AMSCO'S Geometry. New York - Rye High School

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442 The <strong>Geometry</strong> of Three DimensionsDEFINITIONA parallelepiped is a prism that has parallelograms as bases.Examples of parallelepipedsRectangular SolidsDEFINITIONA rectangular parallelepiped is a parallelepiped that has rectangular bases andlateral edges perpendicular to the bases.A rectangular parallelepiped is usually called a rectangular solid. It is themost common prism and is the union of six rectangles. Any two parallel rectanglesof a rectangular solid can be the bases.In the figure, ABCDEFGH is aGrectangular solid. The bases ABCD Hand EFGH are congruent rectanglesEFwith AB EF CD GH 7 and4BC FG DA HE 5. Two ofDthe lateral sides are rectangles ABFEC5and DCGH whose dimensions are 7by 4. The other two lateral sides areABBCGF and ADHE whose dimensions7are 5 by 4.• The area of each base: 7 5 35• The area of each of two lateral sides: 7 4 28• The area of each of the other two lateral sides: 5 4 20The lateral area of the prism is the sum of the areas of the lateral faces. Thetotal surface area is the sum of the lateral area and the areas of the bases.• The lateral area of the prism is 2(28) 2(20) 96.• The area of the bases are each 2(35) 70.• The surface area of the prism is 96 70 166.
Surface Area of a Prism 443EXAMPLE 1The bases of a right prism are regular hexagons. The length of each side of abase is 5 centimeters and the height of the prism is 8 centimeters. Describe thenumber, shape, and size of the lateral sides.SolutionAnswerA hexagon has six sides. Because the base is a regularhexagon, it has six congruent sides, and therefore, theprism has six lateral sides. Since it is a right prism, thelateral sides are rectangles. The length of each of twoedges of a lateral side is the length of an edge of abase, 5 centimeters. The length of each of the othertwo edges of a lateral side is the height of the prism,8 centimeters.There are six lateral sides, each is a rectangle that is8 centimeters by 5 centimeters.8 cm5 cmEXAMPLE 2The bases of a right prism are equilateral triangles. The length of one edge of abase is 4 inches and the height of the prism is 5 inches.a. How many lateral sides does this prism have and what is their shape?b. What is the lateral area of the prism?Solutiona. Because this is a prism with a triangular base, the prism has three lateralsides. Because it is a right prism, the lateral sides are rectangles.b. For each rectangular side, the length of one pair of edges is the length of anedge of the base, 4 inches. The height of the prism, 5 inches, is the length ofa lateral edge. Therefore, the area of each lateral side is 4(5) or 20 squareinches. The lateral area of the prism is 3(20) 60 square inches.Answers a. 3 b. 60 square inchesEXAMPLE 3The lateral sides of a prism are five congruent rectangles.Prove that the bases are equilateral pentagons.ProofThe five lateral sides are congruent rectangles. Two parallelsides of each rectangle are lateral edges. The other twoparallel sides of each rectangle are edges of the bases.Each edge of a base is a side of a rectangle. Therefore, thebase has five sides. The corresponding sides of the congruentrectangles are congruent. Therefore, the edges of eachbase are congruent. The bases are equilateral pentagons.
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AMSCO’SGEOMETRYAnn Xavier Gantert
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PREFACE✔✔✔✔✔✔Geometry i
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CONTENTSChapter 1ESSENTIALS OF GEOM
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viiiCONTENTSReview Exercises 323Cum
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xCONTENTS14-3 Five Fundamental Loci
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2 Essentials of Geometry1-1 UNDEFIN
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4 Essentials of GeometryThe number
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6 Essentials of GeometryEXAMPLE 1In
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8 Essentials of GeometryRecall that
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10 Essentials of GeometryEXAMPLE 1O
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12 Essentials of GeometryOn the num
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14 Essentials of Geometry7. A, B, C
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16 Essentials of Geometry1. By a ca
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18 Essentials of GeometrySolutiona.
