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ANSWER KEY Preparing for the REGENTS EXAMINATION GEOMETRY AMSCO A M S C O S C H O O L P U B L I C AT I O N S , I N C . 315 Hudson Street, New York, N.Y. 10013 N 81 CD Compositor: Monotype LLC Please visit our Web site at: www.amscopub.com Copyright © 2008 by Amsco School Publications, Inc. No part of this book may be reproduced in any form without written permission from the publisher. Printed in the United States of America 1 2 3 4 5 6 7 8 9 10 12 11 10 09 08 07 Contents Chapter 1: Essentials of Geometry Chapter 2: Logic Chapter 3: Introduction to Geometric Proof Chapter 4: Congruence of Lines, Angles, and Triangles Chapter 5: Congruence Based on Triangles Chapter 6: Transformations and the Coordinate Plane Chapter 7: Polygon Sides and Angles Chapter 8: Slopes and Equations of Lines Chapter 9: Parallel Lines Chapter 10: Quadrilaterals Chapter 11: Geometry of Three Dimensions Chapter 12: Ratios, Proportion, and Similarity Chapter 13: Geometry of the Circle Chapter 14: Locus and Constructions Cumulative Reviews Practice Geometry Regents Examinations 1 3 8 12 18 28 34 39 44 56 68 72 79 87 92 138 CHAPTER Essentials of Geometry 1-1 Undefined Terms (page 2) 1 (1) finite 2 (2) infinite 3 (3) empty 4 A B 5 A given line lies on an infinite number of planes. 6 a and b p ᐉ m 1-2 Real Numbers and Their Properties (pages 4–5) 1 (4) 8 2 (4) 5 3 (3) 0n 0 4 (1) a(b c) ab ac 5 (1) additive inverse 6 Commutative 7 Multiplicative identity 8 Commutative 9 Additive identity 10 Associative 11 Associative 12 13 14 15 16 17 18 19 20 1 Commutative Distributive Additive inverse Distributive Multiplicative inverse 1 0 0 Division by zero is undefined. 1-3 Lines and Line Segments (pages 6–7) 1 8 2 4 −− −− 3 BD and CE 4 They must all lie on the same line. 5 They form a triangle. 6 Point S is the midpoint. It is halfway between P and T. −− 7 Correct. The length of AB is equal to the −−− length of CD. 8 Incorrect notation 9 Incorrect notation −−− −− 10 Correct. AB is congruent to CD. −−− −−− 11 CPD bisects APB. It divides it into two segments with the same measure, 7. 12 MS 6 1-3 Lines and Line Segments 1 1-4 Angles (pages 9–10) 1 (4) It is a portion of a line, beginning with point M. 2 (1) DAE 3 (4) XYZ is a 180 angle. 4 (3) equal to 90 5 (2) three acute angles 6 Acute angles: JKM, MKN, NKO, OKL, MKO Obtuse angles: JKO, MKL Right angles: JKN, NKL Straight angle: JKL 7 mLKO 35; mMKN 30; mMKO 85 8 D E F 1-5 Preliminary Angle and Line Relationships (pages 11–12) 1 a mDBC 90 b DBC ABD 2 a mONP 45 b ONP MNO 3 mCBD 4 mCBD 5 mABD mABC mCBD 90 45 135 6 ln 7 x8 8 y 9.5 9 mCAT 28 10 mAFC 46 9 R 1-6 Triangles T B 10 V 11 G H I 12 a and b A E I B C (page 16) 1 False 2 True 3 False 4 False 5 True 6 The length of each leg of the isosceles triangle is 18. 7 a The length of each side of the triangle is 16. b y7 8 mA 40; mB 70; mC 70 9 Check students’ drawings of isosceles triangle ABC with vertex at C. 10 Check students’ drawings ___ of right triangle PQR with hypotenuse PQ. 11 Check students’ drawings of acute triangle FGH that is scalene. −−− −− 12 CD and CE Chapter Review (pages 16–17) 1 Point 2 Line, length 3 Plane, length and width 4 9 5 Only the square roots of perfect square numbers are rational. 2 Chapter 1: Essentials of Geometry −− −− 6 a XZ YZ b ZXY and ZYX 7 Coordinate of B is 2. 8 mKJL 36 −− 9 G is the midpoint of FH. 10 Bisectors pass through the same point, the midpoint of the line segment. Distinct parallel lines have no points in common. 11 12 13 14 15 16 Straight angles: BCE and ACD DEF Right triangles: ABC and CED CFE −− −− −− −− BD, FD, FE, and BE are choices. −− BD CHAPTER 2 Logic 2-1 Statements, Truth Values, and Negations (pages 20–21) 1 (2) a false closed sentence 2 (2) a false closed sentence 3 (4) not a statement 4 (1) a true closed sentence 5 (4) 4 3 is not greater than 2. 6 (2) The coat is not blue. 7 (2) false 8 (1) 3 6 7 9 Any open sentence 10 A triangle has three sides. 11 Statistics is the study of analyzing data. 12 Algebra is the study of operations on sets of numbers. 13 Trigonometry is the study of triangles. 14 Geometry is the study of shapes and sizes. 15 “Chicago is not a city in Indiana.” Original statement is false. Negated statement is true. 16 “The sun does not rise in the west.” Original statement is false. Negated statement is true. 17 “January is not a winter month.” Original statement is true. Negated statement is false. 18 “The area of a rectangle is not length times width.” Original statement is true. Negated statement is false. 19 “Each state does not have two senators.” Original statement is true. Negated statement is false. 