AMSCO'S Geometry. New York - Rye High School

More documents

Recommendations

Info

406 QuadrilateralsProofy D(d, e) C(c, e)M NA(0, 0) B(b, 0) xTheorem 10.23We will use a coordinate proof.Place the trapezoid so that the parallel sides are on horizontal lines. PlaceAB on the x-axis with (0, 0) the coordinates of A and (b, 0) the coordinates ofB, and b 0.Place CD on a line parallel to the x-axis. Every point on a line parallel to thex-axis has the same y-coordinate. Let (c, e) be the coordinates of C and(d, e) be the coordinates of D, and c, d, e 0.Since M is the midpoint of AD, the coordinates of M areA 0 1 2 d , 0 1 2 e B 5 A d 2 , 2 e B .Since N is the midpoint of BC, the coordinates of N areA b 1 c2 , 0 1 e2 B 5 A b 1 c2 , 2 e B .Points that have the same y-coordinate are on the same horizontal line.Therefore, MN is a horizontal line. All horizontal lines are parallel. Therefore,MN AB and MN CD.The length of the median of a trapezoid is equal to one-half the sum of thelengths of the bases.Given Trapezoid ABCD with AB CD, M themidpoint of AD, and N the midpoint ofCDyD(d, e)MC(c, e)NProveMN 12 (AB 1 CD)A(0, 0)B(b, 0)xStrategyUse a coordinate proof. Let the coordinates of A, B, C, D, M, and N be thoseused in the proof of Theorem 10.22. The length of a horizontal line segment isthe absolute value of the difference of the x-coordinates of the endpoints.The proof of this theorem is left to the student. (See exercise 19.)EXAMPLE 1The coordinates of the vertices of ABCD are A(0, 0), B(4, 1), C(5, 2), andD(2, 6). Prove that ABCD is a trapezoid.21 2 0Proof Slope of AB Slope of BC 4 2 0 5 214 521 42 2 (21)5 2 4 5 3 1 5 36 2 2Slope of CD 2 2 5 5 23 4 524 3Slope of DA 0 0 2 2 62 5 2622 5 3
The Trapezoid 407The slopes of BC and DA are equal. Therefore, BC and DA are parallel.The slopes of AB and CD are not equal. Therefore, AB and CD are not parallel.Because quadrilateral ABCD has only one pair of parallel sides, it isa trapezoid.EXAMPLE 2SolutionIn quadrilateral ABCD,mA 105, mB 75, and mC 75.a. Is ABCD a parallelogram? Justify your answer.b. Is ABCD a trapezoid? Justify your answer.c. Is ABCD an isosceles trapezoid? Justify your answer.d. If AB x, BC 2x 1, CD 3x 8, and DA x 1,find the measure of each side of the quadrilateral.a. One pair of opposite angles of ABCD are A and C.Since these angles are not congruent, the quadrilateral isnot a parallelogram. The quadrilateral does not have twopairs of parallel sides.b. A and B are interior angles on the same side of transversal AB andthey are supplementary. Therefore, AD and BC are parallel. The quadrilateralhas only one pair of parallel sides and is therefore a trapezoid.c. Because B and C are congruent base angles of the trapezoid, thetrapezoid is isosceles.d. The congruent legs of the trapezoid are AB and CD.AB CDx 3x 82x 8x 4D105°A75°C75°BAB x BC 2x 1 CD 3x 8 DA x 1 4 2(4) 1 3(4) 8 4 1 8 1 12 8 5 7 4
Page 1 and 2:
AMSCO’SGEOMETRYAnn Xavier Gantert
Page 3 and 4:
PREFACE✔✔✔✔✔✔Geometry i
Page 5:
CONTENTSChapter 1ESSENTIALS OF GEOM
Page 8 and 9:
viiiCONTENTSReview Exercises 323Cum
Page 10 and 11:
xCONTENTS14-3 Five Fundamental Loci
Page 12 and 13:
2 Essentials of Geometry1-1 UNDEFIN
Page 14 and 15:
4 Essentials of GeometryThe number
Page 16 and 17:
6 Essentials of GeometryEXAMPLE 1In
Page 18 and 19:
8 Essentials of GeometryRecall that
Page 20 and 21:
10 Essentials of GeometryEXAMPLE 1O
Page 22 and 23:
12 Essentials of GeometryOn the num
Page 24 and 25:
14 Essentials of Geometry7. A, B, C
Page 26 and 27:
16 Essentials of Geometry1. By a ca
Page 28 and 29:
