AMSCO'S Geometry. New York - Rye High School

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162 Congruence of Line Segments,Angles, and TrianglesSTEP 3. Draw a second angle of the triangle using theother given angle measure. Let B be the vertexof this second angle and let BA hbe one ofthe sides of this angle.CSTEP 4.Let C be the intersection of the rays of A andB that are not on AB g.ABSTEP 5. Repeat steps 1 through 4. Let D the vertex ofthe first angle, E the vertex of the secondangle, and F the intersection of the rays of Dand E.FDEa. Follow the steps to draw two different triangles with each of the givenangle-side-angle measures.(1) 65°, 4 in., 35°(2) 60°, 4 in., 60°(3) 120°, 9 cm, 30°(4) 30°, 9 cm, 30°b. For each pair of triangles, measure the angle and sides that were not given.Do they have equal measures?c. For each pair of triangles, you can conclude that the triangles arecongruent.The triangles formed, ABC and DEF, can be placed on top of oneanother so that A and D coincide, B and E coincide, and C and F coincide.Therefore, ABC DEF.This activity leads to the following statement of angle-side-angle or ASAtriangle congruence, whose truth will be assumed without proof:Postulate 4.15Two triangles are congruent if two angles and the included side of one triangleare congruent, respectively, to two angles and the included side of theother. (ASA)Thus, in ABC and DEF, if B E, BA > ED, and A D, it followsthat ABC DEF.The postulate used here is abbreviated ASA.We willnow use this postulate to prove two triangles congruent.
Proving Triangles Congruent Using Angle, Side,Angle 163EXAMPLE 1Given: AEB and CED intersect at E, E is themidpoint of AEB, AC ' AE, andBD ' BE.DProve: AEC BDEAEBProve the triangles congruent by ASA.CProof Statements Reasonsg1. AEB and CED intersect at E. 1. Given.A 2. AEC BED3. E is the midpoint of AEB. 3. Given.S 4. AE > BE5. AC ' AE and BD ' BE 5. Given.2. If two lines intersect, the verticalangles are congruent.4. The midpoint of a line segmentdivides the segment into two congruentsegments.6. A and B are right angles. 6. Perpendicular lines intersect toform right angles.A 7. A Bg7. If two angles are right angles, theyare congruent.8. AEC BED 8. ASA (steps 2, 4, 7).ExercisesWriting About Mathematics1. Marty said that if two triangles are congruent and one of the triangles is a right triangle,then the other triangle must be a right triangle. Do you agree with Marty? Explain why orwhy not.2. In ABC and DEF, AB > DE, and /B > /E.a. Dora said that if AC is congruent to DF, then itfollows that ABC DEF. Do you agree withCFDora? Justify your answer.A B D Eb. Using only the SAS and ASA postulates, if ABC is not congruent to DEF, whatsides and what angles cannot be congruent?
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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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106 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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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
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Page 220 and 221: 210 Transformations and the Coordin
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212 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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CHAPTER2908CHAPTERTABLE OF CONTENTS
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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
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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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