20 “It is not the case that all real numbers are rational.” Original statement is false. Negated statement is true. 2-2 Compound Statements (pages 23–24) 1 (3) 3 7 10 or 4 7 3 2 (1) Albany is not the capital of New York and is located on Long Island. 3 (2) 5 is not an odd number or 6 4 12. 4 (4) Daylight saving time ends in November or the clock is not turned back one hour. 5 False 2-2 Compound Statments 3 True True True True Answers will vary. One possible answer is: Every square has four sides or every triangle has three sides. True 11 Answers will vary. One possible answer is: The sum of the measures of the angles of every triangle is 180 and every triangle has four angles. False 6 7 8 9 10 2-3 Conditionals (pages 26–27) 1 (2) You take this medicine. 2 (1) I will stay in this afternoon. 3 (1) true 4 (1) true 5 (1) true 6 Conditional 7 Conjunction 8 Conditional 9 Disjunction 10 Negation 11 Answers will vary. x 18 is one possible answer. 12 x 2 13 Any positive number or zero 14 Any perfect square that is not even 2 is one possible 15 Answers will vary. x √ answer. 16 True 17 False 18 The sun is It is not If the sun is shining. Sunny Sun showers Cloudy Stormy 4 raining. shining, it is not raining. T T T T F F F F T F T T Chapter 2: Logic 2-4 Converses, Inverses, and Contrapositives (pages 28–29) 1 (1) If you stop your car, then the traffic light is red. 2 (3) If the sum of two numbers is not even, then the numbers are not both even. 3 (1) If the merrygoround was not oiled, then it will squeak. 4 (2) If we are on vacation, it is summer. 5 (4) If a triangle is not equilateral, then it does not have three congruent angles. 6 (2) If I didn’t turn on the air conditioner, then the house isn’t cool. 7 (2) disjunction, p ∨ q 8 Converse: If school is open, then it is September. Inverse: If it is not September, then school is not open. Contrapositive: If school is not open, then it is not September. 9 Converse: If the figure has four sides, then it is a square. Inverse: If the figure is not a square, then it does not have four sides. Contrapositive: If the figure does not have four sides, then it is not a square. 10 Converse: If someone will trade desserts, then I have cookies with my lunch. Inverse: If I do not have cookies with my lunch, then someone will not trade desserts. Contrapositive: If someone will not trade desserts, then I do not have cookies with my lunch. 11 Converse: If my teacher will be able to read my paper, then I typed it. Inverse: If I do not type my paper, then my teacher will not be able to read it. Contrapositive: If my teacher will not be able to read my paper, then I did not type it. 12 Converse: If I am late to school, then I didn’t set my alarm. Inverse: If I set my alarm, then I will not be late to school. Contrapositive: If I am not late to school, then I set my alarm. 13 Converse: If I am not in the starting lineup, then I will not go to practice. Inverse: If I to go to practice, I will be in the starting lineup. Contrapositive: If I am in the starting lineup, then I will go to practice. 14 Converse: If I am unhappy, then I did not make honor roll. Inverse: If I make honor roll, then I am happy. Contrapositive: If I am happy, then I made honor roll. 15 Converse: If I do not have money for the movies, then I will go shopping. Inverse: If I do not go shopping, then I will have money for the movies. Contrapositive: If I have money for the movies, then I will not go shopping. 16 Converse: If I do not take a school bus, then I live close to school. Inverse: If I do not live close to school, then I take a school bus. Contrapositive: If I take a school bus, then I do not live close to school. 17 Converse: If I don’t study Latin, then I will study French. Inverse: If I don’t study French, then I will study Latin. Contrapositive: If I study Latin, then I will not study French. 18 a If the segments are congruent, then their measures are equal. (True) b If the measures of the segments are equal, then the segments are congruent. (True) 2-5 Biconditionals (pages 31–32) 1 (1) true for any value of x 2 (2) false for any value of x 3 (2) false 4 (1) If I am sleepy, then I did not get eight hours of sleep and if I did not get eight hours of sleep, then I am sleepy. 5 (3) Paul does not buy chicken or he does not roast it. 6 (4) the conjunction of a conditional and its converse 7 If I have money, then I earned it. 8 If tomorrow is Friday, then today is Thursday. (True) 9 Converse: If a number is rational, then it is a repeating decimal. Biconditional: A number is rational if and only if it is a repeating decimal. The biconditional is false for integers and terminating decimals. 