, or a__ , where b ≠ 0. b For example, the ratios 1 to 2, 1 : 2, and 1 __ 2 all represent the same comparison. E X A M P L E 1 Writing Ratios Write a ratio expressing the slope of ℓ. In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined. Slope = rise = run = 3 - (-1) _ 4 - (-2) = 2 _ = 4 _ 3 6 Substitute the given values. Simplify. 1. Given that two points on m are C (-2, 3) and D (6, 5) , write a ratio expressing the slope of m. A ratio can involve more than two numbers. For the rectangle, the ratio of the side lengths may be written as 3 : 7 : 3 : 7. E X A M P L E 2 Using Ratios The ratio of the side lengths of a quadrilateral is 2 : 3 : 5 : 7, and its perimeter is 85 ft. What is the length of the longest side? Let the side lengths be 2x, 3x, 5x, and 7x. Then 2x + 3x + 5x + 7x = 85. After like terms are combined, 17x = 85. So x = 5. The length of the longest side is 7x = 7 (5) = 35 ft. 2. The ratio of the angle measures in a triangle is 1 : 6 : 13. What is the measure of each angle? 454 454 Chapter 7 Similarity �� A proportion is an equation stating that two ratios are equal. In the proportion a __ = c __ , the values a and d are the extremes . The values b and c are d b the means . When the proportion is written as a : b = c : d, the extremes are in the first and last positions. The means are in the two middle positions. In Algebra 1 you learned the Cross Products Property. The product of the extremes ad and the product of the means bc are called the cross products . Cross Products Property = c _ In a proportion, if a _ d b and b and d ≠ 0, then ad = bc. The Cross Products Property can also be stated as, “In a proportion, the product of the extremes is equal to the product of the means.” E X A M P L E 3 Solving Proportions Cross Products Prop. Divide both sides by 45. 5 A Simplify. _ y = Solve each proportion. _ 45 63 5 (63) = y (45) 315 = 45y y = 7 = 24 x + 2) 2 = 6 (24) (x + 2) 2 = 144 x + 2 = ±12 x + 2 = 12 or x + 2 = -12 B Cross Products Prop. Simplify. Find the square root of both sides. Rewrite as two eqns. x = 10 or x = -14 Subtract 2 from both sides. Solve each proportion. = x _ 3a. 3 _ 8 56 3c. d _ = 6 _ 2 3 3b. 3d. 2y = 8 _ _ 4y The following table shows equivalent forms of the Cross Products Property. Properties of Proportions ALGEBRA _ _ c = d a The proportion b the following: is equivalent to NUMBERS = 2 _ 3 6 The proportion 1 _ the following: is equivalent to b a ad = bc 6) = 3 (2- 1 Ratio and Proportion 455 455 �� E X A M P L E 4 Using Properties of Proportions Given that 4x = 10y, find the ratio of x to y in simplest form. Since x comes before y in the sentence, x will be in the numerator of the fraction. 4x = 10y x _ y = 10 _ 4 x _ y = 5 _ 2 Divide both sides by 4y. Simplify. 4. Given that 16s = 20t, find the ratio t : s in simplest form. E X A M P L E 5 Problem-Solving Application During the filming of The Lord of the Rings, the special-effects team built a model of Sauron’s tower with a height of 8 m and a width of 6 m. If the width of the full-size tower is 996 m, what is its height? Understand the Problem The answer will be the height of the tower. Make a Plan Let x be the height of the tower. Write a proportion that compares the ratios of the height to the width. height of model tower ___ = width of model tower = x _ 8 _ 996 6 height of full-size tower ___ width of full-size tower Solve = x _ 8 _ 6 996 6x = 8 (996) 6x = 7968 x = 1328 Cross Products Prop. Simplify. Divide both sides by 6. The height of the full-size tower is 1328 m. Look Back Check the answer in the original problem. The ratio of the height to the width of the model is 8 : 6, or 4 : 3. The ratio of the height to the width of the tower is 1328 : 996. In simplest form, this ratio is also 4 : 3. So the ratios are equal, and the answer is correct. 5. What if...? Suppose the special-effects team made a different model with a height of 9.2 m and a width of 6 m. What is the height of the actual tower? 456 456 Chapter 7 Similarity 1234 THINK AND DISCUSS 1. Is the ratio 6 : 7 the same ratio as 7 : 6? Why or why not? 2. Susan wants to know if the fractions 3 __ 7 and 12 __ 28 are equivalent. Explain how she can use the properties of proportions to find out. 3. GET ORGANIZED Copy and complete the graphic organizer. In the boxes, write the definition of a proportion, the properties of proportions, and examples and nonexamples of a proportion. 7-1 Exercises Exercises KEYWORD: MG7 7-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. = 2 _ 1. Name the means and extremes in the proportion . Write the cross products for the proportion Write a ratio expressing the slope of each line. p. 454 3. ℓ 4. m 5. The ratio of the side lengths of a quadrilateral p. 454 is 2 : 4 : 5 : 7, and its perimeter is 36 m. What is the length of the shortest side? 7. The ratio of the angle measures in a triangle is 5 : 12 : 19. What is the measure of the largest angle. 455 Solve each proportion. = 40 _ 8. x _ 2 16 y = 27 _ y _ 11. 3 9. 7 _ y = 21 _ 27 = x - 1 _ 12. 16 _ 4 x - 1 10. 6 _ 58 13. x 2 _ 18 = t _ 29 = 14. Given that 2a = 8b, find the ratio of a p. 456 to b in simplest form. 15. Given that 6x = 27y, find the ratio y : x in simplest form 16. Architecture The Arkansas State p. 456 Capitol Building is a smaller version of the U.S. Capitol Building. The U.S. Capitol is 752 ft long and 288 ft tall. The Arkansas State Capitol is 564 ft long. What is the height of the Arkansas State Capitol? 7- 1 Ratio and Proportion 457 457 �� PRACTICE AND PROBLEM SOLVING Independent Practice Write a ratio expressing the slope of each line. For See Exercises Example 17. ℓ 18. m 19. n 17–19 20–21 22–27 28–29 30 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S16 Application Practice p. S34 20. The ratio of the side lengths of an isosceles triangle is 4 : 4 : 7, and its perimeter is 52.5 cm. What is the length of the base of the triangle? 21. The ratio of the angle measures in a parallelogram is 2 : 3 : 2 : 3. What is the measure of each angle? Solve each proportion. = 9 _ y 22. 6 _ 8 2m + 2 _ 3 = 12 _ 2m + 2 25. 23. x _ 14 5y _ 16 26. = 50 _ 35 = 125 _ y 24. z _ 12 = 12 27. Travel 28. Given that 5y = 25x, find the ratio of x to y in simplest form. 29. Given that 35b = 21c, find the ratio b : c in simplest form. 30. Travel Madurodam is a park in the Netherlands that contains a complete Dutch city built entirely of miniature models. One of the models of a windmill is 1.2 m tall and 0.8 m wide. The width of the actual windmill is 20 m. What is its height? Given that a __ b the following equations. 32. b _ a = 34. Sports During the 2003 NFL season, the Dallas Cowboys won 10 of their 16 regular-season games. What is their ratio of wins to losses in simplest form? , complete each of 7 33. a _ 5 31. 7a = = 5 __ = For more than 50 years, Madurodam has
been Holland’s smallest city. The canal houses, market, airplanes, and windmills are all replicated on a 1 : 25 scale. Source: madurodam.nl Write a ratio expressing the slope of the line through each pair of points. 35. (-6, -4) and (21, 5) , -2) and (4, 5 1 _ 37. (6 1 _ ) 38. (-6, 1) and (-2, 0) 36. (16, -5) and (6, 1) 2 2 39. This problem will prepare you for the Multi-Step TAKS Prep on page 478. A claymation film is shot on a set that is a scale model of an actual city. On the set, a skyscraper is 1.25 in. wide and 15 in. tall. The actual skyscraper is 800 ft tall. a. Write a proportion that you can use to find the width of the actual skyscraper. b. Solve the proportion from part a. What is the width of the actual skyscraper? 458 458 Chapter 7 Similarity �� 40. Critical Thinking The ratio of the lengths of a quadrilateral’s consecutive sides is 2 : 5 : 2 : 5. The ratio of the lengths of the quadrilateral’s diagonals is 1 : 1. What type of quadrilateral is this? Explain. 41. Multi-Step One square has sides 6 cm long. Another has sides 9 cm long. Find the ratio of the areas of the squares. 42. Photography A photo shop makes prints of photographs in a variety of sizes. Every print has a length-to-width ratio of 5 : 3.5 regardless of its size. A customer wants a print that is 20 in. long. What is the width of this print? 43. Write About It What is the difference between a ratio and a proportion? 44. An 18-inch stick breaks into three pieces. The ratio of the lengths of the pieces is 1 : 4 : 5. Which of these is NOT a length of one of the pieces? 1.8 inches 3.6 inches 7.2 inches 9 inches 45. Which of the following is equivalent to 3 __ 5 = x __ y ? 3 _ y = 5 _ x 3x = 5y x _ 3 y _ = 5 3 (5) = xy 46. A recipe for salad dressing calls for oil and vinegar in a ratio of 5 parts oil to 2 parts 1 _ 2 vinegar. If you use 1 1 __ cups of oil, how many cups of vinegar will you need? 4 2 1 _ 2 5 _ 8 47. Short Response Explain how to solve the proportion 36 __ 72 must assume about x in order to solve the proportion. 6 1 _ 4 = 15 __ x for x. Tell what you CHALLENGE AND EXTEND 48. The ratio of the perimeter of rectangle ABCD to the perimeter of rectangle EFGH is 4 : 7. Find x. and a + b ____ = c __ 49. Explain why a __ b d b are equivalent proportions. = c + d ____ d 50. Probability The numbers 1, 2, 3, and 6 are randomly placed in these four boxes: ___ . What is the probability that the two ratios will form a proportion? ___ ? 51. Express the ratio x 2 + 9x + 18 _________ x 2 - 36 in simplest form. SPIRAL REVIEW Complete each ordered pair so that it is a solution to y - 6x = -3. (Previous course) 52. (0, 54. (-4, 53. ( , 3) ) ) Find each angle measure. (Lesson 3-2) 55. m∠ABD 56. m∠CDB Each set of numbers represents the side lengths of a triangle. Classify each triangle as acute, right, or obtuse. (Lesson 5-7) 57. 5, 8, 9 58. 8, 15, 20 59. 7, 24, 25 7- 1 Ratio and Proportion 459 459 �� 7-2 Use with Lesson 7-2 Activity 1 Explore the Golden Ratio In about 300 B.C.E., Euclid showed in his book Elements how to calculate the golden ratio. It is claimed that this ratio was used in many works of art and architecture to produce rectangles of pleasing proportions. The golden ratio also appears in the natural world and it is said even in the human face. If the ratio of a rectangle’s length to its width is equal to the golden ratio, it is called a golden rectangle. TEKS G.5.B Geometric patterns: use numeric and geometric patterns to make generalizations about geometric properties ... ratios in similar figures .... Also G.1.A, G.2.B, G.5.A KEYWORD: MG7 Lab7 1 Construct a segment and label its endpoints A ̶̶ AP is ̶̶ PB . What are AP, PB, and AB? What and B. Place P on the segment so that longer than is the ratio of AP to PB and the ratio of AB to AP? Drag P along the segment until the ratios are equal. What is the value of the equal ratios to the nearest hundredth? 2 Construct a golden rectangle beginning with ̶̶ AB . Then construct a circle ̶̶ AB . ̶̶ AB through a square. Create with its center at A and a radius of Construct a line perpendicular to A. Where the circle and the perpendicular line intersect, label the point D. Construct perpendicular lines through B and D and label their intersection C. Hide the lines and the circle, leaving only the segments to complete the square. 3 Find the midpoint of ̶̶ AB and label it M. Create a segment from M to C. Construct a ̶̶̶ MC . circle with its center at M and radius of Construct a ray with endpoint A through B. Where the circle and the ray intersect, label the point E. Create a line through E that is AB . Show the previously perpendicular to hidden line through D and C. Label the point of intersection of these two lines F. Hide the lines and circle and create segments to complete golden rectangle AEFD. 4 Measure ̶̶ AE , ̶̶ EF ,and ̶̶ BE . Find the ratio of AE to EF and the ratio of EF to BE. Compare these ratios to those found in Step 1. What do you notice? 460 460 Chapter 7 Similarity Try This 1. Adjust your construction from Step 2 so that the side of the original square is 2 units long. Use the Pythagorean Theorem to find the length of Calculate the length of decimal rounded to the nearest thousandth. ̶̶ AE . Write the ratio of AE to EF as a fraction and as a ̶̶̶ MC . 2. Find the length of ̶̶ BE in your construction from Step 3. Write the ratio of EF to BE as a fraction and as a decimal rounded to the nearest thousandth. Compare your results to those from Try This Problem 1. What do you notice? 3. Each number in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13 …) is created by adding the two preceding numbers together. That is, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, and so on. Investigate the ratios of the numbers in the sequence by finding the ̶̶̶ 666 , 8 __ 5 = 1.6, and so on. What do you notice quotients. 1 __ 1 = 1, 2 __ 1 = 2, 3 __ 2 = 1.5, 5 __ 3 = 1. as you continue to find the quotients? Tell why each of the following is an example of the appearance of the Fibonacci sequence in nature. 4. 5. Determine whether each picture is an example of an application of the golden rectangle. Measure the length and the width of each and decide whether the ratio of the length to the width is approximately the golden ratio. 6. 7. 7- 2 Technology Lab 461 461 �� 7-2 Ratios in Similar Polygons TEKS G.5.B Geometric patterns: use ... geometric patterns to make generalizations about ratios in similar figures ... Objectives Identify similar polygons. Apply properties of similar polygons to solve problems. Vocabulary similar similar polygons similarity ratio Why learn this? Similar polygons are used to build models of actual objects. (See Example 3.) Figures that are similar (∼) have the same shape but not necessarily the same size. △1 is similar to △2 (△1 ∼ △2) . △1 is not similar to △3 (△1 ≁ △3) . Similar Polygons DEFINITION DIAGRAM STATEMENTS Also G.11.A, G.11.B Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding sides are proportional. ABCD ∼ EFGH ∠A ≅ ∠E ∠B ≅ ∠F ∠C ≅ ∠G AB ___ = BC ___ FG EF ∠D ≅ ∠H = CD ___ = DA ___ GH = 1 __ 2 HE E X A M P L E 1 Describing Similar Polygons Identify the pairs of congruent angles and corresponding sides. ∠Z ≅ ∠R and ∠Y ≅ ∠Q. By the Third Angles Theorem, ∠X ≅ ∠S. XY _ SQ = 6 _ , YZ _ = 2 _ 3 9 QR = 12 _ 18 = 2 _ , 3 XZ _ SR = 9 _ 13.5 = 2 _ 3 1. Identify the pairs of congruent angles and corresponding sides. 462 462 Chapter 7 Similarity �� A similarity ratio is the ratio of the lengths of the corresponding sides of two similar polygons. The similarity ratio of △ABC to △DEF is 3 __ 6 , or 1 __ 2 . The similarity ratio of △DEF to △ABC is 6 __ 3 , or 2. E X A M P L E 2 Identifying Similar Polygons Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. A rectangles PQRS and TUVW Step 1 Identify pairs of congruent angles. ∠P ≅ ∠T, ∠Q ≅ ∠U, ∠R ≅ ∠V, and ∠S ≅ ∠W All  of a rect. are rt.  and are ≅. Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order. Step 2 Compare corresponding sides. , PS _ = 3 _ 4 TW = 2 _ = 4 _ 3 6 = 12 _ 16 PQ _ TU Since corresponding sides are not proportional, the rectangles are not similar. B △ABC and △DEF Step 1 Identify pairs of congruent angles. ∠A ≅ ∠D, ∠B ≅ ∠E ∠C ≅ ∠F Given Third  Thm. Step 2 Compare corresponding sides. , AC _ , BC _ = 4 _ = 4 _ 3 3 DF EF = 4 _ 3 Thus the similarity ratio is 4 __ 3 , and △ABC ∼ △DEF. = 24 _ 18 = 20 _ 15 = 16 _ 12 AB _ DE 2. Determine if △JLM ∼ △NPS. If so, write the similarity ratio and a similarity statement. Proportions with Similar Figures When I set up a proportion, I make sure each ratio compares the figures in the same order. To find x, I wrote 10 __ = 6 __ x . 4 This will work because the first ratio compares the lengths starting with rectangle ABCD. The second ratio compares the widths, also starting with rectangle ABCD. Anna Woods Westwood High School ABCD ∼ EFGH 7- 2 Ratios in Similar Polygons 463 463 �� E X A M P L E 3 Hobby Application A Railbox boxcar can be used to transport auto parts. If the length of the actual boxcar is 50 ft, find the width of the actual boxcar to the nearest tenth of a foot. Let x be the width of the actual boxcar in feet. The rectangular model of a boxcar is similar to the rectangular boxcar, so the corresponding lengths are proportional. �� When you work with proportions, be sure the ratios compare corresponding measures. �� = width of boxcar __ width of model length of boxcar __ length of model = x _ 50 _ 7 2 7x = (50) (2) 7x = 100 x ≈ 14.3 Cross Products Prop. Simplify. Divide both sides by 7. The width of the model is approximately 14.3 ft. 3. A boxcar has the dimensions shown. A mode