18 Essentials of GeometrySolutiona.
Page 30 and 31:
20 Essentials of GeometryBisector o
Page 32 and 33:
22 Essentials of GeometryIn 11-13,
Page 34 and 35:
24 Essentials of GeometryIncluded S
Page 36 and 37:
26 Essentials of GeometryRight Tria
Page 38 and 39:
28 Essentials of GeometryIn 5 and 6
Page 40 and 41:
30 Essentials of GeometryPR S TProp
Page 42 and 43:
32 Essentials of Geometry12. In iso
Page 44 and 45:
CHAPTER2CHAPTERTABLE OF CONTENTS2-1
Page 46 and 47:
36 LogicSome sentences are true for
Page 48 and 49:
38 LogicNegationsIn the study of lo
Page 50 and 51:
40 Logic2. a. Give an example of a
Page 52 and 53:
42 Logic2-2 CONJUNCTIONSWe have ide
Page 54 and 55:
44 LogicA compound sentence may con
Page 56 and 57:
46 LogicEXAMPLE 4Three sentences ar
Page 58 and 59:
48 Logic31. I have the hiccups and
Page 60 and 61:
50 LogicAnswers(1) k ∨ q a. Every
Page 62 and 63:
52 Logic8. A gram is a measure of l
Page 64 and 65:
54 LogicParts of a ConditionalThe p
Page 66 and 67:
56 Logic2. “In order to succeed,
Page 68 and 69:
58 LogicExercisesWriting About Math
Page 70 and 71:
60 Logic42. The conditional p → q
Page 72 and 73:
62 Logicp: Angle A and B are congru
Page 74 and 75:
64 LogicOn Friday, p is true and q
Page 76 and 77:
66 LogicConditional If Jessica like
Page 78 and 79:
68 LogicIn 7-10: a. Write the inver
Page 80 and 81:
70 LogicRecall that a conjunction i
Page 82 and 83:
72 LogicEXAMPLE 2The statement “I
Page 84 and 85:
74 LogicApplying Skills17. Let p re
Page 86 and 87:
76 LogicThe Law of Disjunctive Infe
Page 88 and 89:
78 Logic(1) Eliminate the second ro
Page 90 and 91:
80 Logic17. If 2b 6 14, then 2b
Page 92 and 93:
82 Logicglass” is also true. Appl
Page 94 and 95:
84 Logic7. When p → q and p ∧ q
Page 96 and 97:
86 Logic•A hypothesis, also calle
Page 98 and 99:
88 Logic8. July is a winter month i
Page 100 and 101:
90 LogicExploration1. A tautology i
Page 102 and 103:
92 Logic11. The first two sentences
Page 104 and 105:
94 Proving Statements in Geometry3-
Page 106 and 107:
96 Proving Statements in GeometryDe
Page 108 and 109:
98 Proving Statements in GeometryBe
Page 110 and 111:
100 Proving Statements in Geometry1
Page 112 and 113:
102 Proving Statements in GeometryE
Page 114 and 115:
104 Proving Statements in GeometryA
Page 116 and 117:
Page 118 and 119:
108 Proving Statements in GeometryC
Page 120 and 121:
110 Proving Statements in GeometryS
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
116 Proving Statements in GeometryF
Page 128 and 129:
Page 130 and 131:
120 Proving Statements in GeometryJ
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
128 Proving Statements in GeometryC
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
CHAPTER1344CHAPTERTABLE OF CONTENTS
Page 146 and 147:
136 Congruence of Line Segments,Ang
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
176 Congruence Based on TrianglesAC
Page 188 and 189:
178 Congruence Based on Triangles6.
Page 190 and 191:
180 Congruence Based on TrianglesDe
Page 192 and 193:
182 Congruence Based on TrianglesTh
Page 194 and 195:
184 Congruence Based on Triangles(C
Page 196 and 197:
186 Congruence Based on Triangles5-
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
196 Congruence Based on TrianglesAp
Page 208 and 209:
198 Congruence Based on TrianglesCo
Page 210 and 211:
200 Congruence Based on TrianglesCo
Page 212 and 213:
202 Congruence Based on TrianglesEX
Page 214 and 215:
204 Congruence Based on TrianglesCH
Page 216 and 217:
206 Congruence Based on TrianglesCU
Page 218 and 219:
208 Congruence Based on Triangles13
Page 220 and 221:
210 Transformations and the Coordin
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
Page 270 and 271:
Page 272 and 273:
CHAPTER7CHAPTERTABLE OF CONTENTS7-1
Page 274 and 275:
264 Geometric InequalitiesThis corr
Page 276 and 277:
266 Geometric InequalitiesExercises
Page 278 and 279:
268 Geometric InequalitiesSubtracti
Page 280 and 281:
270 Geometric Inequalities7-3 INEQU
Page 282 and 283:
Page 284 and 285:
274 Geometric InequalitiesEXAMPLE 2
Page 286 and 287:
276 Geometric InequalitiesHands-On
Page 288 and 289:
278 Geometric InequalitiesABMCProof
Page 290 and 291:
Page 292 and 293:
282 Geometric InequalitiesGivenTo p
Page 294 and 295:
284 Geometric InequalitiesCIf the m
Page 296 and 297:
286 Geometric Inequalities7.3 If th
Page 298 and 299:
288 Geometric InequalitiesCUMULATIV
Page 300 and 301:
Page 302:
292 Slopes and Equations of LinesAl
Page 305 and 306:
Developing SkillsIn 3-11, in each c
Page 307 and 308:
y 2 bx 2 a 5 m The Equation of a Li
Page 309 and 310:
5 ? 3 2(2) 1 1 5 ? 2(0) 1 7 5 ? 2
Page 311 and 312:
zontal line, they all have the same
Page 313 and 314:
Midpoint of a Line Segment 303Proof
Page 315 and 316:
Midpoint of a Line Segment 305EXAMP
Page 317 and 318:
The Slopes of Perpendicular Lines 3
Page 319 and 320:
The Slopes of Perpendicular Lines 3
Page 321 and 322:
Developing SkillsIn 3-12: a. Find t
Page 323 and 324:
Coordinate Proof 3138-5 COORDINATE
Page 325 and 326:
Coordinate Proof 315ProofThis is a
Page 327 and 328:
Concurrence of the Altitudes of a T
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
VOCABULARY8-1 Slope • x • y8-2
Page 335 and 336:
Cumulative Review 325CUMULATIVE REV
Page 337 and 338:
Cumulative Review 327Part IVAnswer
Page 339 and 340:
Proving Lines Parallel 3299-1 PROVI
Page 341 and 342:
Proving Lines Parallel 331In the di
Page 343 and 344:
Proving Lines Parallel 333The proof
Page 345 and 346:
Properties of Parallel Lines 3359-2
Page 347 and 348:
Properties of Parallel Lines 337The
Page 349 and 350:
gEJ. Therefore, AB g CD gbecause i
Page 351 and 352:
c. Is the measure of an exterior an
Page 353 and 354:
Parallel Lines in the Coordinate Pl
Page 355 and 356:
Parallel Lines in the Coordinate Pl
Page 357 and 358:
. Find the midpoints of two sides o
Page 359 and 360:
The Sum of the Measures of the Angl
Page 361 and 362:
The Sum of the Measures of the Angl
Page 363 and 364:
Proving Triangles Congruent by Angl
Page 365 and 366: Proving Triangles Congruent by Angl
Page 367 and 368: The Converse of the Isosceles Trian
Page 369 and 370: The Converse of the Isosceles Trian
Page 371 and 372: In 3-6, in each case the degree mea
Page 373 and 374: (1) Since any line segment may beex
Page 375 and 376: Proving Right Triangles Congruent b
Page 377 and 378: Interior and Exterior Angles of Pol
Page 383 and 384: Chapter Summary 373CHAPTER SUMMARYD
