AMSCO'S Geometry. New York - Rye High School

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390 QuadrilateralsProperties of a Rectangle1. A rectangle has all the properties of a parallelogram.2. A rectangle has four right angles and is therefore equiangular.3. The diagonals of a rectangle are congruent.Proving That a Quadrilateral Is a RectangleWe prove that a quadrilateral is a rectangle by showing that it has the specialproperties of a rectangle. For example:Theorem 10.10If a quadrilateral is equiangular, then it is a rectangle.GivenQuadrilateral ABCD with A B C D.DCProveABCD is a rectangle.ProofTheorem 10.11Quadrilateral ABCD is a parallelogram because the ABopposite angles are congruent. The sum of the measuresof the angles of a quadrilateral is 360 degrees. Thus, the measure of each of thefour congruent angles is 90 degrees. Therefore, ABCD is a rectangle because itis a parallelogram with a right angle.If the diagonals of a parallelogram are congruent, the parallelogram is arectangle.GivenParallelogram ABCD with AC > BDDCProveABCD is a rectangle.StrategyProve that DAB CBA. Therefore, DAB andCBA are both congruent and supplementary.ABThe proof of Theorem 10.11 is left to the student. (See exercise 13.)SUMMARY To prove that a quadrilateral is a rectangle, prove that any one ofthe following statements is true:1. The quadrilateral is a parallelogram with one right angle.2. The quadrilateral is equiangular.3. The quadrilateral is a parallelogram whose diagonals are congruent.
The Rectangle 391EXAMPLE 1Given: ABCD is a parallelogram with mA mD.Prove: ABCD is a rectangle.DCProofWe will prove that ABCD has a right angle.Statements1. ABCD is a parallelogram. 1. Given.Reasons2. AB CD2. A parallelogram is a quadrilateralin which two pairs of oppositesides are parallel.3. A and D are supplementary. 3. Two consecutive angles of aparallelogram are supplementary.4. mA mD 180 4. Supplementary angles are twoangles the sum of whose measuresis 180.5. mA mD 5. Given.6. mA mA 180 or 6. A quantity may be substituted for2mA 180its equal.7. mA 90 7. Division postulate.8. ABCD is a rectangle. 8. A rectangle is a parallelogramwith a right angle.ABEXAMPLE 2The lengths of diagonals of a rectangle are represented by 7x centimeters and5x 12 centimeters. Find the length of each diagonal.SolutionAnswerThe diagonals of a rectangle are congruent and therefore equal in length.7x 5x 122x 12x 67x 5 7(6) 5x 1 12 5 5(6) 1 125 425 30 1 125 42The length of each diagonal is 42 centimeters.
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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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Page 353 and 354: Parallel Lines in the Coordinate Pl
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Page 357 and 358: . Find the midpoints of two sides o
Page 359 and 360: The Sum of the Measures of the Angl
Page 361 and 362: The Sum of the Measures of the Angl
Page 363 and 364: Proving Triangles Congruent by Angl
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Page 367 and 368: The Converse of the Isosceles Trian
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Page 371 and 372: In 3-6, in each case the degree mea
Page 373 and 374: (1) Since any line segment may beex
Page 375 and 376: Proving Right Triangles Congruent b
Page 377 and 378: Interior and Exterior Angles of Pol
Page 383 and 384: Chapter Summary 373CHAPTER SUMMARYD
Page 385 and 386: AB g In 1-5, CD gand these lines a
Page 387 and 388: 2. The statement “If two angles f
Page 389 and 390: CHAPTERQUADRILATERALSEuclid’s fif
Page 391 and 392: The Parallelogram 381ADBCQuadrilate
Page 393 and 394: The Parallelogram 383DEFINITIONThe
Page 395 and 396: Proving That a Quadrilateral Is a P
Page 397 and 398: Proving That a Quadrilateral Is a P
Page 399: The Rectangle 38910-4 THE RECTANGLE
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
Page 437 and 438: Perpendicular Lines and Planes 427l
Page 439 and 440: Perpendicular Lines and Planes 429G
Page 441 and 442: gPlanes p and q intersect in AB. In
Page 443 and 444: 16. Prove that if two points are ea
Page 445 and 446: Parallel Lines and Planes 435Theore
Page 447 and 448: BF intersecting q at . Since p and
Page 449 and 450: ProofTwo lines perpendicular to the
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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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