10 If the sum of the measures of two angles is not 45, then the two angles are not complementary. Any pair of complementary angles makes this statement and original false. Any pair of angles with a sum of measures not equal to 45 or 90 makes this statement and the original true. 2-6 Laws of Logic (page 36) 1 Margaret is a doctor. (Disjunctive Inference) 2 I will have straight teeth. (Detachment) 3 I did not save money. (Modus Tollens) 4 I helped my friend. (Modus Tollens) 5 It will not rain. (Detachment) 6 Joe is not a teacher. (Disjunctive Inference) 7 I will not eat dinner with Michael. (Modus Tollens) 8 I will not play on Saturday. (Detachment) 9 If I practice the violin, my friend will be jealous. (Chain Rule) 10 If I don’t help my brother, he cannot play football. (Chain Rule) 2-7 Proof in Logic (page 38) 1 1. p ∨ q 2. r → ~q 3. r 4. ~q 5. p 2 1. a → b 2. c ∨ a 3. ~c 4. a 5. b Law of Detachment (2, 3) Law of Disjunctive Inference (1, 4) Law of Disjunctive Inference (2, 3) Law of Detachment (1, 4) 2-7 Proof in Logic 5 3 1. 2. 3. 4. 5. 4 1. 2. 3. 4. 5. 6. 7. 5 1. 2. 3. 4. 5. 6. 7. 8. 9. 6 1. 2. 3. 4. 5. 6. 7. 7 1. 2. 3. 4. 5. 6. 7. 8. 9. 8 1. 2. 3. 4. 5. 6. 7. e ∨ ~f ~f → g ~e ~f g a∨b b→c c→d b→d ~d ~b a s∨t s→r r→q s→q ~q ~s t t→v v d→e d∨f h → ~e h ~e ~d f ~f → g ~f ∨ j g → ~h j→k h ~g f j k x→z x→y ~z ~y ~x x∨t t Law of Disjunctive Inference (1, 3) Law of Detachment (2, 4) Chain Rule (2, 3) Law of Modus Tollens (4, 5) Law of Disjunctive Inference (1, 6) Chain Rule (2, 3) Law of Modus Tollens (4, 5) Law of Disjunctive Inference (1, 6) Law of Detachment (8, 7) Law of Detachment (3, 6) Law of Modus Tollens (1, 5) Law of Disjunctive Inference (2, 6) Law of Modus Tollens (3, 5) Law of Modus Tollens (1, 6) Law of Disjunctive Inference (2, 7) Law of Detachment (4, 8) Law of Modus Tollens (1, 3) Law of Modus Tollens (2, 4) Law of Disjunctive Inference (5, 6) Chapter Review (pages 38–40) 1 (1) true 2 (1) true 3 (2) false 6 Chapter 2: Logic 4 (2) false 5 (1) If I am late for school, I did not set my alarm. 6 (2) If Marie is not bowling today, then today is not Monday. 7 (1) The statement is always true but its converse cannot be determined. 8 Hypothesis: The triangle has a 90 angle. Conclusion: The square of the longest side of a triangle is equal to the sum of the squares of the other sides. 9 Hypothesis: A and B are alternate interior angles. Conclusion: They are congruent. 10 False 11 True 12 True 13 False 14 False 15 Inverse: If the altitude does not bisect the base, the triangle is not isosceles. Converse: If the triangle is isosceles, the altitude bisects the base. Contrapositive: If the triangle is not isosceles, the altitude does not bisect the base. Biconditional: The altitude bisects the base if and only if it is isosceles. 16 True 17 False 18 p is true; q is false. 19 True 20 Converse: If a number is rational, then it is an integer. (True) Conditional statement is true. Biconditional: A number is an integer if and only if it is rational. 21 Converse: If a number is not prime, then a number is a perfect square. (True) Conditional statement is true. Biconditional: A number is a perfect square if and only if it is not prime. 22 Converse: If a parabola is tangent to the x-axis, then its roots are equal. (True) Conditional statement is true. Biconditional: The roots of a quadratic equation are equal if and only if the parabola is tangent to the x-axis. 23 Converse: If a number is real, then it is rational. (False) Conditional statement is true. 2 , then x √ 32 . (True) 24 Converse: If x 4 √ Conditional statement is true. 32 if and only if Biconditional: x √ √ x 4 2. 25 Converse: If a relation is a function, then it passes the vertical line test. (True) Conditional statement is true. Biconditional: A relation passes the vertical line test if and only if it is a function. 26 If I study, then I will not get poor grades. 27 If I get poor grades, then I did not study. 28 If I do not get poor grades, then I study. 29 If my baby cousin cries, then she gets a bottle. 30 If the figure is a rectangle, then two adjacent sides form a right angle. 31 If a triangle is equilateral, then it is equiangular. 32 I like math. 33 The class president is a girl. (Law of Disjunctive Inference) 34 My teacher will be happy. (Law of Detachment) 35 I do not go to the park. (Law of Modus Tollens) 36 If I learn to knit, I will save money. (Chain Rule) 37 I eat lunch. (Law of Modus Tollens) 38 I feel good. (Law of Detachment) 39 Rich likes lacrosse. (Law of Disjunctive Inference) 40 Marlene does not babysit. (Law of Modus Tollens) 41 Jack is using his computer. (Law of Modus Tollens) 42 If I read novels when my teacher assigns them, I watch less television. (Chain Rule) 43 If I like school, I won’t get a job after school. (Chain Rule) 44 1. 