AMSCO'S Geometry. New York - Rye High School

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84 Logic7. When p → q and p ∧ q are both false, what are the truth values of p and of q?8. When p → q is true and p ∧ q is false, what are the truth values of p and of q?9. When p → q is true, p ∨ r is true and q is false, what is the truth value of r?Applying Skills10. Laura, Marta, and Shanti are a lawyer, a doctor, and an investment manager.• The lawyer is Marta’s sister.• Laura is not a doctor.• Either Marta or Shanti is a lawyer.What is the profession of each woman?11. Alex, Tony, and Kevin each have a different job: a plumber, a bookkeeper, and a teacher.• Alex is a plumber or a bookkeeper.• Tony is a bookkeeper or a teacher.• Kevin is a teacher.What is the profession of each person?12. Victoria owns stock in three companies: Alpha, Beta, and Gamma.• Yesterday, Victoria sold her shares of Alpha or Gamma.• If she sold Alpha, then she bought more shares of Beta.• Victoria did not buy more shares of Beta.Which stock did Victoria sell yesterday?13. Ren, Logan, and Kadoogan each had a different lunch. The possible lunches are: a hamsandwich, pizza, and chicken pot pie.• Ren or Logan had chicken pot pie.• Kadoogan did not have pizza.• If Logan did not have pizza, then Kadoogan had pizza.Which lunch did each person have?14. Zach, Steve, and David each play a different sport: basketball, soccer, or baseball. Zachmade each of the following true statements.• I do not play basketball.• If Steve does not play soccer, then David plays baseball.• David does not play baseball.What sport does each person play?
Chapter Summary 8515. Taylor, Melissa, and Lauren each study one language: French, Spanish, and Latin.• If Melissa does not study French, then Lauren studies Latin.• If Lauren studies Latin, then Taylor studies Spanish.• Taylor does not study Spanish.What language does each person study?16. Three friends, Augustus, Brutus, and Caesar, play a game in which each decides to be eithera liar or a truthteller. A liar must always lie and a truthteller must always tell the truth.When you met these friends, you asked Augustus which he had chosen to be. You didn’thear his answer but Brutus volunteered, “Augustus said that he is a liar.” Caesar added, “Ifone of us is a liar, then we are all liars.” Can you determine, for each person, whether he is aliar or a truthteller?CHAPTER SUMMARYDefinitionsto Know• Logic is the study of reasoning.• In logic, a mathematical sentence is a sentence that contains a completethought and can be judged to be true or false.•A phrase is an expression that is only part of a sentence.• An open sentence is any sentence that contains a variable.• The domain or replacement set is the set of numbers that can replace avariable.• The solution set or truth set is the set of all replacements that will changean open sentence to true sentences.•A statement or a closed sentence is a sentence that can be judged to betrue or false.• A closed sentence is said to have a truth value, either true (T) or false (F).•The negation of a statement has the opposite truth value of a given statement.• In logic, a compound sentence is a combination of two or more mathematicalsentences formed by using the connectives not, and, or, if . . . then, or ifand only if.•A conjunction is a compound statement formed by combining two simplestatements, called conjuncts, with the word and. The conjunction p and q iswritten symbolically as p ∧ q.•A disjunction is a compound statement formed by combining two simplestatements, called disjuncts, with or. The disjunction p or q is written symbolicallyas p ∨ q.•A truth table is a summary of all possible truth values of a logic statement.•A conditional is a compound statement formed by using the words if . . .then to combine two simple statements. The conditional if p then q is writtensymbolically as p → q.
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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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Page 46 and 47: 36 LogicSome sentences are true for
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Page 54 and 55: 44 LogicA compound sentence may con
Page 56 and 57: 46 LogicEXAMPLE 4Three sentences ar
Page 58 and 59: 48 Logic31. I have the hiccups and
Page 60 and 61: 50 LogicAnswers(1) k ∨ q a. Every
Page 62 and 63: 52 Logic8. A gram is a measure of l
Page 64 and 65: 54 LogicParts of a ConditionalThe p
Page 66 and 67: 56 Logic2. “In order to succeed,
Page 68 and 69: 58 LogicExercisesWriting About Math
Page 70 and 71: 60 Logic42. The conditional p → q
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Page 74 and 75: 64 LogicOn Friday, p is true and q
Page 76 and 77: 66 LogicConditional If Jessica like
Page 78 and 79: 68 LogicIn 7-10: a. Write the inver
Page 80 and 81: 70 LogicRecall that a conjunction i
Page 82 and 83: 72 LogicEXAMPLE 2The statement “I
Page 84 and 85: 74 LogicApplying Skills17. Let p re
Page 86 and 87: 76 LogicThe Law of Disjunctive Infe
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Page 90 and 91: 80 Logic17. If 2b 6 14, then 2b
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Page 98 and 99: 88 Logic8. July is a winter month i
Page 100 and 101: 90 LogicExploration1. A tautology i
Page 102 and 103: 92 Logic11. The first two sentences
Page 104 and 105: 94 Proving Statements in Geometry3-
Page 106 and 107: 96 Proving Statements in GeometryDe
Page 108 and 109: 98 Proving Statements in GeometryBe
Page 110 and 111: 100 Proving Statements in Geometry1
Page 112 and 113: 102 Proving Statements in GeometryE
Page 114 and 115: 104 Proving Statements in GeometryA
Page 118 and 119: 108 Proving Statements in GeometryC
Page 120 and 121: 110 Proving Statements in GeometryS
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Page 130 and 131: 120 Proving Statements in GeometryJ
Page 138 and 139: 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
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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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