AMSCO'S Geometry. New York - Rye High School

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518 Ratio, Proportion, and SimilarityxxsWhen a diagonal of a square is drawn, the square is separated into twoisosceles right triangles. We can express the length of a leg of the isosceles righttriangle in terms of the length of the hypotenuse or the length of the hypotenusein terms of the length of a leg.Let s be the length of the hypotenuse of an isosceles right triangle and x bethe length of each leg. Use the Pythagorean Theorem to set up two equalities.Solve one for x and the other for s:Solve for x: Solve for s:a 2 1 b 2 5 c 2c 2 5 a 2 1 b 2x 2 1 x 2 5 s 2s 2 5 x 2 1 x 22x 2 5 s 2 s 2 5 2x 2x 2 5 s22x 5 $ s22x 5 $ s22 ? 2 2x 5 "22 ss 5 x"2The 30-60-Degree Right TriangleAn altitude drawn to any side of an equilateral triangle bisects the base and separatesthe triangle into two congruent right triangles. Since the measure of eachangle of an equilateral triangle is 60°, the measure of one of the acute angles ofa right triangle formed by the altitude is 60° and the measure of the other acuteangle is 30°. Each of the congruent right triangles formed by drawing an altitudeto a side of an equilateral triangle is called a 30-60-degree right triangle. If twotriangles are 30-60-degree right triangles, then they are similar by AA.In the diagram, ABC is an equilateral trianglewith s the length of each side and h theClength of an altitude. Then s is the length of thehypotenuse of the 30-60-degree triangle and his the length of a leg. In the diagram, CD ' AB,30°sAC s, AD 2, and CD h.s ha 2 b 2 c 2A s 2 B 2s24 h 2 s 2 h 2 h 2 44 s234 s2h "32 sA60°s2DB
Pythagorean Theorem 519EXAMPLE 3In right triangle ABC, the length of the hypotenuse,AB, is 6 centimeters and the length of one leg is 3 centimeters.Find the length of the other leg.BSolutiona 2 1 b 2 5 c 2a 2 1 (3) 2 5 (6) 2a 2 1 9 5 36a 2 5 27a 5 "27 5 "9 ? 3 5 3"3Answer6 cmC3 cmANote: The measure of one leg is one-half the measure of the hypotenuse and"3the length of the other leg is 2 times the length of the hypotenuse. Therefore,this triangle is a 30-60-degree right triangle. We can use a calculator to verifythis.oppRecall that tan A = adj 5 AC BC 5 3"33 5 "3. Use a graphing calculator tofind the measure of A.ENTER:2ndTAN 12nd¯ENTERThe calculator will return 60 as mA. Therefore, mB 30.3ExercisesWriting About Mathematics1. Ira said that if the lengths of the sides of an obtuse triangle ABC are a, b, and c with copposite the obtuse angle, then c 2 a 2 b 2 . Do you agree with Ira? Explain why or whynot. (Hint: Make use of the altitude from one of the acute angles.)2. Sean said that if the measures of the diagonals of a parallelogram are 6 and 8 and the measureof one side of the parallelogram is 5 then the parallelogram is a rhombus. Do you agreewith Sean? Explain why or why not.Developing SkillsIn 3–8, in each case the lengths of three sides of a triangle are given. Tell whether each triangle is aright triangle.3. 6, 8, 10 4. 7, 8, 12 5. 5, 7, 8 6. 15, 36, 39 7. 14, 48, 50 8. 2, 2"3, 4
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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
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The Rectangle 391EXAMPLE 1Given: AB
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The Rhombus 39316. The coordinates
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The Rhombus 395Theorem 10.15If a qu
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2. Concepta said that if the length
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The Square 39910-6 THE SQUAREDEFINI
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The Square 401EXAMPLE 2In square PQ
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The Trapezoid 403The Isosceles Trap
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The Trapezoid 405Proof We will show
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The Trapezoid 407The slopes of BC a
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18. Prove Theorem 10.23, “The len
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Areas of Polygons 4115. Find the ar
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Chapter Summary 413PostulateTheorem
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Review Exercises 41511. The diagona
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Cumulative Review 417CUMULATIVE REV
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CHAPTERTHEGEOMETRYOF THREEDIMENSION
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Points, Lines, and Planes 421Theore
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Perpendicular Lines and Planes 4231
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Perpendicular Lines and Planes 425G
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Perpendicular Lines and Planes 427l
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Perpendicular Lines and Planes 429G
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gPlanes p and q intersect in AB. In
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16. Prove that if two points are ea
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Parallel Lines and Planes 435Theore
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BF intersecting q at . Since p and
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ProofTwo lines perpendicular to the
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Since the bases are parallel, the c
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Surface Area of a Prism 443EXAMPLE
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10. A prism has bases that are trap
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Volume of a Prism 447The other pris
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Pyramids 44911-6 PYRAMIDSA pyramid
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Pyramids 451For example, consider a
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Cylinders 45314. Let F be the verte
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Express this result as a rational a
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Cones 457We can make a model of a r
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7. The volume of a cone is 127 cubi
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Spheres 461Proof Statements Reasons
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Spheres 463When we round to the nea
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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: Pythagorean Theorem 517EXAMPLE 2Whe
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
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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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