l of the boxcar is 1.25 in. wide. Find the length of the model to the nearest inch. THINK AND DISCUSS 1. If you combine the symbol for similarity with the equal sign, what symbol is formed? 2. The similarity ratio of rectangle ABCD to rectangle EFGH is 1 __ 9 . How do the side lengths of rectangle ABCD compare to the corresponding side lengths of rectangle EFGH? 3. What shape(s) are always similar? 4. GET ORGANIZED Copy and complete the graphic organizer. Write the definition of similar polygons, and a similarity statement. Then draw examples and nonexamples of similar polygons. 464 464 Chapter 7 Similarity �� 7-2 Exercises Exercises KEYWORD: MG7 7-2 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Give an example of similar figures in your classroom Identify the pairs of congruent angles and corresponding sides. p. 462 2. 3. 463 Multi-Step Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 4. rectangles ABCD and EFGH 5. △RMP and △UWX . 464 6. Art The town of Goodland, Kansas, claims that it has one of the world’s largest easels. It holds an enlargement of a van Gogh painting that is 24 ft wide. The original painting is 58 cm wide and 73 cm tall. If the reproduction is similar to the original, what is the height of the reproduction to the nearest foot? Independent Practice For See Exercises Example 7–8 9–10 11 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S16 Application Practice p. S34 PRACTICE AND PROBLEM SOLVING Identify the pairs of congruent angles and corresponding sides. 7. 8. Multi-Step Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 9. △RSQ and △UXZ 10. rectangles ABCD and JKLM 7- 2 Ratios in Similar Polygons 465 465 �� 11. Hobbies The ratio of the model car’s dimensions to the actual car’s dimensions is 1 __ 56 . The model has a length of 3 in. What is the length of the actual car? 12. Square ABCD has an area of 4 m 2 . Square PQRS has an area of 36 m 2 . What is the similarity ratio of square ABCD to square PQRS? What is the similarity ratio of square PQRS to square ABCD? Tell whether each statement is sometimes, always, or never true. 13. Two right triangles are similar. 14. Two squares are similar. 15. A parallelogram and a trapezoid are similar. 16. If two polygons are congruent, they are also similar. 17. If two polygons are similar, they are also congruent. Monument 18. Critical Thinking Explain why any two regular polygons having the same number of sides are similar. Find the value of x. 19. ABCD ∼ EFGH 20. △MNP ∼ △ XYZ The height of the Statue of Liberty from the foundation of the pedestal to the torch is 305 ft. Her index finger measures 8 ft, and the fingernail is 13 in. by 10 in. Source: libertystatepark.org 21. Estimation The Statue of Liberty’s hand is 16.4 ft long. Assume that your own body is similar to that of the Statue of Liberty and estimate the length of the Statue of Liberty’s nose. (Hint : Use a ruler to measure your own hand and nose. Then set up a proportion.) 22. Write the definition of similar polygons as two conditional statements. 23. JKLM ∼ NOPQ. If m∠K = 75°, name two 75° angles in NOPQ. 24. A dining room is 18 ft long and 14 ft wide. On a blueprint for the house, the dining room is 3.5 in. long. To the nearest tenth of an inch, what is the width of the dining room on the blueprint? 25. Write About It Two similar polygons have a similarity ratio of 1 : 1. What can you say about the two polygons? Explain. 26. This problem will prepare you for the Multi-Step TAKS Prep on page 478. A stage set consists of a painted backdrop with some wooden flats in front of it. One of the flats shows a tree that has a similarity ratio of 1 __ 2 to an actual tree. To give an illusion of distance, the backdrop includes a small painted tree that has a similarity ratio of 1 __ 10 to the tree on the flat. a. The tree on the backdrop is 0.9 ft tall. What is the height of the tree on the flat? b. What is the height of the actual tree? c. Find the similarity ratio of the tree on the backdrop to the actual tree. 466 466 Chapter 7 Similarity �� 27. Which value of y makes the two rectangles similar? 3 8.2 25.2 28.8 28. △CGL ∼ △MPS. The similarity ratio of △CGL to △MPS is 5 __ . What is the length of 2 8 ̶̶ PS ? 50 12 75 29. Short Response Explain why 1.5, 2.5, 3.5 and 6, 10, 12 cannot be corresponding sides of similar triangles. CHALLENGE AND EXTEND 30. Architecture An architect is designing a building that is 200 ft long and 140 ft wide. She builds a model so that the similarity ratio of the model to the building is 1 ___ 500 . What is the length and width of the model in inches? 31. Write a paragraph proof. Given: ̶̶ QR ǁ ̶̶ ST Prove: △PQR ∼ △PST 32. In the figure, D is the midpoint of ̶̶ AC . a. Find AC, DC, and DB. b. Use your results from part a to help you explain why △ABC ∼ △CDB. 33. A golden rectangle has the following property: If a square is cut from one end of the rectangle, the rectangle that remains is similar to the original rectangle. a. Rectangle ABCD is a golden rectangle. Write a similarity statement for rectangle ABCD and rectangle BCFE. b. Write a proportion using the corresponding sides of these rectangles. � � � � � � � � �� c. Solve the proportion for ℓ. (Hint : Use the Quadratic Formula.) d. The value of ℓ is known as the golden ratio. Use a calculator to find ℓ to the nearest tenth. SPIRAL REVIEW 34. There are four runners in a 200-meter race. Assuming there are no ties, in how many different orders can the runners finish the race? (Previous course) ̶̶ ̶̶ QP , m∠QPT = 45°, and QR ≅ In kite PQRS, m∠RST = 20°. Find each angle measure. (Lesson 6-6) ̶̶ PS ≅ ̶̶ RS , 36. m∠PST 35. m∠QTR 37. m∠TPS _ x = Complete each of the following equations, given that 4 40. x _ y = 39. 10 _ y = 38. 10x = _ y 10 . (Lesson 7-1) 7- 2 Ratios in Similar Polygons 467 467 �� 7-3 Predict Triangle Similarity Relationships In Chapter 4, you found shortcuts for determining that two triangles are congruent. Now you will use geometry software to find ways to determine that triangles are similar. Use with Lesson 7-3 Activity 1 TEKS G.11.A Similarity and the geometry of shape: use and extend similarity properties and transformations to explore and justify conjectures about geometric figures. Also G.2.A, G.3.B, G.9.B KEYWORD: MG7 Lab7 1 Construct △ABC. Construct ̶̶ DE longer than ̶̶ DE around ̶̶ DE around E by any of the sides of △ABC. Rotate D by rotation ∠BAC. Rotate rotation ∠ABC. Label the intersection point of the two rotated segments as F. 2 Measure angles to confirm that ∠BAC ≅ ∠EDF and ∠ABC ≅ ∠DEF. Drag a vertex of △ABC ̶̶ or an endpoint of DE to show that the two triangles have two pairs of congruent angles. 3 Measure the side lengths of both triangles. Divide each side length of △ABC by the corresponding side length of △DEF. Compare the resulting ratios. What do you notice? Try This 1. What theorem guarantees that the third pair of angles in the triangles are also congruent? 2. Will the ratios of corresponding sides found in Step 3 always be equal? Drag ̶̶ DE to investigate this question. State a a vertex of △ABC or an endpoint of conjecture based on your results. Activity 2 1 Construct a new △ABC. Create P in the interior of the triangle. Create △DEF by enlarging △ABC around P by a multiple of 2 using the Dilation command. Drag P outside of △ABC to separate the triangles. 468 468 Chapter 7 Similarity 2 Measure the side lengths of △DEF to confirm that each side is twice as long as the corresponding side of △ABC. Drag a vertex of △ABC to verify that this relationship is true. 3 Measure the angles of both triangles. What do you notice? Try This 3. Did the construction of the triangles with three pairs of sides in the same ratio guarantee that the corresponding angles would be congruent? State a conjecture based on these results. 4. Compare your conjecture to the SSS Congruence Theorem from Chapter 4. How are they similar and how are they different? Activity 3 1 Construct a different △ABC. Create P in the ̶̶ AB and ̶̶ AC interior of the triangle. Expand around P by a multiple of 2 using the Dilation command. Create an angle congruent to ∠BAC with sides that are each twice as long as ̶̶ AB and ̶̶ AC . 2 Use a segment to create the third side of a new triangle and label it △DEF. Drag P outside of △ABC to separate the triangles. 3 Measure each side length and determine the relationship between corresponding sides of △ABC and △DEF. 4 Measure the angles of both triangles. What do you notice? Try This 5. Tell whether △ABC is similar to △DEF. Explain your reasoning. 6. Write a conjecture based on the activity. What congruency theorem is related to your conjecture? 7- 3 Technology Lab 469 469 7-3 Triangle Similarity: AA, SSS, and SAS TEKS G.11.B Similarity and the geometry of shape: use ratios to solve problems involving similar figures. Also G.5.B, G.11.A Objectives Prove certain triangles are similar by using AA, SSS, and SAS. Use triangle similarity to solve problems. Who uses this? Engineers use similar triangles when designing buildings, such as the Pyramid Building in San Diego, California. (See Example 5.) There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent. Postulate 7-3-1 Angle-Angle (AA) Similarity POSTULATE HYPOTHESIS CONCLUSION If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. △ABC ∼ △DEF E X A M P L E 1 Using the AA
Similarity Postulate Explain why the triangles are similar and write a similarity statement. ̶̶ SR , ∠P ≅ ∠R, and ∠T ≅ ∠S by Since the Alternate Interior Angles Theorem. Therefore △PQT ∼ △RQS by AA ∼. ̶̶ PT ǁ 1. Explain why the triangles are similar and write a similarity statement. Theorem 7-3-2 Side-Side-Side (SSS) Similarity THEOREM HYPOTHESIS CONCLUSION If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. △ABC ∼ △DEF You will prove Theorem 7-3-2 in Exercise 38. 470 470 Chapter 7 Similarity �� Theorem 7-3-3 Side-Angle-Side (SAS) Similarity THEOREM HYPOTHESIS CONCLUSION If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. △ABC ∼ △DEF ∠B ≅ ∠E You will prove Theorem 7-3-3 in Exercise 39. E X A M P L E 2 Verifying Triangle Similarity Verify that the triangles are similar. A △PQR and △PRS QR = 2 _ = 4 _ _ , 3 6 RS PQ = 6 _ = 2 _ _ 3 9 PR Therefore △PQR ∼ △PRS by SSS ∼. , PR _ = 2 _ = 4 _ 3 6 PS B △JKL and △JMN ∠J ≅ ∠J by the Reflexive Property of ≅. JK = 3 _ = 1 _ _ 3 9 JM Therefore △JKL ∼ △JMN by SAS ∼. JL = 1 _ = 2 _ _ , 3 6 JN 2. Verify that △TXU ∼ △VXW. E X A M P L E 3 Finding Lengths in Similar Triangles Explain why △ABC ∼ △DBE and then find BE. Step 1 Prove triangles are similar. ̶̶ AC ǁ ̶̶ ED , ∠A ≅ ∠D, and ∠C ≅ ∠E As shown by the Alternate Interior Angles Theorem. Therefore △ABC ∼ △DBE by AA ∼. Step 2 Find BE. = BC _ AB _ BE DB = 54 _ 36 _ 54 BE 36 (BE) = 54 2 36 (BE) = 2916 Corr. sides are proportional. Substitute 36 for AB, 54 for DB, and 54 for BC. Cross Products Prop. Simplify. BE = 81 Divide both sides by 36. 3. Explain why △RSV ∼ △RTU and then find RT. 7- 3 Triangle Similarity: AA, SSS, and SAS 471 471 �� E X A M P L E 4 Writing Proofs with Similar Triangles Given: A is the midpoint of D is the midpoint of ̶̶ BC . ̶̶ BE . Prove: △BDA ∼ △BEC Proof: Statements ̶̶ BC . ̶̶ BE . 1. A is the mdpt. of D is the mdpt. of ̶̶ ̶̶ DE BA ≅ ̶̶ BD ≅ ̶̶ AC , 2. 3. BA = AC, BD = DE Reasons 1. Given 2. Def. of mdpt. 3. Def. of ≅ seg. 4. BC = BA + AC, BE = BD + DE 4. Seg. Add. Post. 5. BC = BA + BA, BE = BD + BD 5. Subst. Prop. 6. BC = 2BA, BE = 2BD 7. BC _ = 2, BE _ BD BA 8. BC _ = BE _ BA BD 9. ∠B ≅ ∠B = 2 10. △BDA ∼ △BEC 6. Simplify. 7. Div. Prop. of = 8. Trans. Prop. of = 9. Reflex. Prop. of ≅ 10. SAS ∼ Steps 8, 9 ̶̶ JK . 4. Given: M is the midpoint of ̶̶ KL , N is the midpoint of and P is the midpoint of ̶̶ JL . Prove: △JKL ∼ △NPM (Hint : Use the Triangle Midsegment Theorem and SSS ∼.) E X A M P L E 5 Engineering Application The photo shows a gable roof. △ABC ∼ △FBG and then find BF to the nearest tenth of a foot. ̶̶ AC ǁ ̶̶ FG . Use similar triangles to prove � � � �� Step 1 Prove the triangles are similar. ̶̶ ̶̶ FG AC ǁ ∠BFG ≅ ∠BAC ∠B ≅ ∠B Given Corr.  Thm. Reflex. Prop. of ≅ Therefore △ABC ∼ △FBG by AA ∼. 472 472 Chapter 7 Similarity �� Step 2 Find BF. = BF _ BA _ FG AC 17 _ = BF _ 6.5 24 17 (6.5) = 24 (BF) 110.5 = 24 (BF) 4.6 ft ≈ BF Corr. sides are proportional. Substitute the given values. Cross Products Prop. Simplify. Divide both sides by 24. 5. What if…? If AB = 4x, AC = 5x, and BF = 4, find FG. You learned in Chapter 2 that the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. These properties also hold true for similarity of triangles. Properties of Similarity Reflexive Property of Similarity △ABC ∼ △ABC (Reflex. Prop. of ∼) Symmetric Property of Similarity If △ABC ∼ △DEF, then △DEF ∼ △ABC. (Sym. Prop. of ∼) Transitive Property of Similarity If △ABC ∼ △DEF and △DEF ∼ △XYZ, then △ABC ∼ △XYZ. (Trans. Prop. of ∼) THINK AND DISCUSS 1. What additional information, if any, would you you need in order to show that △ABC ∼ △DEF by the AA Similarity Postulate? 