2. 3. 4. 5. a p∨t a → ~t ~t p 45 1. 2. 3. 4. x→y ~z ∨ x z x 5. 46 1. 2. 3. y c∨a ~c a 4. 5. 47 1. 2. 3. 4. 5. 6. 7. a→b b c → ~d d ~c ~b → c b a ∨ ~b a Law of Modus Tollens (1, 2) 48 1. 2. 3. 4. 5. 6. 7. ~m ∨ n ~m → r r → ~q ~m → ~q q m n Chain Rule (2, 3) 8. n → z 9. z Law of Detachment (3, 1) Law of Disjunctive Inference (2, 4) Law of Disjunctive Inference (2, 3) Law of Detachment (1, 4) Law of Disjunctive Inference (1, 2) Law of Detachment (4, 3) Law of Modus Tollens (4, 3) Law of Disjunctive Inference (6, 5) Law of Modus Tollens (4, 5) Law of Disjunctive Inference (1, 6) Law of Detachment (8, 7) Chapter Review 7 CHAPTER Introduction to Geometric Proof 3 3-1 Inductive Reasoning (page 42) 1 For a polygon with n sides and vertices, from each vertex, diagonals can be drawn to n 3 other vertices. This can happen n times. Each diagonal is counted twice from a to b and n(n 3) from b to a. There are _ diagonals. 2 Test: Triangle Quadrilateral Pentagon 0 4(4 3) _ 2 2 5(5 3) _ 5 2 2 n5 12345 54321 −−−−−−−−−−− 6 6 6 6 6 5(6) n(n 1) twice the sum The sum of the integers from 1 through n(n 1) n _. 2 3 Answers will vary. 9, 15, 21 are some possible answers. 4 Answers will vary. 3, 9, 15, 21 are some possible answers. 3-2 Definitions and Logic (page 43) 1 Line segments are congruent if and only if they have the same measure. 2 A point of a line segment is the midpoint if and only if it divides the line segment into two congruent segments. 8 3 A line is an angle bisector if and only if it divides an angle into two congruent parts. 4 Lines are perpendicular if and only if the lines form right angles. 5 A figure is a polygon if and only if it is a closed figure in the plane formed by three or more segments joined at their endpoints. Chapter 3: Introduction to Geometric Proof 3-3 Deductive Reasoning (page 44) 1 1. Given 2. Perpendicular lines form right angles. 3. Right triangles have one right angle. divides FEH into two adjacent angles 2 EG with the same measure. FEG and HEG are congruent because they have the same measure. An angle bisector is the ray that divides an angle into two adjacent congruent angles. 3 1. Given 2. Line segments equal in measure are congruent. 3. A midpoint divides a line segment into two congruent segments. −− 4 Since PR bisects SPQ, it divides the angle into two congruent angles. SPR QPR. If two angles are congruent, they have the same measure. 3-4 Indirect Proof (page 45) 1 Statements −−− −−− 1. CD and HK are not congruent. 2. CD HK −−− −−− 3. CD HK Reasons 1. Assumption. 2. Given. 3. Line segments that are equal in measure are congruent. Therefore, assumption is false. Reasons 2 Statements 1. ABC is not a right angle. −− −−− 2. AB CD 3. ABC is a right angle. 1. Assumption. 1. ABC is not an isosceles triangle. 2. A B 3. ABC is an isosceles triangle. 1. Assumption. 2. Given. 3. Perpendicular lines form right angles. Therefore, the assumption is false. Reasons 3 Statements 2. Given. 3. An isosceles triangle contains two congruent angles. Therefore, the assumption is false. Reasons 4 Statements 1. DB is the angle bi- 1. sector of ADC. 2. 2. mADB mBDC 3. DB 3. is not the bisector of ADC. Assumption. Given. An angle bisector divides an angle into two congruent parts. Therefore, the assumption is false. 3-5 Postulates, Theorems, and Proof (pages 47–49) 1 Yes 2 No 3 No 4 The symmetric property of equality 5 The reflexive property of congruence 6 The symmetric property and transitive property of congruence Reasons 7 Statements −− −−− 1. Given. 1. PQ QR 2. PQR is a right angle. 2. Perpendicular lines are two lines that intersect to form right angles. 3. mPQR 90 3. A right angle is an angle whose degree measure is 90. 4. Given. 5. Perpendicular lines are two lines that intersect to form right angles. 6. A right angle is an angle whose degree measure is 90. 7. Symmetric property of equality. 8. Transitive property of equality. −− −− 4. XY YZ 5. XYZ is a right angle. 6. mXYZ 90 7. 90 mXYZ 8. mPQR mXYZ 8 Statements Reasons is the angle 1. AC bisector of BAD. 1. Given. 2. BAC CAD 2. An angle bisector divides an angle into two congruent parts. 3. If two angles are congruent, they have the same measure. 4. Given. 3. mBAC mCAD is the angle 4. AD bisector of CAE. 5. CAD DAE 5. An angle bisector divides an angle into two congruent parts. 3-5 Postulates, Theorems, and Proof 9 6. BAC DAE 7. mBAC mDAE 9 Statements 6. Transitive property. 7. If two angles are congruent, their measures are equal. Reasons 1. 8 x y 1. Given. 2. y 3 2. Given. 3. 8 x 3 3. Substitution property. 