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20 Essentials of GeometryBisector o
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22 Essentials of GeometryIn 11-13,
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24 Essentials of GeometryIncluded S
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26 Essentials of GeometryRight Tria
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28 Essentials of GeometryIn 5 and 6
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30 Essentials of GeometryPR S TProp
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32 Essentials of Geometry12. In iso
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CHAPTER2CHAPTERTABLE OF CONTENTS2-1
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36 LogicSome sentences are true for
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38 LogicNegationsIn the study of lo
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40 Logic2. a. Give an example of a
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42 Logic2-2 CONJUNCTIONSWe have ide
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44 LogicA compound sentence may con
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46 LogicEXAMPLE 4Three sentences ar
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48 Logic31. I have the hiccups and
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50 LogicAnswers(1) k ∨ q a. Every
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52 Logic8. A gram is a measure of l
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54 LogicParts of a ConditionalThe p
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56 Logic2. “In order to succeed,
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58 LogicExercisesWriting About Math
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60 Logic42. The conditional p → q
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62 Logicp: Angle A and B are congru
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64 LogicOn Friday, p is true and q
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66 LogicConditional If Jessica like
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68 LogicIn 7-10: a. Write the inver
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70 LogicRecall that a conjunction i
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72 LogicEXAMPLE 2The statement “I
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74 LogicApplying Skills17. Let p re
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76 LogicThe Law of Disjunctive Infe
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78 Logic(1) Eliminate the second ro
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80 Logic17. If 2b 6 14, then 2b
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82 Logicglass” is also true. Appl
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84 Logic7. When p → q and p ∧ q
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86 Logic•A hypothesis, also calle
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88 Logic8. July is a winter month i
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90 LogicExploration1. A tautology i
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92 Logic11. The first two sentences
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94 Proving Statements in Geometry3-
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96 Proving Statements in GeometryDe
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98 Proving Statements in GeometryBe
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100 Proving Statements in Geometry1
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102 Proving Statements in GeometryE
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104 Proving Statements in GeometryA
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108 Proving Statements in GeometryC
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110 Proving Statements in GeometryS
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116 Proving Statements in GeometryF
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120 Proving Statements in GeometryJ
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128 Proving Statements in GeometryC
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CHAPTER1344CHAPTERTABLE OF CONTENTS
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136 Congruence of Line Segments,Ang
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176 Congruence Based on TrianglesAC
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178 Congruence Based on Triangles6.
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180 Congruence Based on TrianglesDe
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182 Congruence Based on TrianglesTh
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184 Congruence Based on Triangles(C
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186 Congruence Based on Triangles5-
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196 Congruence Based on TrianglesAp
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198 Congruence Based on TrianglesCo
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200 Congruence Based on TrianglesCo
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202 Congruence Based on TrianglesEX
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204 Congruence Based on TrianglesCH
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206 Congruence Based on TrianglesCU
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208 Congruence Based on Triangles13
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210 Transformations and the Coordin
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CHAPTER7CHAPTERTABLE OF CONTENTS7-1
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264 Geometric InequalitiesThis corr
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266 Geometric InequalitiesExercises
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268 Geometric InequalitiesSubtracti
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270 Geometric Inequalities7-3 INEQU
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274 Geometric InequalitiesEXAMPLE 2
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276 Geometric InequalitiesHands-On
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278 Geometric InequalitiesABMCProof
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282 Geometric InequalitiesGivenTo p
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284 Geometric InequalitiesCIf the m
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286 Geometric Inequalities7.3 If th
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288 Geometric InequalitiesCUMULATIV
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292 Slopes and Equations of LinesAl
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Developing SkillsIn 3-11, in each c
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y 2 bx 2 a 5 m The Equation of a Li
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5 ? 3 2(2) 1 1 5 ? 2(0) 1 7 5 ? 2
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zontal line, they all have the same
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Midpoint of a Line Segment 303Proof
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Midpoint of a Line Segment 305EXAMP
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The Slopes of Perpendicular Lines 3
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The Slopes of Perpendicular Lines 3
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Developing SkillsIn 3-12: a. Find t
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Coordinate Proof 3138-5 COORDINATE
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Coordinate Proof 315ProofThis is a
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Concurrence of the Altitudes of a T
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VOCABULARY8-1 Slope • x • y8-2
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Cumulative Review 325CUMULATIVE REV
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Cumulative Review 327Part IVAnswer
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Proving Lines Parallel 3299-1 PROVI
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Proving Lines Parallel 331In the di
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Proving Lines Parallel 333The proof
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Properties of Parallel Lines 3359-2
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Properties of Parallel Lines 337The
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gEJ. Therefore, AB g CD gbecause i