2. What additional information, if any, would you need in order to show that △ABC ∼ △DEF by the SAS Similarity Theorem? 3. Do corresponding sides of similar triangles need to be proportional and congruent? Explain. 4. GET ORGANIZED Copy and complete the graphic organizer. If possible, write a congruence or similarity theorem or postulate in each section of the table. Include a marked diagram for each. 7- 3 Triangle Similarity: AA, SSS, and SAS 473 473 �� 7-3 Exercises Exercises KEYWORD: MG7 7-3 KEYWORD: MG7 Parent GUIDED PRACTICE Explain why the triangles are similar and write a similarity statement. p. 470 1. 2 Verify that the triangles are similar. p. 471 3. △DEF and △JKL 4. △MNP and △MRQ Multi-Step Explain why the triangles are similar and then find each length. p. 471 5. AB 6. WY . Given: ⎯  MN ǁ ̶̶ KL p. 472 Prove: △JMN ∼ △JKL 8. Given: SQ = 2QP, TR = 2RP Prove: △PQR ∼ △PST 9. The coordinates of A, B, and C are A(0, 0), B(2, 6), and C(8, -2). What theorem or postulate justifies the statement △ABC ∼ △DEF, if the coordinates of D and E are twice the coordinates of B and C. 472 10. Surveying In order to measure the distance AB across the meteorite crater, a surveyor at S locates points A, B, C, and D as shown. What is AB to the nearest meter? nearest kilometer? 474 474 Chapter 7 Similarity ��ge07sec07l03003aaAB533 m733 m800 m586 m644 mCSADB PRACTICE AND PROBLEM SOLVING Explain why the triangles are similar and write a similarity statement. 11. 12. Verify that the given triangles are similar. 13. △KLM and △KNL 14. △UVW and △XYZ Independent Practice For See Exercises Example 11–12 13–14 15–16 17–18 19 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S16 Application Practice p. S34 Multi-Step Explain why the triangles are similar and then find each length. 15. AB 16. PS 17. Given: CD = 3AC, CE = 3BC 18. Given: PR _ MR = QR _ NR Prove: △ABC ∼ △DEC Prove: ∠1 ≅ ∠2 19. Photography The picture shows a person taking a pinhole photograph of himself. Light entering the opening reflects his image on the wall, forming similar triangles. What is the height of the image to the nearest tenth of an inch? Draw △JKL and △MNP. Determine if you can conclude that △JKL ∼ △MNP based on the given information. If so, which postulate or theorem justifies your response? = KL _ = KL _ NP NP = KL _ NP 20. ∠K ≅ ∠N, 22. ∠J ≅ ∠M, JK _ MN JK _ MN JL _ MP JL _ MP 21. = Find the value of x. 23. 24. 7- 3 Triangle Similarity: AA, SSS, and SAS 475 475 ge07sec07l03004aAB5 ft 5 in.4 ft 6 in.15 in.�� 25. This problem will prepare you for the Multi-Step TAKS Prep on page 478. The set for an animated film includes three small triangles that represent pyramids. a. Which pyramids are similar? Why? b. What is the similarity ratio of the similar pyramids? �� 26. Critical Thinking △ABC is not similar to △DEF, and △DEF is not similar to △XYZ. Could △ABC be similar to △XYZ? Why or why not? Make a sketch to support your answer. 27. Recreation To play shuffleboard, two teams take turns sliding disks on a court. The dimensions of the scoring area for a standard shuffleboard court are shown. What are JK and MN? 28. Prove the Transitive Property of Similarity. Given: △ABC ∼ △DEF, △DEF ∼ △XYZ Prove: △ABC ∼ △XYZ 29. 29. Draw and label △PQR and △STU such that PQ ___ ST = QR ___ TU Meteorology but △PQR is NOT similar to △STU. 30. 30. Given: △KNJ is isosceles with ∠N as the vertex angle. ∠H ≅ ∠L Prove: △GHJ ∼ △MLK This satellite image shows Hurricane Lili as it moves across the Gulf of Mexico. In October 2002, an estimated 500,000 people evacuated in advance of Lili’s hitting Texas. 31. Meteorology Satellite photography makes it possible to measure the diameter of a hurricane. The figure shows that a camera’s aperture YX is 35 mm and its focal length WZ is 50 mm. The satellite W holding the camera is 150 mi above the hurricane, centered at C. a. Why is △XYZ ∼ △ABZ ? What assumption must you make about the position of the camera in order to make this conclusion? b. What other triangles in the figure must be similar? Why? c. Find the diameter AB of the hurricane. 32. /////ERROR ANALYSIS///// Which solution for the value of y is incorrect? Explain the error. 33. Write About It Two isosceles triangles have congruent vertex angles. Explain why the two triangles must be similar. 476 476 Chapter 7 Similarity �� 34. What is the length of ̶̶ TU ? 36 40 48 90 35. Which dimensions guarantee that △BCD ∼ △FGH? FG = 11.6, GH = 8.4 FG = 12, GH = 14 FG = 11.4, GH = 11.4 FG = 10.5, GH = 14.5 36. ABCD ∼ EFGH. Which similarity postulate or theorem lets you conclude that △BCD ∼ △FGH? AA SSS SAS None of these 37. Gridded Response If 6, 8, and 12 and 15, 20, and x are the lengths of the corresponding sides of two similar triangles, what is the value of x? CHALLENGE AND EXTEND 38. Prove the SSS Similarity Theorem. = BC _ EF = AC _ DF Prove: △ABC ∼ △DEF Given: AB _ DE (Hint : Assume that AB < DE and choose point X on ̶̶ DF so that ⎯  XY ǁ ̶̶ ̶̶ DX . DE so that ̶̶ EF . Show that △DXY ∼ △DEF ̶̶ AB ≅ Then choose point Y on and that △ABC ≅ △DXY.) 39. Prove the SAS Similarity Theorem. Given: ∠B ≅ ∠E, AB _ DE = BC _ EF Prove: △ABC ∼ △DEF (Hint : Assume that AB < DE and choose point X on ̶̶ EF so that ∠EXY ≅ ∠EDF. Show that △XEY ∼ △DEF ̶̶ DE so that ̶̶ EX ≅ ̶̶ BA . Then choose point Y on and that △ABC ≅ △XEF.) 40. Given △ABC ∼ △XYZ, m∠A = 50°, m∠X = (2x + 5y) °, m∠Z = (5x + y) °, and that m∠B = (102 - x)
°, find m∠Z. SPIRAL REVIEW 41. Jessika’s scores in her last six rounds of golf were 96, 99, 105, 105, 94, and 107. What score must Jessika make on her next round to make her mean score 100? (Previous course) Position each figure in the coordinate plane and give possible coordinates of each vertex. (Lesson 4-7) 42. a right triangle with leg lengths of 4 units and 2 units 43. a rectangle with length 2k and width k Solve each proportion. Check your answer. (Lesson 7-1) 44. 2x _ 10 = 25 _ 10y = 35 _ 25 5y _ 450 45. 46. b - 5 _ 28 = 7 _ b - 5 7- 3 Triangle Similarity: AA, SSS, and SAS 477 477 �� SECTION 7A Similarity Relationships Lights! Camera! Action! Lorenzo, Maria, Sam, and Tia are working on a video project for their history class. They decide to film a scene where the characters in the scene are on a train arriving at a town. Since Lorenzo collects model trains, they decide to use one of his trains and to build a set behind it. To create the set, they use a film technique called forced perspective. They want to use small objects to create an illusion of great distance in a very small space. 1. Lorenzo’s model train is 1 __ 87 the size of the original train. He measures the engine of the model train and finds that it is 2 1 __ 2 in. tall. What is the height of the real engine to the nearest foot? 2. The closest building to the train needs to be made using the same scale as the train. Maria and Sam estimate that the height of an actual station is 20 ft. How tall would they need to build their model of the train station to the nearest 1 __ 4 in.? 3. To give depth to their scene, they want to construct partial buildings behind the train station. Lorenzo decided to build a restaurant. If the height of the restaurant is actually 24 ft, how tall would they need to build their model of the restaurant to the nearest inch? 4. The other buildings on the set will have triangular roofs. Which of the roofs are similar to each other? Why? 478 478 Chapter 7 Similarity �� SECTION 7A Quiz for Lessons 7-1 Through 7-3 7-1 Ratio and Proportion Write a ratio expressing the slope of each line. 1. ℓ 3. n 2. m 4. x-axis Solve each proportion. y = 12 _ _ 5. 9 6 7. 16 _ 24 8. 2 _ 3y = 20 _ t y _ 24 = 9. An architect’s model for a building is 1.4 m long and 0.8 m wide. The actual building is 240 m wide. What is the length of the building? 7-2 Ratios in Similar Polygons Determine whether the two polygons are similar. If so, write the similarity ratio and a similarity statement. 10. rectangles ABCD and WXYZ 11. △JMR and △KNP 12. Leonardo da Vinci’s famous portrait the Mona Lisa is 30 in. long and 21 in. wide. Janelle has a refrigerator magnet of the painting that is 3.5 cm wide. What is the length of the magnet? 7-3 Triangle Similarity: AA, SSS, and SAS 13. Given: ABCD Prove: △EDG ∼ △FBG 14. Given: MQ = 1 __ 3 MN, MR = 1 __ 3 MP Prove: △MQR ∼ △MNP 15. A geologist wants to measure the length XY of a rock formation. To do so, she locates points U, V, X, Y, and Z as shown. What is XY? Ready to Go On? 479 479 �� 7-4 Investigate Angle Bisectors of a Triangle In a triangle, an angle bisector divides the opposite side into two segments. You will use geometry software to explore the relationships between these segments. Use with Lesson 7-4 Activity 1 TEKS G.5.B Geometric patterns: use... geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures... Also G.2.A, G.3.B, G.9.B KEYWORD: MG7 Lab7 1 Construct △ABC. Bisect ∠BAC and create the point of intersection of the angle bisector and ̶̶ BC . Label the intersection D. 2 Measure ̶̶ AB , ̶̶ AC , ̶̶ BD , and measurements to write ratios. What are the results? Drag a vertex of △ABC and examine the ratios again. What do you notice? ̶̶ CD . Use these Try This 1. Choose Tabulate and create a table using the four lengths and the ratios from Step 2. Drag a vertex of △ABC and add the new measurements to the table. What conjecture can you make about the segments created by an angle bisector? 2. Write a proportion based on your conjecture. Activity 2 1 Construct △DEF. Create the incenter of the triangle and label it I. Hide the angle bisectors of ̶̶ ∠E and ∠F. Find the point of intersection of EF and the bisector of ∠D. Label the intersection G. 2 Find DI, DG, and the perimeter of △DEF. ̶̶ 3 Divide the length of DI by the length of DG. ̶̶ ̶̶ Add the lengths of DF . Then divide DE and this sum by the perimeter of △DEF. Compare the two quotients. Drag a vertex of △DEF and examine the quotients again. What do you notice? 4 Write a proportion based on your quotients. What conjecture can you make about this relationship? Try This 3. Show the hidden angle bisector of ∠E or ∠F. Confirm that your conjecture is true for this bisector. Drag a vertex of △DEF and observe the results. 4. Choose Tabulate and create a table with the measurements you used in your proportion in Step 4. 480 480 Chapter 7 Similarity 7-4 Applying Properties of Similar Triangles TEKS G.11.B Similarity and the geometry of shape: use ratios to solve problems involving similar figures. Also G.2.A, G.3.B, G.5.B, G.9.B, G.11.A Objectives Use properties of similar triangles to find segment lengths. Apply proportionality and triangle angle bisector theorems. Who uses this? Artists use similarity and proportionality to give paintings an illusion of depth. (See Example 3.) Artists use mathematical techniques to make two-dimensional paintings appear three-dimensional. The invention of perspective was based on the observation that far away objects look smaller and closer objects look larger. Mathematical theorems like the Triangle Proportionality Theorem are important in making perspective drawings. Theorem 7-4-1 Triangle Proportionality Theorem THEOREM HYPOTHESIS CONCLUSION If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally. AE _ EB = AF _ FC ̶̶ EF ǁ ̶̶ BC You can use a compass-and-straightedge construction to verify this theorem. Although the construction is not a proof, it should help convince you that the theorem is true. After you have completed the construction, use a ruler to measure ̶̶ AF , and ̶̶ EB , ̶̶ AE , ̶̶ FC to see that AE ___ EB = AF ___ . FC Construction Triangle Proportionality Theorem Construct a line parallel to a side of a triangle.    Use a straightedge to draw △ABC. Label E on AB. Construct ∠E ≅ ∠B. Label the ̶̶ AC as F. intersection of   EF and ̶̶   EF ǁ BC by the Converse of the Corresponding Angles Postulate. 7- 4 Applying Properties of Similar Triangles 481 481 �� E X A M P L E 1 Finding the Length of a Segment Find CY. It is given that ̶̶ XY ǁ ̶̶ BC , so AX ___ XB = AY ___ YC by the Triangle Proportionality Theorem. = 10 _ 9 _ 4 CY 9 (CY ) = 40 CY = 40 _ , or 4 4 _ 9 9 1. Find PN. Substitute 9 for AX, 4 for XB, and 10 for AY. Cross Products Prop. Divide both sides by 9. Theorem 7-4-2 Converse of the Triangle Proportionality Theorem THEOREM HYPOTHESIS CONCLUSION If a line divides two sides of a triangle proportionally, then it is parallel to the third side. AE _ EB = AF _ FC   EF ǁ ̶̶ BC You will prove Theorem 7-4-2 in Exercise 23. E X A M P L E 2 Verifying Segments are Parallel ̶̶̶ MN ǁ ̶̶ KL . = 2 Verify that = 42 _ 21 = 30 _ 15 = JN ___ , NL JM _ MK JN _ NL Since JM ___ MK Triangle Proportionality Theorem. ̶̶̶ MN ǁ = 2 ̶̶ KL by the Converse of the 2. AC = 36 cm, and BC = 27 cm. ̶̶ DE ǁ Verify that ̶̶ AB . Corollary 7-4-3 Two-Transversal Proportionality THEOREM HYPOTHESIS CONCLUSION If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. AC _ CE = BD _ DF You will prove Corollary 7-4-3 in Exercise 24. 482 482 Chapter 7 Similarity �� E X A M P L E 3 Art Application ̶̶ AK ǁ An artist used perspective to draw guidelines to help her sketch a row of parallel trees. She then checked the drawing by measuring the distances between the trees. What is LN ? ̶̶̶ ̶̶ CM ǁ BL ǁ KL _ = AB _ LN BD BD = BC + CD BD = 1.4 + 2.2 = 3.6 cm 2.6 _ LN = 2.4 _ 3.6 ̶̶̶ DN Given 2-Transv. Proportionality Corollary Seg. Add. Post. Substitute 1.4 for BC and 2.2 for CD. Substitute the given values. 2.4 (LN) = 3.6 (2.6) LN = 3.9 cm Cross Products Prop. Divide both sides by 2.4. 3. Use the diagram to find LM and MN to the nearest tenth. The previous theorems and corollary lead to the following conclusion. Theorem 7-4-4 Triangle Angle Bisector Theorem THEOREM HYPOTHESIS CONCLUSION An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. (△ ∠ Bisector Thm.) BD _ DC = AB _ AC You will prove Theorem 7-4-4 in Exercise 38. E X A M P L E 4 Using the Triangle Angle Bisector Theorem Find RV and VT. RV ___ VT = SR ___ ST x + 2 _ 2x + 1 = 10 _ 14 by the △ ∠ Bisector Thm. Substitute the given values. You can check your answer by substituting the values into the proportion. = SR __ RV ___ VT ST = 10 __ 5 __ 14 7 = 5 __ 5 __ 7 7 14 (x + 2) = 10 (2x + 1) 14x + 28 = 20x + 10 18 = 6x x = 3 Cross Products Prop. Dist. Prop. Simplify. Divide both sides by 6. RV = x + 2 = 3 + 2 = 5 VT = 2x + 1 Substitute 3 for x. = 2 (3) + 1 = 7 4. Find AC and DC. 7- 4 Applying Properties of Similar Triangles 483 483 �� THINK AND DISCUSS 1. ̶̶ BC . Use what you know about similarity ̶̶ XY ǁ and proportionality to state as many different proportions as possible. 2. GET ORGANIZED Copy and complete the graphic organizer. Draw a figure for each proportionality theorem or corollary and then measure it. Use your measurements to wri