4. x 3 8 4. Symmetric property. Reasons 10 Statements 1. M is the midpoint 1. Given. −− of AB. −−− −−− 2. A midpoint 2. AM MB divides a line segment into two congruent line segments. −−− −− 3. Given. 3. MB BC −−− −− 4. Transitive 4. AM BC property. 3-6 Remaining Postulates of Equality (pages 51–52) 1 Partition postulate of equality 2 Division postulate of equality 3 Addition postulate 4 Subtraction postulate 5 Division postulate of equality Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 10 Chapter 3: Introduction to Geometric Proof 6 1. 2. 3. 4. 5. 7 1. 2. 3. 4. 5. 6. 7. 8 1. m1 m2 m3 m4 m1 m3 m2 m4 mDAB m1 m3 mBCD m2 m4 mDAB mBCD (Substitution postulate) −− −− AB CB −−− −− AD CE AB CB AD CE AB AD CB CE DB EB −− −− DB EB (Line segments that are equal in measure are congruent.) AB AC 1 AC 2. AD _ 3 1 _ 3. AE AB 3 4. AD AE (Division postulate) 9 1. mEAB mFBC is the angle bisector of EAB. 2. AG 3. BH is the angle bisector of FBC. 1 mEAB 4. m1 _ 2 1 mFBC 5. m2 _ 2 6. m1 m2 (Division postulate) 10 1. AB DE 2. AC 3AB 3. DF 3DE 4. AC DF (Multiplication postulate) Chapter Review (page 52) 1 Reflexive property of equality 2 Transitive property of equality 3 Symmetric property of equality 1 mBAC 4 mBAD _ 2 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −−−− 5 1. CMD is a line segment. −−− −−− 2. CM MD 3. M is the mid- (Definition of a midpoint) −−− point of CD. 6 1. 2. 3. 4. 5. 6. 7. 7 1. 2. 3. 4. 5. 6. 7. 8 1. 2. 3. 4. 5. −−−− ABCD is a line segment. −− −−− AB CD BC BC AB CD AB BC CD BC AC BC −− −− AC BD (Line segments that are equal in measure are congruent.) −− −−− AB CD AB CD −−− −− AB and CD bisect each other at E. 1 AB AE _ 2 1 _ CE CD 2 AE CE −− −− AE CE (Line segments that are equal in measure are congruent.) −− −−− PQ QR mPQR 90 mPQR mABC mABC 90 −− −− AB BC (Perpendicular lines form right angles.) 9 1. 2. 3. 4. 5. 6. 7. 10 1. 2. 3. 4. 5. 6. 11 1. 2. 4. 5. 6. 7. 8. 12 1. 2. 3. 4. 5. 6. 7. 8. PQ XY QR YZ PQ QR XY YZ PQ QR PR XY YZ XZ PR XZ −− −− PR XZ (Line segments that are equal in measure are congruent.) DAB and EBA are straight angles. mDAB 180 mEBA 180 m1 m2 mDAB m1 mEBA m2 m3 m4 (Substitution postulate) −− −− AC BC AC BC −−− −− DC EC DC EC AD DC BC EC AD BE −−− −− AD BE (Line segments that are equal in measure are congruent.) −−− −− AD BC AD BC −−− E is the midpoint of AD. 1 AD DE _ 2 −− F is the midpoint of BC. 1 BC BF _ 2 BF DE −− −− BF DE (Line segments that are equal in measure are congruent.) Chapter Review 11 CHAPTER Congruence of Lines, Angles, and Triangles 4 4-1 Setting Up a Valid Proof (pages 56–57) , BD 1 Given: ABC AC Statements 1. PQRS −− −− 2. PR QS −−− −−− 3. QR QR D 4. PR QR QS QR −− −− 5. PQ RS A B C Prove: ABD CBD Statements 1. ABC 2. BD AC 3. ABD is a right angle. 4. CBD is a right angle. 5. ABD CBD −− −− , PR QS 2 Given: PQRS P Q R S Reasons 1. Given. 2. Given. 3. Definition of perpendicular lines. 4. Definition of perpendicular lines. 5. Congruent angles are angles that have the same measure. −−− −− 3 Given: AD BC −−− −− EF bisects AD −− −− GF bisects BC −− −− Prove: AE BG Statements −−− −− 1. AD BC −−− −− 2. EF bisects AD −− −− 3. GF bisects BC −− −−− 4. AE AD −− −− 5. BG GC 6. AE ED 7. BG GC −− −− Prove: PQ RS 8. AE ED AD 12 Chapter 4: Congruence of Lines, Angles, and Triangles Reasons 1. Given. 2. Given. 3. Reflexive property of congruence. 4. Subtraction postulate 5. Substitution postulate Reasons 1. Given. 2. Given. 3. Given. 4. Definition of a bisector. 5. Definition of a bisector. 6. Definition of congruent segments. 7. Definition of congruent segments. 8. Partition postulate. 9. BG GC BC 10. AE AE AD or 2AE AD 9. Partition postulate. 10. Substitution postulate. 11. BG BG BC or 11. Substitution postulate. 2BG BC 1 12. Division 12. AE _AD 2 postulate. 1 _ 13. Division 13. BG BC 2 postulate. 14. Definition of 14. AD BC congruent segments. 1 _ 15. Substitution 15. AE BC 2 postulate. 1 _ 16. Symmetric 16. BC BG 2 property. 17. Transitive 17. AE BG property. −− −− 18. Segments with 18. AE BG equal measures are congruent. 4 Given: URS UTS −− US bisects RUT −− US bisects RST Prove: 1 2 Statements Reasons 1. Given. 1. URS UTS −− 2. US bisects RUT 2. Given. −− 3. US bisects RST 3. Given. 4. Definition of an 4. 1 TUS angle bisector. 5. Definition of an 5. RSU 2 angle bisector. 6. Congruent 6. m1 mTUS angles are equal in measure. 7. Congruent 7. mRSU m2 angles are equal in measure. 8. Partition 8. m1 mTUS postulate. mRUT 9. mRSU m2 mRST 9. Partition postulate. 10. m1 m1 mRUT or 2(m1) mRUT 10. Substitution postulate. 11. Substitution 11. m2 m2 postulate. mRST or 2(m2) mRST 1 mRUT 12. Division 12. m1 _ 2 postulate. 1 13. m2 _mRST 13. Division 2 postulate. 