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c. Is the measure of an exterior an
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Parallel Lines in the Coordinate Pl
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Parallel Lines in the Coordinate Pl
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. Find the midpoints of two sides o
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The Sum of the Measures of the Angl
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The Sum of the Measures of the Angl
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Proving Triangles Congruent by Angl
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Proving Triangles Congruent by Angl
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The Converse of the Isosceles Trian
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The Converse of the Isosceles Trian
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In 3-6, in each case the degree mea
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(1) Since any line segment may beex
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Proving Right Triangles Congruent b
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Interior and Exterior Angles of Pol
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Chapter Summary 373CHAPTER SUMMARYD
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AB g In 1-5, CD gand these lines a
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2. The statement “If two angles f
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CHAPTERQUADRILATERALSEuclid’s fif
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The Parallelogram 381ADBCQuadrilate
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The Parallelogram 383DEFINITIONThe
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Proving That a Quadrilateral Is a P
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Proving That a Quadrilateral Is a P
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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 and 416: The Trapezoid 405Proof We will show
Page 417 and 418: The Trapezoid 407The slopes of BC a
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
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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: Since the bases are parallel, the c
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
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Proving Triangles Similar 493EXAMPL
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Dilations 49511. If CA 48, DA 12,
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Dilations 497Slope ofAB 5 d b 2 2 a
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EXAMPLE 2SolutionDilations 499Find
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Dilations 50129. The vertices of oc
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Proportional Relations Among Segmen
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Proportional Relations Among Segmen
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Concurrence of the Medians of a Tri
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(2) Find the equation of AM g and t
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Proportions in a Right Triangle 511
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Proportions in a Right Triangle 513
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Pythagorean Theorem 51512-9 PYTHAGO
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Pythagorean Theorem 517EXAMPLE 2Whe
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Pythagorean Theorem 519EXAMPLE 3In
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The Distance Formula 52123. A young
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The lengths of the sides of the qua
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Developing SkillsIn 3-10, the coord
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Chapter Summary 527CHAPTER SUMMARYD
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Review Exercises 529FormulasIn the
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14. The length of a side of a rhomb
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Cumulative Review 5334. The measure
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CHAPTERGEOMETRYOF THECIRCLEEarly ge
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Arcs and Angles 537In the diagram,
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If (O > (Or and mCDX 5 mCrDr 60, t
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Arcs and Angles 541b. mACX mAOC 75
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Arcs and Chords 54313-2 ARCS AND CH
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Arcs and Chords 545ProofFirst draw
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Arcs and Chords 547Since a diameter
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Arcs and Chords 549(5) Since AB CD
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Arcs and Chords 551EXAMPLE 3ProofPr
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Inscribed Angles and Their Measures
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. m/A 5 X 12 mBC c. m/C 5 180 2 (m/
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Inscribed Angles and Their Measures
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Tangents and Secants 559Theorem 13.
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ProofTangents and Secants 561We wil
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Tangents and Secants 563The proofs
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Developing SkillsIn 3 and 4, ABC is
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Hands-On ActivityConsider any regul
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Angles Formed by Tangents, Chords,
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Measures of Tangent Segments, Chord
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Circles in the Coordinate Plane 581
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T 3, 22lation :Circles in the Coord
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Developing SkillsIn 3-8, write an e
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Circles in the Coordinate Plane 587
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How to Proceed(1) Solve the pair of
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Tangents and Secants in the Coordin
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Chapter Summary 59321. a. Write an
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Chapter Summary 59513.12 The measur
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Vocabulary 597Formulas(Continued)Ty
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Review Exercises 59911. Two circles
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Cumulative Review 6012. The coordin
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Find the coordinatesR 90+ r y5x.Cum
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Constructing Parallel Lines 60514-1
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Constructing Parallel Lines 607Exer
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PL h b. Choose point N on . Constru
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The Meaning of Locus 611Note that i
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Five Fundamental Loci 613Applying S
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SolutionAnswerBCAB g CD g . The lo
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Points at a Fixed Distance in Coord
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Equidistant Lines in Coordinate Geo
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Points Equidistant from a Point and
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Review Exercises 631VOCABULARY14-2
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CUMULATIVE REVIEW Chapters 1-14Part
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Part IVAnswer all questions in this
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Index 637Biconditional(s), 69-73app
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Index 639multiplicative, 5Identity
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Index 641Projectionof point on a li
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Index 643Trichotomy postulate, 264-
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AMSCO'S Geometry. New York - Rye High School

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