te an if-then statement about each figure. 7-4 Exercises Exercises GUIDED PRACTICE . 482 . 482 Find the length of each segment. ̶̶̶ DG 1. Verify that the given segments are parallel. ̶̶ AB and ̶̶ CD 3. KEYWORD: MG7 7-4 KEYWORD: MG7 Parent ̶̶ RN 2. ̶̶ TU and ̶̶ RS 4. Travel The map shows the area around p. 483 Herald Square in Manhattan, New York, and the approximate length of several streets. If the numbered streets are parallel, what is the length of Broadway between 34th St. and 35th St. to the nearest foot? 484 484 Chapter 7 Similarity �� Find the length of each segment. p. 483 6. ̶̶ QR and ̶̶ RS ̶̶ CD and ̶̶ AD 7. PRACTICE AND PROBLEM SOLVING Find the length of each segment. ̶̶ KL 8. ̶̶ XZ 9. Independent Practice For See Exercises Example 8–9 10–11 12 13–14 1 2 3 4 TEKS TEKS TAKS TAKS Verify that the given segments are parallel. Skills Practice p. S17 Application Practice p. S34 10. ̶̶ AB and ̶̶ CD 11. ̶̶̶ MN and ̶̶ QR � �� 12. Architecture The wooden treehouse has horizontal siding that is parallel to the base. What are LM and MN to the nearest hundredth? �� Find the length of each segment. 13. ̶̶ BC and ̶̶ CD 14. ̶̶ ST and ̶̶ TU = AC_ In the figure,    BC ǁ    DE ǁ    FG . Complete each proportion. 15. AB_ BD 17. DF _ = EG _ CE = AE_ EG = 16. _ DF 18. AF _ AB 20. AB_ AC _ AC = BF_ 19. BD_ CE = _ EG 21. The bisector of an angle of a triangle divides the opposite side of the triangle into segments that are 12 in. and 16 in. long. Another side of the triangle is 20 in. long. What are two possible lengths for the third side? 7- 4 Applying Properties of Similar Triangles 485 485 �� 22. This problem will prepare you for the Multi-Step TAKS Prep on page 502. Jaclyn is building a slide rail, the narrow, slanted beam found in skateboard parks. a. Write a proportion that Jaclyn can use to calculate the length of ̶̶ CE . b. Find CE. c. What is the overall length of the slide rail AJ? 23. Prove the Converse of the Triangle Proportionality Theorem. Given: AE _ EB Prove:   EF ǁ = AF _ FC ̶̶ BC 24. Prove the Two-Transversal Proportionality Corollary. Given:   AB ǁ   CD ,   CD ǁ   EF Prove: AC _ = BD _ DF CE (Hint : Draw   BE through X.) 25. Given that   PQ ǁ   RS ǁ   TU a. Find PR, RT, QS, and SU. b. Use your results from part b to write a proportion relating the segment lengths. Find the length of each segment. 26. ̶̶ EF 27. ̶̶ ST 28. Real Estate A developer is laying out lots along Grant Rd. whose total width is 500 ft. Given the width of each lot along Chavez St., what is the width of each of the lots along Grant Rd. to the nearest foot? 29. Critical Thinking Explain how to use a sheet of lined notebook paper to divide a segment into five congruent segments. Which theorem or corollary do you use? ̶̶ ̶̶ XY ǁ DE ǁ ̶̶ BC , ̶̶ AD 30. Given that Find EC. 31. Write About It In △ABC,  AD bisects ∠BAC. Write a proportionality statement for the triangle. What theorem supports your conclusion? 486 486 Chapter 7 Similarity �� 32. Which dimensions let you conclude that SR = 12, TR = 9 SR = 16, TR = 20 ̶̶ UV ǁ ̶̶ ST ? SR = 35, TR = 28 SR = 50, TR = 48 33. In △ABC, the bisector of ∠A divides ̶̶ BC into segments with lengths 16 and 20. AC = 25. Which of these could be the length of ̶̶ AB ? 20 12.8 16 18.75 34. On the map, 1st St. and 2nd St. are parallel. What is the distance from City Hall to 2nd St. along Cedar Rd.? 1.8 mi 3.2 mi 4.2 mi 5.6 mi 35. Extended Response Two segments are divided proportionally. The first segment is divided into lengths 20, 15, and x. The corresponding lengths in the second segment are 16, y, and 24. Find the value of x and y. Use these values and write six proportions. CHALLENGE AND EXTEND 36. The perimeter of △ABC is 29 m. ̶̶ AD bisects ∠A. Find AB and AC. 37. Prove that if two triangles are similar, then the ratio of their corresponding angle bisectors is the same as the ratio of their corresponding sides. 38. Prove the Triangle Angle Bisector Theorem. ̶̶ AD bisects ∠A. Given: In △ABC, = AB _ Prove: BD _ AC DC ̶̶ BX ǁ ̶̶ AD and extend ̶̶ AC to X. Use properties Plan: Draw of parallel lines and the Converse of the Isosceles Triangle Theorem to show that Then apply the Triangle Proportionality Theorem. ̶̶ AX ≅ ̶̶ AB . 39. Construction Draw of similarity to divide ̶̶ AB any length. Use parallel lines and the properties ̶̶ AB into three congruent parts. SPIRAL REVIEW Write an algebraic expression that can be used to find the nth term of each sequence. (Previous course) 40. 5, 6, 7, 8,… 41. 3, 6, 9, 12,… 42. 1, 4, 9, 16,… 43. B is the midpoint of ̶̶ AC . A has coordinates (1, 4) , and B has coordinates (3, -7) . Find the coordinates of C. (Lesson 1-6) Verify that the given triangles are similar. (Lesson 7-3) 44. △ABC and △ADE 45. △JKL and △MLN 7- 4 Applying Properties of Similar Triangles 487 487 2.4 mi2.1 mi2.8 miCedar Rd.Aspen Rd.1st St.2nd St.CityHallLibraryge07se_c07l04007a�� 7-5 Using Proportional Relationships TEKS G.11.D Similarity and the geometry of shape: describe the effect on perimeter, area ... when ... dimensions of a figure are changed .... Also G.1.B, G.5.A, G.11.A, G.11.B Objectives Use ratios to make indirect measurements. Use scale drawings to solve problems. Vocabulary indirect measurement scale drawing scale Why learn this? Proportional relationships help you find distances that cannot be measured directly. Indirect measurement is any method that uses formulas, similar figures, and/or proportions to measure an object. The following example shows one indirect measurement technique. E X A M P L E 1 Measurement Application Eiffel Tower replica in Paris, Texas A student wanted to find the height of a statue of a pineapple in Nambour, Australia. She measured the pineapple’s shadow and her own shadow. The student’s height is 5 ft 4 in. What is the height of the pineapple? Step 1 Convert the measurements to inches. AC = 5 ft 4 in. = (5 ⋅ 12) in. + 4 in. = 64 in. BC = 2 ft = (2 ⋅ 12) in. = 24 in. EF = 8 ft 9 in. = (8 ⋅ 12) in. + 9 in. = 105 in. Step 2 Find similar triangles. Because the sun’s rays are parallel, ∠1 ≅ ∠2. Therefore △ABC ∼ △DEF by AA ∼. Step 3 Find DF. = BC_ AC_ EF DF 64 _ = 24 _ 105 DF 24 (DF) = 64 ⋅ 105 DF = 280 Corr. sides are proportional. Substitute 64 for AC, 24 for BC, and 105 for EF. Cross Products Prop. Divide both sides by 24. The height of the pineapple is 280 in., or 23 ft 4 in. 1. A student who is 5 ft 6 in. tall measured shadows to find the height LM of a flagpole. What is LM? Whenever dimensions are given in both feet and inches, you must convert them to either feet or inches before doing any calculations. 488 488 Chapter 7 Similarity 8 ft 9 in.DFEge07se_ c07105002aaAB22 ftAB1C�� A scale drawing represents an object as smaller than or larger than its actual size. The drawing’s scale is the ratio of any length in the drawing to the corresponding actual length. For example, on a map with a scale of 1 cm : 1500 m, one centimeter on the map represents 1500 m in actual distance. E X A M P L E 2 Solving for a Dimension A proportion may compare measurements that have different units. The scale of this map of downtown Dallas is 1.5 cm : 300 m. Find the actual distance between Union Station and the Dallas Public Library. Use a ruler to measure the distance between Union Station and the Dallas Public Library. The distance is 6 cm. To find the actual distance x write a proportion comparing the map distance to the actual distance. 6 _ x = 1.5 _ 300 1.5x = 6 (300) 1.5x = 1800 x = 1200 Cross Products Prop. Simplify. Divide both sides by 1.5. The actual distance is 1200 m, or 1.2 km. 2. Find the actual distance between City Hall and El Centro College. E X A M P L E 3 Making a Scale Drawing The Lincoln Memorial in Washington, D.C., is approximately 57 m long and 36 m wide. Make a scale drawing of the base of the building using a scale of 1 cm : 15 m. Step 1 Set up proportions to find the length ℓ and width w of the scale drawing. = 1 _ 15 15w = 36 ℓ _ = 1 _ 57 15 15ℓ = 57 w _ 36 ℓ = 3.8 m w = 2.4 cm Step 2 Use a ruler to draw a rectangle with these dimensions. 3. The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1 in. : 20 ft. 7- 5 Using Proportional Relationships 489 489 S. LamarElmS. MarketS. HoustonS. GriffinFieldS. AustinCantonYoungWoodS. ErvayS. Akard JacksonCommerceMainCity HallDallas PublicLibraryUnion StationEl CentroCollege0300 mScale30Holt, Rinehart & WinstonGeometry © 2007ge07sec07105003a Dallas Area Street map2nd proof�� Similar Triangles Similarity, Perimeter, and Area Ratios STATEMENT RATIO △ABC ∼ △DEF = BC _ EF Perimeter ratio: Similarity ratio: AB _ = AC _ DE DF perimeter △ABC __ perimeter △DEF = 6 _ = 1 _ 4 24 Area ratio: area △ABC __ = 1 _ 2 = 12 _ 24 = ( 1 _ ) area △DEF 2 2 = 1 _ 2 The comparison of the similarity ratio and the ratio of perimeters and areas of similar triangles leads to the following theorem. Theorem 7-5-1 Proportional Perimeters and Areas Theorem If the similarity ratio of two similar figures is a __ , then the ratio of their perimeters b , and the ratio of their areas is a 2 __ is a __ b 2 b , or ( a __ ) b . 2 Y
ou will prove Theorem 7-5-1 in Exercises 44 and 45. E X A M P L E 4 Using Ratios to Find Perimeters and Areas Given that △RST ∼ △UVW, find the perimeter P and area A of △UVW. The similarity ratio of △RST to △UVW is 16 __ 20 , or 4 __ 5 . By the Proportional Perimeters and Areas Theorem, the ratio of the triangles’ perimeters is also 4 __ 5 , and the ratio of the triangles’ areas is ( 4 __ 5 ) , or 16 __ 25 . 2 Perimeter 36 _ = 4 _ 5 P 4P = 5 (36) P = 45 ft Area = 16 _ 48 _ 25 A 16A = 25 ⋅ 48 A = 75 ft 2 The perimeter of △UVW is 45 ft, and the area is 75 ft 2 . 4. △ABC ∼ △DEF, BC = 4 mm, and EF = 12 mm. If P = 42 mm and A = 96 mm 2 for △DEF, find the perimeter and area of △ABC. THINK AND DISCUSS 1. Explain how to find the actual distance between two cities 5.5 in. apart on a map that has a scale of 1 in. : 25 mi. 2. GET ORGANIZED Copy and complete the graphic organizer. Draw and measure two similar figures. Then write their ratios. 490 490 Chapter 7 Similarity �� 7-5 Exercises Exercises KEYWORD: MG7 7-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Finding distances using similar triangles is called ? . ̶̶̶̶ (indirect measurement or scale drawing ) . Measurement To find the height of a dinosaur in p. 488 a museum, Amir placed a mirror on the ground 40 ft from its base. Then he stepped back 4 ft so that he could see the top of the dinosaur in the mirror. Amir’s eyes were approximately 5 ft 6 in. above the ground. What is the height of the dinosaur. 489 The scale of this blueprint of an art gallery is 1 in. : 48 ft. Find the actual lengths of the following walls. ̶̶ AB ̶̶ EF 3. 5. ̶̶ CD ̶̶ FG 4. 6. 489 Multi-Step A rectangular classroom is 10 m long and 4.6 m wide. Make a scale drawing of the classroom using the following scales. 7. 1 cm : 1 m 8. 1 cm : 2 m 9. 1 cm : 2. Given: rectangle MNPQ ∼ rectangle RSTU p. 490 10. Find the perimeter of rectangle RSTU. 11. Find the area of rectangle RSTU. Independent Practice For See Exercises Example 12 13–14 15–17 18–19 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S17 Application Practice p. S34 PRACTICE AND PROBLEM SOLVING 12. Measurement Jenny is 5 ft 2 in. tall. To find the height of a light pole, she measured her shadow and the pole’s shadow. What is the height of the pole? Space Exploration Use the following information for Exercises 13 and 14. This is a map of the Mars Exploration Rover Opportunity’s predicted landing site on Mars. The scale is 1 cm : 9.4 km. What are the approximate measures of the actual length and width of the ellipse? 13. ̶̶ KJ 14. ̶̶ NP � � � � Multi-Step A park at the end of a city block is a right triangle with legs 150 ft and 200 ft long. Make a scale drawing of the park using the following scales. 15. 1.5 in. : 100 ft 16. 1 in. : 300 ft 17. 1 in. : 150 ft 7- 5 Using Proportional Relationships 491 491 �� Given that pentagon ABCDE ∼ pentagon FGHJK, find each of the following. 18. perimeter of pentagon FGHJK 19. area of pentagon FGHJK Estimation Use the scale on the map for Exercises 20–23. Give the approximate distance of the shortest route between each pair of sites. 20. campfire and the lake 21. lookout point and the campfire 22. cabins and the dining hall 23. lookout point and the lake Given: △ABC ∼ △DEF 24. The ratio of the perimeter of △ABC to the perimeter of △DEF is 8 __ 9 . What is the similarity ratio of △ABC to △DEF ? 25. The ratio of the area of △ABC to the area of △DEF is 16 __ 25 . What is the similarity ratio of △ABC to △DEF? 26. The ratio of the area of △ABC to the area of △DEF is 4 __ 81 . What is the ratio of the perimeter of △ABC to the perimeter of △DEF? 27. Space Exploration The scale of this model of the space shuttle is 1 ft : 50 ft. In the actual space shuttle, the main cargo bay measures 15 ft wide by 60 ft long. What are the dimensions of the cargo bay in the model? 28. Given that △PQR ∼ △WXY, find each ratio. a. perimeter of △PQR__ perimeter of △WXY b. area of △PQR __ area of △WXY c. How does the result in part a compare with the result in part b? 29. Given that rectangle ABCD ∼ EFGH . The area of rectangle ABCD is 135 in 2 . The area of rectangle EFGH is 240 in 2 . If the width of rectangle ABCD is 9 in., what is the length and width of rectangle EFGH? 30. Sports An NBA basketball court is 94 ft long and 50 ft wide. Make a scale drawing of a court using a scale of 1 __ 4 in. : 10 ft. 31. This problem will prepare you for the Multi-Step TAKS Prep on page 502. A blueprint for a skateboard ramp has a scale of 1 in. : 2 ft. On the blueprint, the rectangular piece of wood that forms the ramp measures 2 in. by 3 in. a. What is the similarity ratio of the blueprint to the actual ramp? b. What is the ratio of the area of the ramp on the blueprint to its actual area? c. Find the area of the actual ramp. 492 492 Chapter 7 Similarity �� Math History In 1075 C.E., Shen Kua created a calendar for the emperor by measuring the positions of the moon and planets. He plotted exact coordinates three times a night for five years. Source: history.mcs. st-andrews.ac.uk 32. Estimation The photo shows a person who is 5 ft 1 in. tall standing by a statue in Jamestown, North Dakota. Estimate the actual height of the statue by using a ruler to measure her height and the height of the statue in the photo. 