14. Congruent an14. mRUT gles are equal in mRST measure. 1 _ 15. m1 mRST 15. Substitution 2 postulate. 1 _ 16. Symmetric 16. mRST 2 2 property. 17. Transitive 17. m1 m2 property. 18. Congruent an18. 1 2 gles are equal in measure. −− −− 5 Given: HJ FK −−− −− Prove: HK FJ Statements −− −− 1. HJ FK 2. HJ FK 3. JK JK 4. HJ JK HK 5. FK JK FJ 6. HJ JK FK JK 7. HK FJ −−− −− 8. HK FJ Reasons 1. Given. 2. Definition of congruent segments. 3. Reflexive property. 4. Partition postulate. 5. Partition postulate. 6. Addition postulate. 7. Substitution postulate. 8. Segments with equal measures are congruent. 4-1 Setting Up a Valid Proof 13 bisects CBE 6 Given: BA 1 2 2 4 Prove: 3 4 Narrative Proof: A bisector divides an angle into two congruent angles, so 1 2. Using the transitive property of congruence, 1 4. By substitution, 3 4. 7 4x 3 2x 21 x 12 mBCA 4x 3 4(12) 3 45 mBCE 2(45) 90 8 4x 4 2(3x 21) x 23 AB 4x 4 4(23) 4 96 9 Since DE CE, then 2x y 5y, and x 2y. AC BD 6 5y (2x y) (x y) 6 5y 3x 2y 6 5y 3(2y) 2y 6 5y 6y 2y 6 3y 2y x 2(2) 4 BD 2(4) 2 4 2 16 10 AB DE 8, so BD 4 and CD 2. Therefore, CE 8 2 10. 4-2 Proving Theorems About Angles (pages 59–61) 1 If two angles are right angles, they both have a measurement of 90. Angles with the same measure are congruent. 2 If two angles are straight angles, they both have a measurement of 180. Angles with the same measure are congruent. 3 If 1 is a complement of A, then m1 90 mA. If 2 is a complement of A, then m2 90 mA. Then m1 m2 by the symmetric property and transition, and 1 2 by definition of congruence. 14 4 If 1 2, 3 is the complement of 1, and 4 is the complement of 2, then m3 90 m1 and m4 90 m2. Since m1 m2 by congruence, m3 m4 by symmetric property and transition, and 3 4 by congruence. 5 If 1 is a supplement of A, then m1 180 mA. If 2 is a supplement of A, then m2 180 mA. Then m1 m2 by the symmetric property and transition, and 1 2 by definition of congruence. 6 If 1 2 and 3 is the supplement of 1 and 4 is the supplement of 2, then m3 90 m1 and m4 90 m2. Since m1 m2 by congruence, m3 m4 by symmetric property and transition, and 3 4 by congruence. 7 By definition the sum of their measures is a straight angle or 180. 8 The two angles are a linear pair so the sum of their measures is 180. If they are congruent, each measure is 90; they form right angles and are therefore perpendicular. 9 Each angle forms a linear pair with the same angle. They are supplementary to the same angle and are therefore congruent. 10 m2 m3 because they are vertical angles. By substitution, 1 is complementary to 3. 1 4 because complements of the same angle are congruent. 11 3 forms a linear pair with 1, 3 is supplementary to 1. Since 1 2, 3 is supplementary to 2. 4 forms a linear pair with 2, 4 is supplementary to 2. 3 4 because supplements of the same angle are congruent. 12 Since they form a linear pair, 2 is supplementary to EDG. Since 1 and 2 have the same measure (they are congruent), 1 is supplementary to EGD. 13 Since 1 is complementary to 3, m1 m3 90. Since ACB is a right angle, m1 m2 90. By subtraction and substitution, m2 m3, so they are congruent angles. 14 By the transitive postulate, 1 3. Because they are vertical angle pairs, 1 4, and 3 6. By substitution and transition, 4 6. Chapter 4: Congruence of Lines, Angles, and Triangles 15 x 25 180 x 155 16 2x x 180 x 60 17 x 85 18 (x 10) x 90 x 40 19 60 x x x 180 x 40 20 6x 20 4x 10 x 15 21 40 mz 180 mz 140 22 a mDOE 75 b mAOB 15 c mDOC 105 23 2x 15 4x 1 x7 mPRT 2(7) 15 29 mRTQ 180 mPRT 180 29 151 24 3x 25 10x 4 x3 mWVY 3(3) 25 34 mWVZ 180 mWVY 180 34 146 25 (5x 2y) (5x 2y) 180 x 18 mAEC mBED 11y 5x 2y 11y 5(18) 2y 9y 90 y 10 mAED mCEB 5x 2y 5(18) 2(10) 70 4-3 Congruent Polygons (pages 64–66) 1 (3) IHG MNL 2 (4) a side of one is congruent to a side of the other 3 (4) IV 4 (1) I 5 (2) II −− 6 a AC b BAC −− c BC d CBA 7 a ACD BCD b ADC CBA 8 a mE 44 b mB 110 c mC 26 9 Answers will vary. For instance, a 3-4-5 triangle has the same angle measure as a 6-8-10 triangle. They are not congruent. 10 Answers will vary. 11 Answers will vary. 12 a Vertical angles are congruent; the triangles are congruent by SAS. b AC BD 2x x 5 x5 AC BD 10 AE BE 11 CE DE 12 13 The triangles are congruent by SAS. ACE BDE by corresponding parts of congruent triangles are congruent. Therefore, ACE BDE. So 5x 7x 18, and x 9. mACE 45; mBDE 45; mBED 45. mDBE 180 45 45 90. Therefore, DBE is a right triangle. 2 14 x 4x x 14 2 x 5x 14 0 (x 7)(x 2) 0 x 7 and x 2 When x 7, AB AB 21 When x 2, AB AB 12 15 No, the angle must be included between the pairs of sides. (page 67) Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 4-3 Congruent Polygons 15 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 2 1. 