Page 385 and 386: AB g In 1-5, CD gand these lines a
Page 387 and 388: 2. The statement “If two angles f
Page 389 and 390: CHAPTERQUADRILATERALSEuclid’s fif
Page 391 and 392: The Parallelogram 381ADBCQuadrilate
Page 393 and 394: The Parallelogram 383DEFINITIONThe
Page 395 and 396: Proving That a Quadrilateral Is a P
Page 397 and 398: Proving That a Quadrilateral Is a P
Page 399 and 400: The Rectangle 38910-4 THE RECTANGLE
Page 401 and 402: The Rectangle 391EXAMPLE 1Given: AB
Page 403 and 404: The Rhombus 39316. The coordinates
Page 405 and 406: The Rhombus 395Theorem 10.15If a qu
Page 407 and 408: 2. Concepta said that if the length
Page 409 and 410: The Square 39910-6 THE SQUAREDEFINI
Page 411 and 412: The Square 401EXAMPLE 2In square PQ
Page 413 and 414: The Trapezoid 403The Isosceles Trap
Page 415: The Trapezoid 405Proof We will show
Page 419 and 420: 18. Prove Theorem 10.23, “The len
Page 421 and 422: Areas of Polygons 4115. Find the ar
Page 423 and 424: Chapter Summary 413PostulateTheorem
Page 425 and 426: Review Exercises 41511. The diagona
Page 427 and 428: Cumulative Review 417CUMULATIVE REV
Page 429 and 430: CHAPTERTHEGEOMETRYOF THREEDIMENSION
Page 431 and 432: Points, Lines, and Planes 421Theore
Page 433 and 434: Perpendicular Lines and Planes 4231
Page 435 and 436: Perpendicular Lines and Planes 425G
Page 437 and 438: Perpendicular Lines and Planes 427l
Page 439 and 440: Perpendicular Lines and Planes 429G
Page 441 and 442: gPlanes p and q intersect in AB. In
Page 443 and 444: 16. Prove that if two points are ea
Page 445 and 446: Parallel Lines and Planes 435Theore
Page 447 and 448: BF intersecting q at . Since p and
Page 449 and 450: ProofTwo lines perpendicular to the
Page 451 and 452: Since the bases are parallel, the c
Page 453 and 454: Surface Area of a Prism 443EXAMPLE
Page 455 and 456: 10. A prism has bases that are trap
Page 457 and 458: Volume of a Prism 447The other pris
Page 459 and 460: Pyramids 44911-6 PYRAMIDSA pyramid
Page 461 and 462: Pyramids 451For example, consider a
Page 463 and 464: Cylinders 45314. Let F be the verte
Page 465 and 466: Express this result as a rational a
Page 467 and 468:
Cones 457We can make a model of a r
Page 469 and 470:
7. The volume of a cone is 127 cubi
Page 471 and 472:
Spheres 461Proof Statements Reasons
Page 473 and 474:
Spheres 463When we round to the nea
Page 475 and 476:
Chapter Summary 465• Perpendicula
Page 477 and 478:
Vocabulary 46711.14 The intersectio
Page 479 and 480:
Review Exercises 46914. A prism and
Page 481 and 482:
Cumulative Review 471b. Using the s
Page 483 and 484:
Cumulative Review 47314. Line segme
Page 485 and 486:
Ratio and Proportion 47512-1 RATIO
Page 487 and 488:
Ratio and Proportion 477ProofWe can
Page 489 and 490:
Ratio and Proportion 479EXAMPLE 3Th
Page 491 and 492:
Given ABC, D is the midpoint of AC,
Page 493 and 494:
Proportions Involving Line Segments
Page 495 and 496:
Proportions Involving Line Segments
Page 497 and 498:
If two polygons are similar, then t
Page 499 and 500:
Proving Triangles Similar 4899. Tri
Page 501 and 502:
Proving Triangles Similar 491We can
Page 503 and 504:
Proving Triangles Similar 493EXAMPL
Page 505 and 506:
Dilations 49511. If CA 48, DA 12,
Page 507 and 508:
Dilations 497Slope ofAB 5 d b 2 2 a
Page 509 and 510:
EXAMPLE 2SolutionDilations 499Find
Page 511 and 512:
Dilations 50129. The vertices of oc
Page 513 and 514:
Proportional Relations Among Segmen
Page 515 and 516:
Proportional Relations Among Segmen
Page 517 and 518:
Concurrence of the Medians of a Tri
Page 519 and 520:
(2) Find the equation of AM g and t
Page 521 and 522:
Proportions in a Right Triangle 511
Page 523 and 524:
Proportions in a Right Triangle 513
Page 525 and 526:
Pythagorean Theorem 51512-9 PYTHAGO
Page 527 and 528:
Pythagorean Theorem 517EXAMPLE 2Whe
Page 529 and 530:
Pythagorean Theorem 519EXAMPLE 3In
Page 531 and 532:
The Distance Formula 52123. A young
Page 533 and 534:
The lengths of the sides of the qua
Page 535 and 536:
Developing SkillsIn 3-10, the coord
Page 537 and 538:
Chapter Summary 527CHAPTER SUMMARYD
Page 539 and 540:
Review Exercises 529FormulasIn the
Page 541 and 542:
14. The length of a side of a rhomb
Page 543 and 544:
Cumulative Review 5334. The measure
Page 545 and 546:
CHAPTERGEOMETRYOF THECIRCLEEarly ge
Page 547 and 548:
Arcs and Angles 537In the diagram,
Page 549 and 550:
If (O > (Or and mCDX 5 mCrDr 60, t
Page 551 and 552:
Arcs and Angles 541b. mACX mAOC 75
Page 553 and 554:
Arcs and Chords 54313-2 ARCS AND CH
Page 555 and 556:
Arcs and Chords 545ProofFirst draw
Page 557 and 558:
Arcs and Chords 547Since a diameter
Page 559 and 560:
Arcs and Chords 549(5) Since AB CD
Page 561 and 562:
Arcs and Chords 551EXAMPLE 3ProofPr
Page 563 and 564:
Inscribed Angles and Their Measures
Page 565 and 566:
. m/A 5 X 12 mBC c. m/C 5 180 2 (m/
Page 567 and 568:
Inscribed Angles and Their Measures
Page 569 and 570:
Tangents and Secants 559Theorem 13.
Page 571 and 572:
ProofTangents and Secants 561We wil
Page 573 and 574:
Tangents and Secants 563The proofs
Page 575 and 576:
Developing SkillsIn 3 and 4, ABC is
Page 577 and 578:
Hands-On ActivityConsider any regul
Page 579 and 580:
Angles Formed by Tangents, Chords,
Page 581 and 582:
Page 583 and 584:
Page 585 and 586:
Measures of Tangent Segments, Chord
Page 587 and 588:
Page 589 and 590:
Page 591 and 592:
Circles in the Coordinate Plane 581
Page 593 and 594:
T 3, 22lation :Circles in the Coord
Page 595 and 596:
Developing SkillsIn 3-8, write an e
Page 597 and 598:
Circles in the Coordinate Plane 587
Page 599 and 600:
How to Proceed(1) Solve the pair of
Page 601 and 602:
Tangents and Secants in the Coordin
Page 603 and 604:
Chapter Summary 59321. a. Write an
Page 605 and 606:
Chapter Summary 59513.12 The measur
Page 607 and 608:
Vocabulary 597Formulas(Continued)Ty
Page 609 and 610:
Review Exercises 59911. Two circles
Page 611 and 612:
Cumulative Review 6012. The coordin
Page 613 and 614:
Find the coordinatesR 90+ r y5x.Cum
Page 615 and 616:
Constructing Parallel Lines 60514-1
Page 617 and 618:
Constructing Parallel Lines 607Exer
Page 619 and 620:
PL h b. Choose point N on . Constru
Page 621 and 622:
The Meaning of Locus 611Note that i
Page 623 and 624:
Five Fundamental Loci 613Applying S
Page 625 and 626:
SolutionAnswerBCAB g CD g . The lo
Page 627 and 628:
Points at a Fixed Distance in Coord
Page 629 and 630:
Equidistant Lines in Coordinate Geo
Page 631 and 632:
Page 633 and 634:
Page 635 and 636:
Points Equidistant from a Point and
Page 637 and 638:
Page 639 and 640:
Page 641 and 642:
Review Exercises 631VOCABULARY14-2
Page 643 and 644:
CUMULATIVE REVIEW Chapters 1-14Part
Page 645 and 646:
Part IVAnswer all questions in this
Page 647 and 648:
Index 637Biconditional(s), 69-73app
Page 649 and 650:
Index 639multiplicative, 5Identity
Page 651 and 652:
Index 641Projectionof point on a li
Page 653:
Index 643Trichotomy postulate, 264-
show all

AMSCO'S Geometry. New York - Rye High School

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?