33. Math History In A.D. 1076, the mathematician Shen Kua was asked by the emperor of China to produce maps of all Chinese territories. Shen created 23 maps, each drawn with a scale of 1 cm : 900,000 cm. How many centimeters long would a 1 km road be on such a map? 34. Points X, Y, and Z are the midpoints of ̶̶ JK , ̶̶ KL , and ̶̶ LJ , respectively. What is the ratio of the area of △JKL to the area of △XYZ? 35. Critical Thinking Keisha is making two scale drawings of her school. In one drawing, she uses a scale of 1 cm : 1 m. In the other drawing, she uses a scale of 1 cm : 5 m. Which of these scales will produce a smaller drawing? Explain. 36. The ratio of the perimeter of square ABCD to the perimeter of square EFGH is 4 __ 9 . Find the side lengths of each square. 37. Write About It Explain what it would mean to make a scale drawing with a scale of 1 : 1. 38. Write About It One square has twice the area of another square. Explain why it is impossible for both squares to have side lengths that are whole numbers. 39. △ABC ∼ △RST, and the area of △ABC is 24 m 2 . What is the area of △RST ? 16 m 2 29 m 2 36 m 2 54 m 2 40. A blueprint for a museum uses a scale of 1 __ in. : 1 ft. 4 One of the rooms on the blueprint is 3 3 __ in. long. 4 How long is the actual room? 4 ft 15 ft 45 ft 180 ft 41. The similarity ratio of two similar pentagons is 9 __ . What is the ratio of the 4 perimeters of the pentagons 81 _ 16 42. Of two similar triangles, the second triangle has sides half the length of the first. Given that the area of the first triangle is 16 ft 2 , find the area of the second. 32 ft 2 16 ft 2 4 ft 2 8 ft 2 7- 5 Using Proportional Relationships 493 493 �� CHALLENGE AND EXTEND 43. Astronomy The city of Eugene, Oregon, has a scale model of the solar system nearly 6 km long. The model’s scale is 1 km : 1 billion km. a. Earth is 150,000,000 km from the Sun. How many meters apart are Earth and the Sun in the model? b. The diameter of Earth is 12,800 km. What is the diameter, in centimeters, of Earth in the model? �� 44. Given: △ABC ∼ △DEF AB + BC + AC __ DE + EF +DF Prove: = AB _ DE 45. Given: △PQR ∼ △WXY Area △PQR __ Area △WXY Prove: = PR 2 _ WY 2 46. Quadrilateral PQRS has side lengths of 6 m, 7 m, 10 m, and 12 m. The similarity ratio of quadrilateral PQRS to quadrilateral WXYZ is 1 : 2. a. Find the lengths of the sides of quadrilateral WXYZ. b. Make a table of the lengths of the sides of both figures. c. Graph the data in the table. d. Determine an equation that relates the lengths of the sides of quadrilateral PQRS to the lengths of the sides of quadrilateral WXYZ. SPIRAL REVIEW Solve each equation. Round to the nearest hundredth if necessary. (Previous course) 47. (x - 3) 2 = 49 49. 4 (x + 2) 2 - 28 = 0 48. (x + 1) 2 - 4 = 0 Show that the quadrilateral with the given vertices is a parallelogram. (Lesson 6-3) 50. A (-2, -2) , B (1, 0) , C (5, 0) , D (2, -2) 51. J (1, 3) , K (3, 5) , L (6, 2) , M (4, 0) 52. Given that 58x = 26y, find the ratio y : x in simplest form. (Lesson 7-1) KEYWORD: MG7 Career Q: What math classes did you take in high school? A: Algebra, Geometry, and Probability and Statistics Q: What math-related classes did you take in college? A: Trigonometry, Precalculus, Drafting, and System Design Q: How do photogrammetrists use math? A: Photogrammetrists use aerial photographs to make detailed maps. To prepare maps, I use computers and perform a lot of scale measures to make sure the maps are accurate. Q: What are your future plans? A: My favorite part of making maps is designing scale drawings. Someday I’d like to apply these skills toward architectural work. Elaine Koch Photogrammetrist 494 494 Chapter 7 Similarity �� 7-6 Dilations and Similarity in the Coordinate Plane TEKS G.11.A Similarity and the geometry of shape: use ... properties and transformations to ... justify conjectures .... Objectives Apply similarity properties in the coordinate plane. Use coordinate proof to prove figures similar. Vocabulary dilation scale factor Also G.2.B, G.9.B Who uses this? Computer programmers use coordinates to enlarge or reduce images. Many photographs on the Web are in JPEG format, which is short for Joint Photographic Experts Group. When you drag a corner of a JPEG image in order to enlarge it
or reduce it, the underlying program uses coordinates and similarity to change the image’s size. A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar. A scale factor describes how much the figure is enlarged or reduced. For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b) → (ka, kb). E X A M P L E 1 Computer Graphics Application If the scale factor of a dilation is greater than 1 (k > 1) , it is an enlargement. If the scale factor is less than 1 (k < 1) , it is a reduction. The figure shows the position of a JPEG photo. Draw the border of the photo after a dilation with scale factor 3 __ . 2 Step 1 Multiply the vertices of the photo A (0, 0) , B (0, 4) , C (3, 4) , and D (3, 0) by 3 __ 2 . Rectangle ABCD Rectangle A'B'C'D' _ _ A (0, 0) → A' (0 ⋅ 3 ) → A' (0, 00, 4) → B' (0 ⋅ 3 ) → B' (0, 63, 4) → C' (3 ⋅ 3 ) → C' (4.5, 63, 0) → D' (3 ⋅ 3 ) → D' (4.5, 0) , 0 ⋅ 3 2 2 Step 2 Plot points A' (0, 0) , B' (0, 6), C' (4.5, 6), and D' (4.5, 0). Draw the rectangle. 1. What if…? Draw the border of the original photo after a dilation with scale factor 1 __ 2 . 7- 6 Dilations and Similarity in the Coordinate Plane 495 495 �� E X A M P L E 2 Finding Coordinates of Similar Triangles Given that △AOB ∼ △COD, find the coordinates of D and the scale factor. Since △AOB ∼ △COD, AO _ = OB _ CO OD = 3 _ 2 _ 4 OD Substitute 2 for AO, 4 for CO, and 3 for OB. 2OD = 12 OD = 6 Cross Products Prop. Divide both sides by 2. D lies on the x-axis, so its y-coordinate is 0. Since OD = 6, its x-coordinate must be 6. The coordinates of D are (6, 0) . (3, 0) → (3 ⋅ 2, 0 ⋅ 2) → (6, 0) , so the scale factor is 2. 2. Given that △MON ∼ △POQ and coordinates P (-15, 0) , M (-10, 0) , and Q (0, -30) , find the coordinates of N and the scale factor. E X A M P L E 3 Proving Triangles Are Similar Given: A (1, 5) , B (-1, 3) , C (3, 4) , D (-3, 1) , and E (5, 3) Prove: △ABC ∼ △ADE Step 1 Plot the points and draw the triangles. Step 2 Use the Distance Formula to find the side lengths. AB = √  (-1 - 1) 2 + (3 - 5) 2 AC = √  (3 - 1) 2 + (4 - 5 AD = √  (-3 - 1) 2 + (1 - 5) 2 AE = √  (5 - 1) 2 + (3 - 5) 2 = √  32 = 4 √  2 = √  20 = 2 √  5 Step 3 Find the similarity ratio. AB _ AD = AC _ AE √  Since AB ___ AD by SAS ∼. = AC ___ AE and ∠A ≅ ∠A by the Reflexive Property, △ABC ∼ △ADE 3. Given: R (-2, 0) , S (-3, 1) , T (0, 1) , U (-5, 3) , and V (4, 3) Prove: △RST ∼ △RUV 496 496 Chapter 7 Similarity �� E X A M P L E 4 Using the SSS Similarity Theorem Graph the image of △ABC after a dilation with scale factor 2. Verify that △A'B'C ' ∼ △ABC. Step 1 Multiply each coordinate by 2 to find the coordinates of the vertices of △A'B'C '. A (2, 3) → A' (2 ⋅ 2, 3 ⋅ 2) = A' (4, 6) B (0, 1) → B' (0 ⋅ 2, 1 ⋅ 2) = B' (0, 2) C (3, 0) → C' (3 ⋅ 2, 0 ⋅ 2) = C' (6, 0) Step 2 Graph △A'B'C '. Step 3 Use the Distance Formula to find the side lengths. AB = √  (2 - 0) 2 + (3 - 1) 2 A'B' = √  (4 - 0) 2 + (6 - 2 = √  32 = 4 √  2 BC = √  (3 - 0) 2 + (0 - 1) 2 B'C ' = √  (6 - 0) 2 + (0 - 2) 2 = √  10 = √  40 = 2 √  10 AC = √  (3 - 2) 2 + (0 - 3) 2 A'C ' = √  (6 - 4) 2 + (0 - 6) 2 = √  10 = √  40 = 2 √  10 Step 4 Find the similarity ratio. A'B' _ AB = , B'C ' _ = BC 2 √  10 _ √  10 = 2, A'C ' _ = AC 2 √  10 _ √  10 = 2 Since A'B' _ = B'C' _ BC = A'C' _ AC AB , △ABC ∼ △A'B'C ' by SSS ∼. 4. Graph the image of △MNP after a dilation with scale factor 3. Verify that △M'N'P' ∼ △MNP. THINK AND DISCUSS 1. △JKL has coordinates J (0, 0) , K (0, 2) , and L (3, 0) . Its image after a dilation has coordinates J' (0, 0) , K ' (0, 8) , and L' (12, 0) . Explain how to find the scale factor of the dilation. 2. GET ORGANIZED Copy and complete the graphic organizer. Write the definition of a dilation, a property of dilations, and an example and nonexample of a dilation. 7- 6 Dilations and Similarity in the Coordinate Plane 497 497 �� 7-6 Exercises Exercises KEYWORD: MG7 7-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. A ? is a transformation that proportionally reduces or enlarges a figure, ̶̶̶̶ such as the pupil of an eye. (dilation or scale factor) 2. A ratio that describes or determines the dimensional relationship of a figure to that which it represents, such as a map scale of 1 in. : 45 ft, is called a (dilation or scale factor) ? . ̶̶̶̶ . Graphic Design A designer created p. 495 this logo for a real estate agent but needs to make the logo twice as large for use on a sign. Draw the logo after a dilation with scale factor 2. Given that △AOB ∼ △COD, p. 496 find the coordinates of C and the scale factor. 5. Given that △ROS ∼ △POQ, find the coordinates of S and the scale factor. Given: A (0, 0) , B (-1, 1) , C (3, 2) , D (-2, 2) , and E (6, 4) p. 496 Prove: △ ABC ∼ △ADE 7. Given: J (-1, 0) , K (-3, -4) , L (3, -2) , M (-4, -6) , and N (5, -3) Prove: △ JKL ∼ △JMN . 497 Multi-Step Graph the image of each triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. 9. scale factor 3 __ 2 8. scale factor 2 498 498 Chapter 7 Similarity xy440ge07se_c07l06004a�� Independent Practice For See Exercises Example 10 11–12 13–14 15–16 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S17 Application Practice p. S34 PRACTICE AND PROBLEM SOLVING 10. Advertising A promoter produced this design for a street festival. She now wants to make the design smaller to use on postcards. Sketch the design after a dilation with scale factor 1 __ 2 . 11. Given that △UOV ∼ △XOY, find the coordinates of X and the scale factor. 12. Given that △MON ∼ △KOL, find the coordinates of K and the scale factor. 13. Given: D (-1, 3) , E (-3, -1) , F (3, -1) , G (-4, -3) , and H (5, -3) Prove: △DEF ∼ △DGH 14. Given: M (0, 10) , N (5, 0) , P (15, 15) , Q (10, -10) , and R (30, 20) Prove: △MNP ∼ △MQR Multi-Step Graph the image of each triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. 15. J (-2, 0) and K (-1, -1) , and L (-3, -2) with scale factor 3 16. M (0, 4) , N (4, 2) , and P (2, -2) with scale factor 1 __ 2 17. Critical Thinking Consider the transformation given by the mapping (x, y) → (2x, 4y) . Is this transformation a dilation? Why or why not? 18. /////ERROR ANALYSIS///// Which solution to find the scale factor of the dilation that maps △RST to △UVW is incorrect? Explain the error. 19. Write About It A dilation maps △ABC to △A'B 'C '. How is the scale factor of the dilation related to the similarity ratio of △ABC to △A'B 'C ' ? Explain. 20. This problem will prepare you for the Multi-Step TAKS Prep on page 502. a. In order to build a skateboard ramp, Miles draws △JKL on a coordinate plane. One unit on the drawing represents 60 cm of actual distance. Explain how he should assign coordinates for the vertices of △JKL. b. Graph the image of △JKL after a dilation with scale factor 3. 7- 6 Dilations and Similarity in the Coordinate Plane 499 499 yxc07106005a48840�� 21. Which coordinates for C make △COD similar to △AOB? (0, 2.4) (0, 2.5) (0, 3) (0, 3.6) 22. A dilation with scale factor 2 maps △RST to △R'S'T'. The perimeter of △RST is 60. What is the perimeter of △R'S'T'? 30 60 120 240 23. Which triangle with vertices D, E, and F is similar to △ABC? D (1, 2) , E (3, 2) , F (2, 0) D (-1, -2) , E (2, -2) , F (1, -5) D (1, 2) , E (5, 2) , F (3, 0) D (-2, -2) , E (0, 2) , F (-1, 0) 24. Gridded Resonse ̶̶ AB with endpoints A (3, 2) and B (7, 5) is dilated by a scale factor of 3. Find the length of ̶̶̶ A'B' . CHALLENGE AND EXTEND 25. How many different triangles having ̶̶ XY as a side are similar to △MNP? 26. △XYZ ∼ △MPN. Find the coordinates of Z. 27. A rectangle has two of its sides on the x- and y-axes, a vertex at the origin, and a vertex on the line y = 2x. Prove that any two such rectangles are similar. 28. △ ABC has vertices A (0, 1) , B (3, 1) , and C (1, 3) . △DEF has vertices D (1, -1) and E (7, -1) . Find two different locations for vertex F so that △ABC ∼ △DEF. SPIRAL REVIEW Write an inequality to represent the situation. (Previous course) 29. A weight lifter must lift at least 250 pounds. There are two 50-pound weights on a bar that weighs 5 pounds. Let w represent the additional weight that must be added to the bar. Find the length of each segment, given that (Lesson 5-2) ̶̶ HF 30. 31. ̶̶ JF ̶̶ DE ≅ ̶̶ FE . 32. ̶̶ CF △SUV ∼ △SRT. Find the length of each segment. (Lesson 7-4) 33. ̶̶ RT 34. ̶̶ V T 35. ̶̶ ST 500 500 Chapter 7 Similarity �� Direct Variation Algebra In Lesson 7-6 you learned that for two similar figures, the measure of each point was multiplied by the same scale factor. Is the relationship between the scale factor and the perimeter of the figure a direct variation? y Recall from algebra that if y varies directly as x, then y = kx, or where k is the constant of variation. __ x = k, See Skills Bank page S62 Example A rectangle has a length of 4 ft and a width of 2 ft. Find the relationship between the scale factors of similar rectangles and their corresponding perimeters. If the relationship is a direct variation, find the constant of variation. Step 1 Make a table to record