2. 3. 4. 5. 1 6. 1. 2. 3. 4. 5. 6. 7. 8. 4 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 5 1. 2. 3. 4. 5. 6. 6 1. 2. 3. 3 16 −−−− ABCD 1 ___ 2 −− AF DE −− −− AC BD AC BD BC BC (Reflexive property) AC BC BD BC AB CD −− −−− AB CD ABF DCE (SAS SAS) −−− −−− ABC, DEF 3 4 −− −− BF EC 1 2 −−− −−− FBC CEF (If two angles are congruent, their supplements are congruent.) BCF EFL (ASA ASA) −− −− BE is a median to FD. −− −− FE DE −− −− BE BE −−− −− AD CF −− −− AB CB AD AB CF CB −− −− BD BF FBE DBE (SSS SSS) −− −−− AE DC −− −− DE DE AE DE DC DE −−− −− AD EC 3 4 ADB and 3 are linear pairs. ADB and 3 are supplements. CEB and 4 are linear pairs. CEB and 4 are supplements. ADB CEB 1 2 ADB CEB (ASA ASA) −− D is the midpoint of AB. AD DB −−− −− AD DB −− −− AC BC −−− −−− DC DC ADC BDC (SSS SSS) −− −− AB BC −− −− EF AB −− −− EF BC 4. 5. 6. 7. 7 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 8 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. −− −− BE bisects CF at D. −−− −− CD DF 1 2 BCD EFD (SAS SAS) −− −− AC BD −− −− BC BC AC BC BD BC −− −−− AB DC 3 4 3 and FBA are linear pairs. 3 and FBA are supplements. 4 and ECD are linear pairs. 4 and ECD are supplements. FBA ECD 1 2 EDC FAB (ASA ASA) −− is the perpendicular bisector of AB. EG −− −− AF BF EFA is a right angle. EFB is a right angle. EFA EFB mEFA m1 mEFB m1 1 2 mEFA m1 mEFB m2 DFA CFB −− −− DF CF ADF BCF (SAS SAS) Chapter Review (pages 68–69) 1 Given: ABC and DBE are vertical angles. PQR ABC Prove: PQR DBE intersect at E. and CD 2 Given: AB AEC FGH Prove: CEB is supplementary to FGH. are perpendicular lines. and CED 3 Given: AEB Prove: AEC AED are perpendicular lines. and CED 4 Given: AEB . F is not on CD . Prove: FE is not perpendicular to AB −− is a bisec5 Since E is the midpoint of AB, CD −− tor of AB. Since AEC BEC and they are a linear pair, the measure of each must be 180 _ 90. 2 Chapter 4: Congruence of Lines, Angles, and Triangles 6 1 3 because they are vertical angles. 4 2 because they are vertical angles. By the substitution postulate, 2 3. By the transitive postulate, 4 3, or 3 4 by the symmetric property. −− 7 Since PQ bisects ABC, then 2 CBQ. Since CBQ and 1 are vertical pairs, CBQ 1. By the transitive postulate, 2 1, or 1 2 by the symmetric property. 8 Since CDE is a right triangle, the two angles that are not the right angle are complementary because there sum is 180 (whole triangle) 90 (right angle) 90. Therefore, FED is complementary to FCD. By substitution, EDF is complementary to FCD. 9 mAED mBED 180 4x 20 9x 48 180 x 16 mCEB mAED 4x 20 4(16) 20 84 10 mAED mCEB 5x 7x 26 x 13 mAED 5(13) 65 mAED mDEB 180 65 mDEB 180 mDEB 115 11 mAED mCEB 12x 6x 10y 6x 10y mAED mDEB 180 12x 6x 180 18x 180 x 10 6x 10y 60 10y y6 mAEC mDEB 10y 10(6) 60 12 mCEB mDEB 180 9y 12y 12 180 y8 mAEC mDEB 12(8) 12 108 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −− −− 13 1. AC DF 2. A D 3. C F 4. ABC DEF (ASA ASA) −− −− 14 1. HE FE −−− −− 2. HG FE −− −− 3. HF HF 4. EFH GHF (SSS SSS) 15 1. A, B, C, D lie on circle O. −−−− −−−− 2. AOC; BOD −−− −− −−− −−− 3. AO, BO, CO, DO are radii. −−− −−− 4. AD CO −− −−− 5. BO DO 6. AOB COD (If two lines intersect, the vertical angles are congruent.) 7. ABO CDO (SAS SAS) −− −− 16 1. QT RT 2. QT RT −− −− 3. TV TU 4. TV TU 5. QT TV RT TV 6. QT TV RT TU −−− −−− 7. QTV; RTU 8. QU RU −−− −−− 9. QV RU −− −− 10. PU VS −− −−− −− −−− 11. PU UV VS UV −− −− 12. PV SU −− −− 13. PQ RS 14. PQU SRU (SSS SSS) −− 17 1. C is the midpoint of AE. −− −− 2. AC CE 3. 1 2 4. 3 BCD 5. 4 BCD 6. 3 4 7. ABC EDC (ASA ASA) Chapter Review 17 18 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. −− I is the midpoint of EG. −− −− EI IG EI IG −− −− −− EG EI IG EG EI IG EG EI EI or EG 2EI EH 2EI EG EH −− −− EG EH FGE IEH FEG HIE FGE IEH (ASA ASA) 19 1. 2. 3. 4. 5. 20 1. 2. 3. 4. 5. 6. 7. 8. 9. AEB, CED are right angles. AEB CED A C −− −− AE CE ABC CDE (ASA ASA) −−− −−− DC AD −− −−− AB AD −−− −− DC AB −− −− DF BE −− −− EF EF DF EF BE EF −− −− DE BF 1 2 ABF DCE (SAS SAS) CHAPTER Congruence Based on Triangles 5 5-1 Line Segments Associated With Triangles (page 71) 1 One line 2 C Median A 3 4 5 6 18 Altitude E F D Angle Bisector B −− −− AE and BE −−− −− AB and CD ACF and BCF ADC and BDC Chapter 5: Congruence Based on Triangles 5-2 Using Congruent Triangles to Prove Line Segments Congruent and Angles Congruent (pages 72–73) Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 1 1. 2. 3. 4. 5. −− −−− AB CD −− −−− BC DA −− −− AC AC ABC CDA BAC DCA 2 1. 2. 3. 4. 5. −− −− BA BC −−− −−− DA DC −− −− DB DB ABD CBD ABC CBD 3 1. 2. 3. 4. 5. 