AMSCO'S Geometry. New York - Rye High School

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542 <strong>Geometry</strong> of the CircleIn 23–32, P, Q, S, and R are points on circle O,mPOQ 100, mQOS 110, and mSOR 35.Find each measure.23. mPQX24. mQS XP25. mSRX26. mROP27. 28. mPQS XOmRPXR 100°35°29. mQOR 30. mQSR XS 110° Q31. mSRP X32. mRPQ XApplying Skills33. Given: Circle O with ABX > CD X.Prove: AOB CODOABDC34. Given: AB and CD intersect at O, and the endpoints of AB and CDAare on circle O.DProve: AC > BDC O35. In circle O, AOB ' COD. Find mACX and mADC.36. Points A, B, C, and D lie on circle O, and AC ' BD at O. Prove that quadrilateral ABCD isa square.XBHands-On ActivityFor this activity, you may use compass, protractor, and straightedge, or geometrysoftware.1. Draw circle O with a radius of 2 inches and circle O with a radius of 3 inches.2. Draw points A and B on the circle O so that mABX 60 and points A and B on circle Oso that mArBr X 60.3. Show that AOB AOB.
Arcs and Chords 54313-2 ARCS AND CHORDSDEFINITIONA chord of a circle is a line segment whose endpoints are points of the circle.A diameter of a circle is a chord that has the center of the circle as one of itspoints.In the diagram, AB and AOC are chords of circle O.Since O is a point of AOC, AOC is a diameter. Since OAand OC are the lengths of the radius of circle O, OA OCand O is the midpoint of AOC.If the length of the radius of circle O is r, and thelength of the diameter is d, thenCOBAd 5 AOC5 OA 1 OC5 r 1 r5 2rThat is:d 2rThe endpoints of a chord are points on a circle and,therefore, determine two arcs of a circle, a minor arc anda major arc. In the diagram, chord AB, central AOB,minor ABX , and major ABX are all determined by points Aand B. We proved in the previous section that in a circle,congruent central angles intercept congruent arcs. Nowwe can prove that in a circle, congruent central angleshave congruent chords and that congruent arcs havecongruent chords.AOBTheorem 13.3aIn a circle or in congruent circles, congruent central angles have congruentchords.Given(O > (Or andCOD AOB AOBABABProveCD > AB > ArBrCOOD
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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
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Vocabulary 46711.14 The intersectio
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Review Exercises 46914. A prism and
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Cumulative Review 471b. Using the s
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Cumulative Review 47314. Line segme
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Ratio and Proportion 47512-1 RATIO
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Ratio and Proportion 477ProofWe can
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Ratio and Proportion 479EXAMPLE 3Th
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Given ABC, D is the midpoint of AC,
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Proportions Involving Line Segments
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Proportions Involving Line Segments
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If two polygons are similar, then t
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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: Arcs and Angles 541b. mACX mAOC 75
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 585 and 586: Measures of Tangent Segments, Chord
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
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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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