data. Scale Factor k 1 _ 2 2 3 4 5 Length ℓ = k (4) ℓ = 1 _ (4) = 2 2 Width w = k (2) w = 1 _ (2) = 1 2 Perimeter P = 2ℓ + 2w 2 (2) + 2 (1) = 6 8 12 16 20 4 6 8 10 24 36 48 60 Step 2 Graph the points ( 1 _ , 6) , (2, 24) , (3, 36) , (4, 48) , and (5, 60) . 2 Since the points are collinear and the line that contains them includes the origin, the relationship is a direct variation. Step 3 Find the equation of direct variation. y = kx 60 = k (5) Substitute 60 for y and 5 for x. 12 = k Divide both sides by 5. y = 12 x Substitute 12 for k. Thus the constant of variation is 12. Try This TAKS Grades 9–11 Obj. 3, 8, 10 Use the scale factors given in the above table. Find the relationship between the scale factors of similar figures and their corresponding perimeters. If the relationship is a direct variation, find the constant of variation. 1. regular hexagon with side length 6 2. triangle with side lengths 3, 6, and 7 3. square with side length 3 On Track for TAKS 501 501 �� SECTION 7B Applying Similarity Ramp It Up Many companies sell plans for build-it-yourself skateboard ramps. The figures below show a ramp and the plan for the triangular support structure at the ̶̶̶ GH , side of the ramp. In the plan, ̶̶ BC . and ̶̶ EF , ̶̶ JK are perpendicular to the base ̶̶ AB , 1. The instructions call for extra pieces of wood to ̶̶ GJ , and ̶̶ JC . Given AE = 42.2 cm, reinforce find EG, GJ, and JC to the nearest tenth. ̶̶ EG , ̶̶ AE , 2. Once the support structure is built, it is covered with a triangular piece of plywood. Find the area of the piece of wood needed to cover △ABC. A separate blueprint for the ramp uses a scale of 1 cm : 25 cm. What is the area of △ABC in the blueprint? 3. Before building the ramp, you transfer the plan to a coordinate plane. Draw △ABC on a coordinate plane so that 1 unit represents 25 cm and B is at the origin. Then draw the image of △ABC after a dilation with scale factor 3 __ 2 . 502 502 Chapter 7 Similarity �� Quiz for Lessons 7-4 Through 7-6 SECTION 7B 7-4 Applying Properties of Similar Triangles Find the length of each segment. ̶̶ ST 1. ̶̶ AB and ̶̶ AC 2. 3. An artist drew a picture of railroad tracks ̶̶ EF , such that the ties What is the length of ̶̶̶ GH , and ̶̶ FH ? ̶̶ JK are parallel. 7-5 Using Proportional Relationships The plan for a restaurant uses the scale of 1.5 in. : 60 ft. Find the actual length of the following walls. ̶̶ AB ̶̶ CD 4. 6. ̶̶ BC ̶̶ EF 5. 7. 8. A student who is 5 ft 3 in. tall measured her shadow and the shadow cast by a water tower shaped like a golf ball. What is the height of the tower? 7-6 Dilations and Similarity in the Coordinate Plane 9. Given: A (-1, 2) , B (-3, -2) , C (3, 0) , D (-2, 0) , and E (1, 1) Prove: △ADE ∼ △ABC 10. Given: R (0, 0) , S (-2, -1) , T (0, -3) , U (4, 2) , and V (0, 6) Prove: △RST ∼ △RUV Graph the image of each triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. 11. scale factor 3 12. scale factor 1.5 Ready to Go On? 503 503 ��ge07sec07rg2001aaAB40 ft5 ft 10 in.�� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary cross products . . . . . . . . . . . . . . 455 proportion . . . . . . . . . . . . . . . . . 455 scale factor . . . . . . . . . . . . . . . . . 495 dilation . . . . . . . . . . . . . . . . . . . . 495 ratio . . . . . . . . . . . . . . . . . . . . . . . 454 similar . . . . . . . . . . . . . . . . . . . . . 462 extremes . . . . . . . . . . . . . . . . . . . 455 scale . . . . . . . . . . . . . . . . . . . . . . . 489 similar polygons . . . . . . . . . . . . 462 indirect measurement . . . . . . . 488 scale drawing . . . . . . . . . . . . . . . 489 similarity ratio . . . . . . . . . . . . . 463 means . . . . . . . . . . . . . . . . . . . . . 455 Complete the sentences below with vocabulary words from the list above. 1. An equation stating that two ratios are equal is called a(n) ? . ̶̶̶̶ ? is a transformation that changes the size of a figure but not its shape. ̶̶̶̶ 2. A(n) 3. In the proportion u _ v = x _ y , the 4. A(n) ? compares two numbers by division. ̶̶̶̶ ? are v and x. ̶̶̶̶ 7-1 Ratio and Proportion (pp. 454–459) TEKS G.5.B, G.7.B, G.7.C, G.11.B E X A M P L E S EXERCISES ■ Write a ratio expressing the slope of ℓ. slope = rise _ run 1 - 3 = = 2 _ -4 = - 1 _ 2 Write a ratio expressing the slope of each line. 5. m 6. n 7. p ■ Solve the proportion. _ x - 3 50 2 _ = 4 (x - 3) 2 = 2 (50) 4 (x - 3) 2 4 (x - 3) = 100 2 (x - 3) = 25 x - 3 = ±5 Cross Products Prop. Simplify. Divide both sides by 4. Find the square root of both sides. x - 3 = 5 or x - 3 = -5 Rewrite as two eqns. x = 8 or x = -2 Add 3 to both sides. 504 504 Chapter 7 Similarity 8. If 84 is divided into three parts in the ratio 3 : 5 : 6, what is the sum of the smallest and the largest part? 9. The ratio of the measures of a pair of sides of a rectangle is 7 : 12. If the perimeter of the rectangle is 95, what is the length of each side? Solve each proportion. y = 9 _ _ 10. 7 3 = 9 _ x 12. x _ 4 = 3x _ 14. 12 _ 32 2x = 25 _ s 11. 10 _ 4 13 24 15. = = z - 1 _ 36 2 _ 3 (y + 1) �� 7-2 Ratios in Similar Polygons (pp. 462–467) TEKS G.5.B, G.11.A, G.11.B E X A M P L E EXERCISES ■ Determine whether △ABC and △DEF are similar. If so, write the similarity ratio and a similarity statement. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 16. rectangles JKLM and PQRS 17. △TUV and △WXY It is given that ∠A ≅ ∠D and ∠B ≅ ∠E. ∠C ≅ ∠F by the Third Angles Theorem. AB ___ = AC ___ DE DF is 2 __ 3 , and △ABC ∼ △DEF. = 2 __ 3 . Thus the similarity ratio = BC ___ EF 7-3 Triangle Similarity: AA, SSS, and SAS (pp. 470–477) TEKS G.5.B, G.9.B, G.11.A, G.11.B E X A M P L E ■ Given: ̶̶ AB ǁ ̶̶ CD , AB = 2CD, AC = 2CE Prove: △ABC ∼ △CDE Proof: Statements Reasons ̶̶ AB ǁ ̶̶ CD 1. 1. Given 2. ∠BAC ≅ ∠DCE 2. Corr.  Post. 3. AB = 2CD, AC = 2CE 3. Given = 2 4. AB ___ CD 5. AB ___ CD = 2, AC ___ CE = AC ___ CE 6. △ABC ∼ △CDE 4. Division Prop. 5. Trans. Prop. of = 6. SAS ∼ (Steps 2, 5) EXERCISES 18. Given: JL = 1 _ JN, JK = 1 _ 3 3 Prove: △JKL ∼ △JMN JM 19. Given: ̶̶ QR ǁ ̶̶ ST Prove: △PQR ∼ △PTS 20. Given: ̶̶ BD ǁ ̶̶ CE Prove: AB (CE) = AC (BD) (Hint: After you have proved the triangles similar, look for a proportion using AB, AC, CE, and BD, the lengths of corresponding sides.) Study Guide: Review 505 505 �� 7-4 Applying Properties of Similar Triangles (pp. 481–487) TEKS G.2.A, G.3.B, G.5.B, G.9.B, G.11.A, G.11.B EXERCISES Find each length. 21. CE 22. ST Verify that the given segments are parallel. 23. ̶̶ KL and ̶̶̶ MN 24. ̶̶ AB and ̶̶ CD 25. Find SU and SV. 26. Find the length of the third side of △ABC. 27. One side of a triangle is x inches longer than another side. The ray bisecting the angle formed by these sides divides the opposite side into 3-inch and 5-inch segments. Find the perimeter of the triangle in terms of x. E X A M P L E S ■ Find PQ. ̶̶ = PR ___ QR ǁ It is given that RT Triangle Proportionality Theorem. ̶̶ ST , so PQ ___ QS by the PQ = 15 _ _ 5 6 Substitute 5 for QS, 15 for PR, and 6 for RT. 6 (PQ) = 75 Cross Products Prop. PQ = 12.5 Divide both sides by 6. ̶̶ AB ǁ ̶̶ CD . = 1.5 EC _ CA ED _ DB ■ Verify that = 6 _ 4 = 4.5 _ 3 Since EC ___ = ED ___ , DB CA = 1.5 ̶̶ AB ǁ ̶̶ CD by the Converse of the Triangle Proportionality Theorem. ■ Find JL and LK. Since ̶̶ JK bisects ∠LJM, JL ___ LK = JM ___ MK by the Triangle Angle Bisector Theorem. 3x - 2 _ 2x = 12.5 _ 10 Substitute the given values. 10 (3x - 2) = 12.5 (2x) Cross Products Prop. 30x - 20 = 25x Simplify. 30x = 25x + 20 Add 20 to both sides. 5x = 20 Subtract 25x from both sides. x = 4 Divide both sides by 5. JL = 3x - 2 = 3 (4) - 2 = 10 LK = 2x = 2 (4) = 8 506 506 Chapter 7 Similarity �� 7-5 Using Proportional Relationships (pp. 488–494) TEKS G.1.B, G.5.A, G.11.A, G.11.B, G.11.D E X A M P L E EXERCISES ■ Use the dimensions in the diagram to find the height h of the tower. A student who is 5 ft 5 in. tall measured his shadow and a tower’s shadow to find the height of the tower. 28. To find the height of a flagpole, Casey measured her own shadow and the flagpole’s shadow. Given that Casey’s height is 5 ft 4 in., what is the height x of the flagpole? 5 ft 5 in. = 65 in. 1 ft 3 in. = 15 in. 11 ft 3 in. = 135 in. h _ 135 = 65 _ 15 15h = 65 (135) 15h = 8775 h = 585 in. Corr. sides are proportional. Cross Products Prop. Simplify. Divide both sides by 15. The height of the tower is 48 ft 9 in. 29. Jonathan is 3 ft from a lamppost that is 12 ft high. The lamppost and its shadow form the legs of a right triangle. Jonathan is 6 ft tall and is standing parallel to the lamppost. How long is Jonathan’s shadow? 7-6 Dilations and Similarity in the Coordinate Plane (pp. 495–500) E X A M P L E EXERCISES TEKS G.2.B, G.9.B, G.11.A ■ Given: A (5, -4) , B (-1, -2) , C (3, 0) , D (-4, -1) and E (2, 2) Prove: △ABC ∼ △ADE Proof: Plot the points and draw the triangles. 30. Given: R (1, -3) , S (-1, -1) , T (2, 0) , U (-3, 1) , and V (3, 3) Prove: △RST ∼ △RUV 31. Given: J (4, 4) , K (2, 3) , L (4, 2) , M (-4, 0) , and N (4, -4) Prove: △JKL ∼ △JMN 32. Given that △AOB ∼ △COD, find the coordinates of B and the scale factor. Use the Distance Formula to find the side lengths. AC = 2 √  5 , AE = 3 √  5 AB = 2 √  10 , AD = 3 √  10 = 2 _ . 3 Therefore AB _ AD = AC _ AE Since corresponding sides are proportional and ∠A ≅ ∠A by the Reflexive Property, △ABC ∼ △ADE by SAS ∼. 33. Graph the image of the triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangl
e. K (0, 3) , L (0, 0) , and M (4, 0) with scale factor 3. Study Guide: Review 507 507 �� 1. Two points on ℓ are A (-6, 4) and B (10, -6) . Write a ratio expressing the slope of ℓ. 2. Alana has a photograph that is 5 in. long and 3.5 in. wide. She enlarges it so that its length is 8 in. What is the width of the enlarged photograph? Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 3. △ABC and △MNP 4. rectangle DEFG and rectangle HJKL 5. Given: RSTU Prove: △RWV ∼ △SWT 6. Derrick is building a skateboard ramp as shown. Given that BD = DF = FG = 3 ft, find CD and EF to the nearest tenth. Find the length of each segment. 7. ̶̶ PR ̶̶̶ YW and ̶̶̶ WZ 8. 9. To find the height of a tree, a student measured the tree’s shadow and her own shadow. If the student’s height is 5 ft 8 in., what is the height of the tree? 10. The plan for a living room uses the scale of 1.5 in. : 30 ft. Use a ruler and find the length of the actual room’s diagonal ̶̶ AB . �� 11. Given: A (6, 5) , B (3, 4) , C (6, 3) , D (-3, 2) , and E (6, -1) Prove: △ABC ∼ △ADE 12. A quilter designed this patch for a quilt but needs a larger version for a different project. Draw the quilt patch after a dilation with scale factor 3 __ 2 . 508 508 Chapter 7 Similarity �� FOCUS ON SAT The SAT consists of seven test sections: three verbal, three math, and one more verbal or math section not used to compute your final score. The “extra” section is used to try out questions for future tests and to compare your score to previous tests. Read each question carefully and make sure you answer the question being asked. Check that your answer makes sense in the context of the problem. If you have time, check your work. You may want to time yourself as you take this practice test. It should take you about 8 minutes to complete. 1. In the figure below, the coordinates of the vertices are A (1, 5) , B (1, 1) , D (10, 1) , and E (10, -7) . If the length of the coordinates of C? ̶̶ CE is 10, what are Note: Figure not drawn to scale. (A) (4, 1) (B) (1, 4) (C) (7, 1) (D) (1, 7) (E) (6, 1) 2. In the figure below, triangles JKL and MKN are similar, and ℓ is parallel to segment JL. What is the length of ̶̶̶ KM ? Note: Figure not drawn to scale. (A) 4 (B) 8 (C) 9 (D) 13 (E) 18 3. Three siblings are to share an inheritance of $750,000 in the ratio 4 : 5 : 6. What is the amount of the greatest share? (A) $125,000 (B) $187,500 (C) $250,000 (D) $300,000 (E) $450,000 4. A 35-foot flagpole casts a 9-foot shadow at the same time that a girl casts a 1.2-foot shadow. How tall is the girl? (A) 3 feet 8 inches (B) 4 feet 6 inches (C) 4 feet 7 inches (D) 4 feet 8 inches (E) 5 feet 6 inches 5. What polygon is similar to every other polygon of the same name? (A) Triangle (B) Parallelogram (C) Rectangle (D) Square (E) Trapezoid College Entrance Exam Practice 509 509 �� Any Question Type: Interpret A Diagram When a diagram is included as part of a test question, do not make any assumptions about the diagram. Diagrams are not always drawn to scale and can be misleading if you are not careful. Multiple Choice What is DE ? 3.6 4 4.8 9 Make your own sketch of the diagram. Separate the two triangles so that you are able to find the side length measures. By redrawing the diagram, it is clear that the two triangles are similar. Set up a proportion to find DE. AB _ BC 6 _ 10 48 _ 10 DE = 4.8 = DE _ EF = DE _ 8 = DE The correct choice is C. Gridded Response △X′ Y ′Z′ is the image of △XYZ after a dilation with scale factor 1 __ . Find X ′Z′. 2 Before you begin, look at the scale of both the x-axis and the y-axis. Do not assume that the scale is always 1. At first glance, you might assume that XZ is 4. But by looking closely at the x-axis, notice that each increment represents 2 units. So XZ is actually 8. When △XYZ is dilated by a factor of 1 _ 2 , X′Z′ will be half of XZ. XZ = 1 _ X′Z′ = 1 _ 2 2 (8) = 4 510 510 Chapter 7 Similarity �� If the diagram does not match the given information, draw one that is more accurate. Item C Gridded Response What is the measure of MN? Read each test item and answer the questions that follow. Item A Multiple Choice Which ratio is the slope of m? 1 _ 15 1 _ 3 3 15 1. What is the scale of the y-axis? Use this scale to determine the rise of the slope. 2. What is the scale of the x-axis? Use this scale to determine the run of the slope. 3. Write the ratio that represents the slope of m. 4. Anna selected choice B as her answer. Is she correct? If not, what do you think she did wrong? Item B Gridded Response If ABCD ∼ MNOP and AC is 6, what is AB? 5. Examine the figures. Do you think longer or shorter than ̶̶̶ MN ? ̶̶ AB is 8. Describe how redrawing the figure can help you better understand the given information. 9. After reading this test question, a student redrew the figure as shown below. Explain if it is a correct interpretation of the original figure. If it is not, redraw and/or relabel it so that it is correct. Item D Multiple Choice Which is a similarity ratio for the triangles shown? 20 _ 1 10 _ 1 2 _ 1 15 _ 1 6. Do you think the drawings actually represent the given information? If not, explain why. 7. Create your own sketch of the figures to more accurately match the given information. 10. Chad determined that choice D was correct. Do you agree? If not, what do you think he did wrong? 11. Redraw the figures so that they are easier to understand. Write three statements that describe which vertices correspond to each other and three statements that describe which sides correspond to each other. TAKS Tackler 511 511 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–7 Multiple Choice 1. Which similarity statement is true for rectangles ABCD and MNPQ, given that AB = 3, AD = 4, MN = 6, and NP = 4.5? Rectangle ABCD ∼ rectangle MNPQ 5. If 12x = 16y, what is the ratio of x to y in simplest form Rectangle ABCD ∼ rectangle PQMN Use the diagram for Items 6 and 7. Rectangle ABCD ∼ rectangle MPNQ Rectangle ABCD ∼ rectangle QMNP 2. △ ABC has perpendicular bisectors If AP = 6 and ZP = 4.5, what is the length of to the nearest tenth? ̶̶ XP , ̶̶ YP , and ̶̶ BC ̶̶ ZP . 6. Given that ̶̶ AB ≅ ̶̶ CD , which additional information would be sufficient to prove that ABCD is a parallelogram? ̶̶ AB ǁ ̶̶ AC ǁ ̶̶ CD ̶̶ BD ∠CAB ≅ ∠CDB E is the midpoint of ̶̶̶ AD . 7. If   AC is parallel to   BD and m∠1 + m∠2 = 140°, what is the measure of ∠3? 20° 40° 50° 70° 8. If of ̶̶ AC is twice as long as ̶̶ DC ? ̶̶ AB , what is the length 2.5 centimeters 3.75 centimeters 5 centimeters 15 centimeters 4.0 7.9 9.0 12.7 3. What is the converse of the statement “If a quadrilateral has 4 congruent sides, then it is a rhombus”? If a quadrilateral is a rhombus, then it has 4 congruent sides. If a quadrilateral does not have 4 congruent sides, then it is not a rhombus. If a quadrilateral is not a rhombus, then it does not have 4 congruent sides. If a rhombus has 4 congruent sides, then it is a quadrilateral. 4. A blueprint for a hotel uses a scale of 3 in. : 100 ft. On the blueprint, the lobby has a width of 1.5 in. and a length of 2.25 in. If the carpeting for the lobby costs $1.25 per square foot, how much will the carpeting for the entire lobby cost? $312.50 $1406.25 $3000.00 $4687.50 512 512 Chapter 7 Similarity �� When writing proportions for similar figures, make sure that each ratio compares corresponding side lengths in each figure. STANDARDIZED TEST PREP Short Response 17. △ ABC has vertices A (-2, 0) , B (2, 2) , and C (2, -2) . △DEC has vertices D (0, -1) , E (2, 0) , and C (2, -2) . Prove that △ ABC ∼ △DEC. 9. What type of triangle has angles that measure (2x) °, (3x - 9) °, and (x + 27) °? 18. ∠TUV in the diagram below is an obtuse angle. Isosceles acute triangle Isosceles right triangle Scalene acute triangle Scalene obtuse triangle Use the diagram for Items 10 and 11. 10. Which of these points is the orthocenter of △FGH? F G H J Write an inequality showing the range of possible measurements for ∠TUW. Show your work or explain your answer. 19. △ ABC and △ ABD share side ̶̶ AB . Given that △ABC ∼ △ABD, use AAS to explain why these two triangles must also be congruent. 20. Rectangle ABCD has a length of 2.6 cm and a width of 1.8 cm. Rectangle WXYZ has a length of 7.8 cm and a width of 5.4 cm. Determine whether rectangle ABCD is similar to rectangle WXYZ. Explain your reasoning. 11. Which of the following could be the side lengths 21. If △ABC and △XYZ are similar triangles, there are of △FGH? FG = 2, GH = 3, and FH = 4 FG = 4, GH = 5, and FH = 6 FG = 5, GH = 4, and FH = 3 FG = 6, GH = 8, and FH = 10 12. The measure of one of the exterior angles of a right triangle is 120°. What are the measures of the acute interior angles of the triangle? 30° and 60° 40° and 50° 40° and 80° 60° and 60° Gridded Response 13. The ratio of a football field’s length to its width is 9 : 4. If the length of the field is 360 ft, what is the width of the field in feet? 14. The sum of the measures of the interior angles of a convex polygon is 1260°. How many sides does the polygon have? 15. In kite PQRS, ∠P and ∠R are opposite angles. If m∠P = 25° and m∠R = 75°, what is the measure of ∠Q in degrees? 16. Heather is 1.6 m tall and casts a shadow of 3.5 m. At the same time, a barn casts a shadow of 17.5 m. Find the height of the barn in meters. six possible similarity statements. a. What is the probability that △ABC ∼ △XYZ is correct? b. If △ABC and △XYZ are isosceles, what is the probability that △ABC ∼ △XYZ? c. If △ABC and △XYZ are equilateral, what is the probability that △ABC ∼ △XYZ? Explain. Extend
ed Response 22. a. Given: △SRT ∼ △VUW and ̶̶ VU ≅ ̶̶̶ VW Prove: ̶̶ SR ≅ ̶̶ ST b. Explain in words how you determine the possible values for x and y that would make the two triangles below similar. Note: Triangles not drawn to scale. c. Explain why x cannot have a value of 1 if the two triangles in the diagram above are similar. Cumulative Assessment, Chapters 1–7 513 513 �� Right Triangles and Trigonometry 8A Trigonometric Ratios 8-1 Similarity in Right Triangles Lab Explore Trigonometric Ratios 8-2 Trigonometric Ratios 8-3 Solving Right Triangles 8B Applying Trigonometric Ratios 8-4 Angles of Elevation and Depression Lab Indirect Measurement Using Trigonometry 8-5 Law of Sines and Law of Cosines 8-6 Vectors Ext Trigonometry and the Unit Circle KEYWORD: MG7 ChProj Standing over 567 feet tall, the San Jacinto Monument in LaPorte is the tallest memorial column in the world. 514 514 Chapter 8 Vocabulary Match each term on the left with a definition on the right. 1. altitude A. a comparison of two numbers by division 2. proportion 3. ratio 4. right triangle B. a segment from a vertex to the midpoint of the opposite side of a triangle C. an equation stating that two ratios are equal D. a perpendicular segment from the vertex of a triangle to a line containing the base E. a triangle that contains a right angle Identify Similar Figures Determine if the two triangles are similar. 5. 6. Special Right Triangles Find the value of x. Give the answer in simplest radical form. 7. 8. 9. 10. Solve Multi-Step Equations Solve each equation. 11. 3 (x - 1) = 12 13. 6 = 8 (x - 3) 12. -2 (y + 5) = -1 14. 2 = -1 (z + 4) Solve Proportions Solve each proportion. 15. 4_ y = 6_ 18 16. 5_ 8 = x_ 32 17. m_ 9 = 8_ 12 18. y_ 4 = 9_ y Rounding and Estimation Round each decimal to the indicated place value. 19. 13.118; hundredth 20. 37.91; tenth 21. 15.992; tenth 22. 173.05; whole number Right Triangles and Trigonometry 515515 �� Key Vocabulary/Vocabulario angle of depression ángulo de depresión angle of elevation ángulo de elevación cosine coseno geometric mean media geométrica sine tangent seno tangente trigonometric ratio razón trigonométrica vector vector Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The term angle of elevation includes the word elevation. What does elevate mean in everyday usage? What do you think an angle of elevation might be? 2. A vector is sometimes defined as “a directed line segment.” How can you use this definition to understand this term? 3. The word trigonometric comes from the Greek word trigonon, which means “triangle,” and the suffix metric, which means “measurement.” Based on this, how do you think you might use a trigonometric ratio ? Geometry TEKS 8-2 Tech. Lab Les. 8-1 Les. 8-2 Les. 8-3 Les. 8-4 8-4 Geo. Lab Les. 8-5 Les. 8-6 Ext. G.1.B Geometric structure* recognize the historical development of ★ geometric systems ... G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships ★ G.5.B Geometric patterns* use … numeric and geometric patterns to make generalizations about geometric properties, including … ratios in similar figures ... ★ ★ ★ ★ ★ G.5.D Geometric patterns* identify and apply patterns from right ★ ★ ★ ★ ★ ★ triangles to solve meaningful problems … G.7.A Dimensionality and the geometry of location* use … two- ★ ★ ★ ★ dimensional coordinate systems … G.8.C Congruence and the geometry of size* … use the ★ ★ ★ Pythagorean Theorem G.10.A Congruence and geometry of size* use congruence transformations to … justify properties of geometric figures including figures represented on a coordinate plane G.11.A Similarity and the geometry of shape* use and extend similarity properties and transformations to explore and justify conjectures about geometric figures G.11.B Similarity and the geometry of shape* use ratios to solve problems involving similar figures G.11.C Similarity and the geometry of shape* develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, … ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 516 516 Chapter 8 Reading Strategy: Read to Understand As you read a lesson, read with a purpose. Lessons are about one or two specific objectives. These objectives are at the top of the first page of every lesson. Reading with the objectives in mind can help you understand the lesson. Identify similar polygons. Lesson 7-2 Ratios in Similar Polygons Figures that are similar (∼) have the same shape but not necessarily the same size. △1 is similar to △2 (△1 ∼ △2) . △1 is not similar to △3 (△1 ≁ △3) . Identify the objectives of the lesson. Read through the lesson to find where the objectives are explained. • Can two polygons be both similar and congruent? • In Example 1, the triangles are not oriented the same. How can you tell which angles are congruent and which sides are corresponding? List any questions, problems, or trouble spots you may have. • Similarity is represented by the symbol ∼. Congruence is represented by the symbol ≅. • Similar: same shape but not necessarily the same size Write down any new vocabulary or symbols. Try This Use Lesson 8-1 to complete each of the following. 1. What are the objectives of the lesson? 2. Identify any new vocabulary, formulas, and symbols. 3. Identify any examples that you need clarified. 4. Make a list of questions you need answered during class. Right Triangles and Trigonometry 517 517 �� 8-1 Similarity in Right Triangles TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as right triangle ratios .... Objectives Use geometric mean to find segment lengths in right triangles. Apply similarity relationships in right triangles to solve problems. Vocabulary geometric mean Also G.5.B, G.5.D, G.8.A, G.8.C, G.11.A, G.11.B Why learn this? You can use similarity relationships in right triangles to find the height of Big Tex. Big Tex debuted as the official symbol of the State Fair of Texas in 1952. This 6000-pound cowboy wears size 70 boots and a 75-gallon hat. In this lesson, you will learn how to use right triangle relationships to find Big Tex’s height. In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles. Theorem 8-1-1 The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. △ABC ∼ △ACD ∼ △CBD PROOF PROOF Theorem 8-1-1 ̶̶ CD . Given: △ABC is a right triangle with altitude Prove: △ABC ∼ △ACD ∼ △CBD Proof: The right angles in △ABC, △ACD, and △CBD are all congruent. By the Reflexive Property of Congruence, ∠A ≅ ∠A. Therefore △ABC ∼ △ACD by the AA Similarity Theorem. Similarly, ∠B ≅ ∠B, so △ABC ∼ △CBD. By the Transitive Property of Similarity, △ABC ∼ △ACD ∼ △CBD. E X A M P L E 1 Identifying Similar Right Triangles Write a similarity statement comparing the three triangles. Sketch the three right triangles with the angles of the triangles in corresponding positions. By Theorem 8-1-1, △RST ∼ △SPT ∼ △RPS. 1. Write a similarity statement comparing the three triangles. 518 518 Chapter 8 Right Triangles and Trigonometry �� Consider the proportion a __ x = x __ the same number, and that number is the geometric mean of the extremes. The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that x = √  ab , or x 2 = ab. . In this case, the means of the proportion are b E X A M P L E 2 Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. A 4 and 9 Let x be the geometric mean. x 2 = (4) (9) = 36 x = 6 Def. of geometric mean Find the positive square root. B 6 and 15 Let x be the geometric mean. x 2 = (6) (15) = 90 x = √  90 = 3 √  10 Def. of geometric mean Find the positive square root. Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 2a. 2 and 8 2b. 10 and 30 2c. 8 and 9 You can use Theorem 8-1-1 to write proportions comparing the side lengths of the triangles formed by the altitude to the hypotenuse of a right triangle. All the relationships in red involve geometric means Corollaries Geometric Means COROLLARY EXAMPLE DIAGRAM 8-1-2 The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. 8-1-3 The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. h 2 = xy a 2 = xc b 2 = yc 8-1 Similarity in Right Triangles 519 519 �� E X A M P L E 3 Finding Side Lengths in Right Triangles Once you’ve found the unknown side lengths, you can use the Pythagorean Theorem to check your answers. Find x, y, and z. x 2 = (2) (10) = 20 x = √  20 = 2 √  5 y 2 = (12) (10) = 120 120 = 2 √  30 y = √  z 2 = (12) (2) = 24 z = √  24 = 2 √  6 x is the geometric mean of 2 and 10. Find the positive square root. y is the geometric mean of 12 and 10. Find the positive square root. z is the geometric mean of 12 and 2. Find the positive square root. 3. Find u, v, and w. E X A M P L E 4 Measurement Application To estimate the height of Big Tex at the State Fair of Texas, Michael steps away from the statue until his line of sight to the top of the statue and his line of sight to the bottom of the statue form a 90° angle. His eyes are 5 ft above the ground, and he is standing 15 ft 3 in. from Big Tex. How tall is Big Tex to the ne
arest foot? Let x be the height of Big Tex above eye level. 15 ft 3 in. = 15.25 ft Convert 3 in. to 0.25 ft. (15.25) 2 = 5x 15.25 is the geometric mean of 5 and x. x = 46.5125 ≈ 47 Solve for x and round. Big Tex is about 47 + 5, or 52 ft tall. 4. A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom form a right angle as shown. What is the height of the cliff to the nearest foot? THINK AND DISCUSS 1. Explain how to find the geometric mean of 7 and 21. 2. GET ORGANIZED Copy and complete the graphic organizer. Label the right triangle and draw the altitude to the hypotenuse. In each box, write a proportion in which the given segment is a geometric mean. 520 520 Chapter 8 Right Triangles and Trigonometry �� 8-1 Exercises Exercises KEYWORD: MG7 8-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In the proportion 2 __ 8 = 8 __ 32 , which number is the geometric mean of the other two numbers Write a similarity statement comparing the three triangles in each diagram. p. 518 2. 3. 4. 519 Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 5. 2 and 50 8. 9 and 12 6. 4 and 16 9. 16 and 25 7. 1 _ 2 and 8 10. 7 and 11 Find x, y, and z. p. 520 11. 12. 13. 520 14. Measurement To estimate the length of the USS Constitution in Boston harbor, a student locates points T and U as shown. What is RS to the nearest tenth? PRACTICE AND PROBLEM SOLVING Independent Practice Write a similarity statement comparing the three triangles in each diagram. For See Exercises Example 15. 16. 17. 15–17 18–23 24–26 27 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S18 Application Practice p. S35 Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 18. 5 and 45 21. 1 _ 4 and 80 Find x, y, and z. 24. 19. 3 and 15 22. 1.5 and 12 20. 5 and 8 and 27 _ 23. 2 _ 40 3 25. 26. 8-1 Similarity in Right Triangles 521 521 �� 27. Measurement To estimate the height of the Taipei 101 tower, Andrew stands so that his lines of sight to the top and bottom of the tower form a 90° angle. What is the height of the tower to the nearest foot? 28. The geometric mean of two numbers is 8. One of the numbers is 2. Find the other number. 29. The geometric mean of two numbers is 2 √  5 . One of the numbers is 6. Find the other number. Use the diagram to complete each equation. 30 33. ? 31. ? _ u = u _ x 34. (?) 2 = y (x + y) 32. x + y _ v 35. u 2 = (x + y) (?) = v _ ? Give each answer in simplest radical form. 36. AD = 12, and CD = 8. Find BD. 37. AC = 16, and CD = 5. Find BC. 38. AD = CD = √  2 . Find BD. 39. BC = √  5 , and AC = √  10 . Find CD. 40. Finance An investment returns 3% one year and 10% the next year. The average rate of return is the geometric mean of the two annual rates. What is the average rate of return for this investment to the nearest tenth of a percent? 41. /////ERROR ANALYSIS///// Two students were asked to find EF. Which solution is incorrect? Explain the error. 42. The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments that are 2 cm long and 5 cm long. Find the length of the altitude to the nearest tenth of a centimeter. 43. Critical Thinking Use the figure to show how Corollary 8-1-3 can be used to derive the Pythagorean Theorem. (Hint: Use the corollary to write expressions for a 2 and b 2 . Then add the expressions.) 44. This problem will prepare you for the Multi-Step TAKS Prep on page 542. Before installing a utility pole, a crew must first dig a hole and install the anchor for the guy wire ̶̶ RT , that supports the pole. In the diagram, ̶̶̶ ̶̶̶ WT , RS = 4 ft, and ST = 3 ft. RW ⊥ a. Find the depth of the anchor ̶̶̶ SW to the ̶̶̶ SW ⊥ nearest inch. b. Find the length of the rod ̶̶̶ RW to the nearest inch. 522 522 Chapter 8 Right Triangles and Trigonometry � � � � �� 5 ft91 ft 3 in.ge07sec08l01003a�� 45. Write About It Suppose the rectangle and square have the same area. Explain why s must be the geometric mean of a and b. 46. Write About It Explain why the geometric mean of two perfect squares must be a whole number. 47. Lee is building a skateboard ramp based on the plan shown. Which is closest to the length of the ramp from point X to point Y? 4.9 feet 5.7 feet 8.5 feet 9.4 feet 48. What is the area of △ABC? 18 square meters 36 square meters 39 square meters 78 square meters 49. Which expression represents the length of ̶̶ RS ? √  y + 1) CHALLENGE AND EXTEND 50. Algebra An 8-inch-long altitude of a right triangle divides the hypotenuse into two segments. One segment is 4 times as long as the other. What are the lengths of the segments of the hypotenuse? 51. Use similarity in right triangles to find x, y, and z. 52. Prove the following. If the altitude to the hypotenuse of a right triangle bisects the hypotenuse, then the triangle is a 45°-45°-90° right triangle. 53. Multi-Step Find AC and AB to the nearest hundredth. SPIRAL REVIEW Find the x-intercept and y-intercept for each equation. (Previous course) 54. 3y + 4 = 6x 55. x + 4 = 2y 56. 3y - 15 = 15x The leg lengths of a 30°-60°-90° triangle are given. Find the length of the hypotenuse. (Lesson 5-8) 57. 3 and √  27 58. 7 and 7 √  3 59. 2 and 2 √  3 For rhombus ABCD, find each measure, given that m∠DEC = 30y°, m∠EDC = (8y + 15) °, AB = 2x + 8, and BC = 4x. (Lesson 6-4) 61. m∠EDA 60. m∠EDC 62. AB 8-1 Similarity in Right Triangles 523 523 �� 8-2 Explore Trigonometric Ratios In a right triangle, the ratio of two side lengths is known as a trigonometric ratio. Use with Lesson 8-2 Activity TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.5.B, G.9.B, G.11.A KEYWORD: MG7 Lab8 1 Construct three points and label them A, B, and C. Construct rays endpoint A. Move C so that ∠A is an acute angle.  AC with common  AB and 2 Construct point D on  AC . Construct a line  AB . Label the through D perpendicular to intersection of the perpendicular line and  AB as E. 3 Measure ∠A. Measure DE, AE, and AD, the side lengths of △AED. 4 Calculate the ratios DE _ AD , AE _ AD , and DE _ . AE Try This 1. Drag D along  AC . What happens to the measure of ∠A as D moves? What postulate or theorem guarantees that the different triangles formed are similar to each other? 2. As you move D, what happens to the values of the three ratios you calculated? Use the properties of similar triangles to explain this result. 3. Move C. What happens to the measure of ∠A? With a new value for m∠A, note the values of the three ratios. What happens to the ratios if you drag D? 4. Move C until DE ___ AD = AE ___ AD . What is the value of DE ___ AE ? What is the measure of ∠A? Use the properties of special right triangles to justify this result. 524 524 Chapter 8 Right Triangles and Trigonometry 8-2 Trigonometric Ratios TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as ... trigonometric ratios .... Objectives Find the sine, cosine, and tangent of an acute angle. Use trigonometric ratios to find side lengths in right triangles and to solve real-world problems. Vocabulary trigonometric ratio sine cosine tangent Also G.5.B, G.5.D, G.8.C, G.11.B Who uses this? Contractors use trigonometric ratios to build ramps that meet legal requirements. According to the Americans with Disabilities Act (ADA), the maximum slope allowed for a wheelchair ramp is 1__ 12 , which is an angle of about 4.8°. Properties of right triangles help builders construct ramps that meet this requirement. By the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure. So △ABC ∼ △DEF ∼△XYZ, and BC ___ . These are trigonometric AC ratios. A trigonometric ratio is a ratio of two sides of a right triangle. = EF ___ DF = YZ ___ XZ Trigonometric Ratios DEFINITION The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example, the sine of ∠A is written as sin A. DIAGRAM SYMBOLS __ _ a = c _ __ b = c opposite leg hypotenuse opposite leg hypotenuse sin A = sin B = cos A = cos B = _ __ b = c _ __ a = c adjacent leg hypotenuse adjacent leg hypotenuse tan A = tan B = _ __ a = b _ __ b = a opposite leg adjacent leg opposite leg adjacent leg E X A M P L E 1 Finding Trigonometric Ratios Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. A sin R sin R = 12 _ 13 ≈ 0.92 The sine of an ∠ is opp. leg _ hyp. . 8-2 Trigonometric Ratios 525 525 �� Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. B cos R cos R = 5 _ 13 ≈ 0.38 C tan S tan S = 5 _ 12 ≈ 0.42 The cosine of an ∠ is adj. leg _ . hyp. The tangent of an ∠ is opp. leg _ . adj. leg Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. 1a. c
os A 1b. tan B 1c. sin B E X A M P L E 2 Finding Trigonometric Ratios in Special Right Triangles Use a special right triangle to write sin 60° as a fraction. Draw and label a 30°-60°-90° △. sin 60° = s √  3 _ 2s = √  3 _ 2 The sine of an ∠ is opp. leg _ hyp. . 2. Use a special right triangle to write tan 45° as a fraction. E X A M P L E 3 Calculating Trigonometric Ratios Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. A cos 76° B sin 8° C tan 82° Be sure your calculator is in degree mode, not radian mode. cos 76° ≈ 0.24 sin 8° ≈ 0.14 tan 82° ≈ 7.12 Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 3a. tan 11° 3b. sin 62° 3c. cos 30° The hypotenuse is always the longest side of a right triangle. So the denominator of a sine or cosine ratio is always greater than the numerator. Therefore the sine and cosine of an acute angle are always positive numbers less than 1. Since the tangent of an acute angle is the ratio of the lengths of the legs, it can have any value greater than 0. 526 526 Chapter 8 Right Triangles and Trigonometry �� E X A M P L E 4 Using Trigonometric Ratios to Find Lengths Find each length. Round to the nearest hundredth. A AB ̶̶ AB is adjacent to the given angle, ∠A. You are given BC, which is opposite ∠A. Since the adjacent and opposite legs are involved, use a tangent ratio. Do not round until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator. tan A = opp. leg _ adj. leg = BC _ AB Write a trigonometric ratio. tan 41° = 6.1 _ AB AB = 6.1 _ tan 41° AB ≈ 7.02 in. Substitute the given values. Multiply both sides by AB and divide by tan 41°. Simplify the expression. B MP ̶̶̶ MP is opposite the given angle, ∠N. You are given NP, which is the hypotenuse. Since the opposite side and hypotenuse are involved, use a sine ratio. = MP _ NP sin N = opp. leg _ hyp. sin 20° = MP _ 8.7 8.7 (sin 20°) = MP Write a trigonometric ratio. Substitute the given values. Multiply both sides by 8.7. MP ≈ 2.98 cm Simplify the expression. C YZ YZ is the hypotenuse. You are given XZ, which is adjacent to the given angle, ∠Z. Since the adjacent side and hypotenuse are involved, use a cosine ratio. cos Z = adj. leg _ hyp. cos 38° = 12.6 _ = XZ _ YZ YZ YZ = 12.6 _ cos 38° Write a trigonometric ratio. Substitute the given values. Multiply both sides by YZ and divide by cos 38°. YZ ≈ 15.99 cm Simplify the expression. Find each length. Round to the nearest hundredth. 4a. DF 4b. ST 4c. BC 4d. JL 8-2 Trigonometric Ratios 527 527 �� E X A M P L E 5 Problem Solving Application A contractor is building a wheelchair ramp for a doorway that is 1.2 ft above the ground. To meet ADA guidelines, the ramp will make an angle of 4.8° with the ground. To the nearest hundredth of a foot, what is the horizontal distance covered by the ramp? Understand the Problem Make a sketch. The answer is BC. Make a Plan ̶̶ BC is the leg adjacent to ∠C. You are given AB, which is the leg opposite ∠C. Since the opposite and adjacent legs are involved, write an equation using the tangent ratio. Solve tan C = AB _ BC tan 4.8° = 1.2 _ BC BC = 1.2 _ tan 4.8° Write a trigonometric ratio. Substitute the given values. Multiply both sides by BC and divide by tan 4.8°. BC ≈ 14.2904 ft Simplify the expression. Look Back The problem asks for BC rounded to the nearest hundredth, so round the length to 14.29. The ramp covers a horizontal distance of 14.29 ft. 5. Find AC, the length of the ramp in Example 5, to the nearest hundredth of a foot. THINK AND DISCUSS 1. Tell how you could use a sine ratio to find AB. 2. Tell how you could use a cosine ratio to find AB. 3. GET ORGANIZED Copy and complete the graphic organizer. In each cell, write the meaning of each abbreviation and draw a diagram for each. 528 528 Chapter 8 Right Triangles and Trigonometry ge07sec08l02002aABCAB12��34�� 8-2 Exercises Exercises KEYWORD: MG7 8-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In △JKL, ∠K is a right angle. Write the sine of ∠J as a ratio of side lengths. 2. In △MNP, ∠M is a right angle. Write the tangent of ∠N as a ratio of side lengths. 525 Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. 3. sin C 6. cos C 4. tan A 7. tan C 5. cos A 8. sin Use a special right triangle to write each trigonometric ratio as a fraction. p. 526 9. cos 60° 10. tan 30° 11. sin 45. 526 Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 12. tan 67° 15. cos 88° 13. sin 23° 16. cos 12° 14. sin 49° 17. tan 9 Find each length. Round to the nearest hundredth. p. 527 18. BC 19. QR 20. KL . 528 21. Architecture A pediment has a pitch of 15°, as shown. If the width of the pediment, WZ, is 56 ft, what is XY to the nearest inch? �� Independent Practice Write each trigonometric ratio as a fraction and as a PRACTICE AND PROBLEM SOLVING For See Exercises Example 22–27 28–30 31–36 37–42 43 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S18 Application Practice p. S35 decimal rounded to the nearest hundredth. 22. cos D 25. cos F 23. tan D 26. sin F 24. tan F 27. sin D Use a special right triangle to write each trigonometric ratio as a fraction. 28. tan 60° 29. sin 30° 30. cos 45° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 31. tan 51° 34. tan 14° 32. sin 80° 35. sin 55° 33. cos 77° 36. cos 48° 8-2 Trigonometric Ratios 529 529 �� Find each length. Round to the nearest hundredth. 37. PQ 38. AC 39. GH 40. 40. XZ 41. KL 42. EF 43. Sports A jump ramp for waterskiing makes an angle of 15° with the surface of the water. The ramp rises 1.58 m above the surface. What is the length of the ramp to the nearest hundredth of a meter? �� Use special right triangles to complete each statement. �� 44. An angle that measures 45. For a 45° angle, the 46. The sine of a ? has a tangent of 1. ̶̶̶̶ ? and ̶̶̶̶ ? angle is 0.5. ̶̶̶̶ ? ratios are equal. ̶̶̶̶ Sports The Aquaplex Ski Lake in Austin hosted the 2004 and 2005 Barefoot National Championships. Barefoot skiing began in 1947. The first U.S. national competition was held in Waco in September 1978. 47. The cosine of a 30° angle is equal to the sine of a ? angle. ̶̶̶̶ 48. Safety According to the Occupational Safety and Health Administration (OSHA), a ladder that is placed against a wall should make a 75.5° angle with the ground for optimal safety. To the nearest tenth of a foot, what is the maximum height that a 10-ft ladder can safely reach? Find the indicated length in each rectangle. Round to the nearest tenth. 49. BC 50. SU 51. Critical Thinking For what angle measures is the tangent ratio less than 1? greater than 1? Explain. 52. This problem will prepare you for the Multi-Step TAKS Prep on page 542. ̶̶ AB at A utility worker is installing a 25-foot pole ̶̶ AD , will help ̶̶ AC and the foot of a hill. Two guy wires, keep the pole vertical. a. To the nearest inch, how long should b. ̶̶ AD is perpendicular to the hill, which makes an angle of 28° with a horizontal line. To the nearest inch, how long should this guy wire be? ̶̶ AC be? 530 530 Chapter 8 Right Triangles and Trigonometry �� 53. Find the sine of the smaller acute angle in a triangle with side lengths of 3, 4, and 5 inches. History 54. Find the tangent of the greater acute angle in a triangle with side lengths of 7, 24, and 25 centimeters. The Pyramid of Cheops consists of more than 2,000,000 blocks of stone with an average weight of 2.5 tons each. 55. History The Great Pyramid of Cheops in Giza, Egypt, was completed around 2566 B.C.E. Its original height was 482 ft. Each face of the pyramid forms a 52° angle with the ground. To the nearest foot, how long is the base of the pyramid? 56. Measurement Follow these steps to calculate trigonometric ratios. a. Use a centimeter ruler to find AB, BC, and AC. b. Use your measurements from part a to find the sine, cosine, and tangent of ∠A. c. Use a protractor to find m∠A. d. Use a calculator to find the sine, cosine, and tangent of ∠A. e. How do the values in part d compare to the ones you found in part b? 57. Algebra Recall from Algebra I that an identity is an equation that is true for all values of the variables. a. Show that the identity tan A = sin A _ is true when m∠A = 30°. cos A b. Write tan A, sin A, and cos A in terms of a, b, and c. c. Use your results from part b to prove the identity tan A = sin A _ . cos A Verify that (sin A) 2 + (cos A) 2 = 1 for each angle measure. 58. m∠A = 45° 61. Multi-Step The equation (sin A) 2 + (cos A) 2 = 1 is known as a 59. m∠ A = 30° 60. m∠A = 60° Pythagorean Identity. a. Write sin A and cos A in terms of a, b, and c. b. Use your results from part a to prove the identity (sin A) 2 + (cos A) 2 = 1. c. Write About It Why do you think this identity is called a Pythagorean identity? Find the perimeter and area of each triangle. Round to the nearest hundredth. 62. 64. 63. 65. 66. Critical Thinking Draw △ABC with ∠C a right angle. Write sin A and cos B in terms of the side lengths of the triangle. What do you notice? How are ∠A and ∠B related? Make a conjecture based on your observations. 67. Write About It Explain how the tangent of an acute angle changes as the angle measure increases. 8-2 Trigonometric Ratios 531 531 �� 68. Which expression can be used to find AB? 7.1 (sin 25°) 7.1 (cos 25°) 7.1 (sin 65°) 7.1 (tan 65°) 69. A steel cable supports an electrical tower as shown. The cable makes a

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