1 3 2 4 −− −− AC AC DAC BCA −−− −− AD CB 4 5 6 7 (SSS SSS) (Corresponding parts of congruent triangles are congruent.) (SSS SSS) (Corresponding parts of congruent triangles are congruent.) (ASA ASA) (Corresponding parts of congruent triangles are congruent.) −−− 1. CD is the median drawn from C. −−− −− 2. AD DB −−− −− 3. CD AB 4. ADC is a right angle. 5. BCD is a right angle. 6. ADC BDC −−− −−− 7. CD CD 8. ADC BDC (SAS SAS) −− −− (Corresponding 9. CA CB parts of congruent triangles are congruent.) RS RS 4x 1 3x 3 x4 RT RT x 6 4 6 10 AD CB 2x 5 3x 7 x 12 AB x 10 12 10 22 CD AB 22 AD 2x 5 2(12) 5 29 CB 3x 7 3(12) 7 29 The triangle is isosceles and RS RT. 6x 4 3x 11 x5 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 8 1. 1 3 2. 2 4 −− −−− 3. AB CD 4. ABE CDE (ASA ASA) −− −− 5. AE CE −− −− 6. BD bisects AC. −− −− 7. BE DE −− −− (Definition of a 8. AC bisects BD. bisector) −−− 9 1. DA bisects BDF. 2. FDA BDA 3. 1 2 4. m1 mFDA m2 mFDA 5. m1 mFDA m2 mBDA 6. EDA CDA −−− −− 7. CD DE −−− −−− 8. AD AD 9. EDA CDA (SAS SAS) −− −− (Corresponding 10. AE AC parts of congruent triangles are congruent.) −− 10 1. BA is a median of CBF. −− −− 2. CA FA −−− −− 3. CD FE −− −−− 4. FC CD −− −− 5. FC EF 6. DCA is a right angle. 7. EFA is a right angle. 8. DCA EFA 9. DCA EFA (SAS SAS) −−− −− (Corresponding 10. DA EA parts of congruent triangles are congruent.) 5-2 Using Congruent Triangles to Prove Line Segments Congruent and Angles Congruent 19 5-3 Isosceles and Equilateral Triangles 6. SRU TRU (page 74) −−− 7. RU bisects SRT. 1 Statements 1. Construct a median from the vertex formed by the two congruent sides of the triangle, to form RST with −−− median RU. −− −− 2. RS RT −− −− 3. SU TU −−− −−− 4. RU RU 5. RSU RTU 6. RSU RTU 2 Statements 1. Construct a median from the vertex formed by the two congruent sides of the triangle, to form RST with −−− median RU. −− −− 2. RS RT −− −− 3. SU TU −−− −−− 4. RU RU 5. RSU RTU 20 Reasons 1. A median of a triangle is a line segment with one endpoint at any vertex of the triangle, extending to the midpoint of the opposite side. 2. Definition of an isosceles triangle. 3. Definition of a median. 4. Reflexive property of congruence. 5. SSS SSS. 6. Corresponding parts of congruent triangles are congruent. Reasons 1. A median of a triangle is a line segment with one endpoint at any vertex of the triangle, extending to the midpoint of the opposite side. 2. Definition of an isosceles triangle. 3. Definition of a median. 4. Reflexive property of congruence. 5. SSS SSS. Chapter 5: Congruence Based on Triangles 3 Statements 1. Construct a median from the vertex formed by the two congruent sides of the triangle, to form RST with −−− median RU. −− −− 2. RS RT −− −− 3. SU TU −−− −−− 4. RU RU 5. RSU RTU 6. SUR TUR 7. SUR is a right angle; TUR is a right angle. −−− −− 8. RU ST 4 Statements 1. Construct a median from the vertex formed by the two congruent sides of the triangle, to form ABC with −−− median AX. −− −− 2. AB AC 6. Corresponding parts of congruent triangles are congruent. 7. Definition of angle bisector. Reasons 1. A median of a triangle is a line segment with one endpoint at any vertex of the triangle, extending to the midpoint of the opposite side. 2. Definition of an isosceles triangle. 3. Definition of a median. 4. Reflexive property of congruence. 5. SSS SSS. 6. Corresponding parts of congruent triangles are congruent. 7. Adjacent congruent angles are supplementary. 8. Definition of perpendicular lines. Reasons 1. A median of a triangle is a line segment with one endpoint at any vertex of the triangle, extending to the midpoint of the opposite side. 2. Definition of an equilateral triangle. −− −− 3. BX XC −−− −−− 4. AX AX 5. ABX ACX 6. ABX ACX 7. Construct a sec−− ond median, BY, from vertex B. −− −− 8. BA BC −− −− 9. AY YC −− −− 10. BY BY 11. BAY BCY 12. BAY BCY 13. ABC is equiangular. 3. Definition of a median. 4. Reflexive property of congruence. 5. SSS SSS. 6. Corresponding parts of congruent triangles are congruent. 7. A median of a triangle is a line segment with one endpoint at any vertex of the triangle, extending to the midpoint of the opposite side. 8. Definition of an equilateral triangle. 9. Definition of a median. 10. Reflexive property of congruence. 11. SSS SSS. 12. Corresponding angles of congruent triangles are congruent. 13. BCY is the same as ACX, and all three angles of the triangle are equal. 5 mA mC 3x 5 mA mB mC 180 (3x 5) (4x 10) (3x 5) 180 10x 180 x 18 mC 3(18) 5 59 6 The measure of each angle is 60. So, 3x 6 60; x 22. And 3y 6 60; y 18. 7 Show that RXZ TYZ SYX by −− −− −− SAS SAS, and XY YZ ZX because corresponding parts of congruent triangles are congruent. −− −− 8 Construct AC and BD, and label their inter−− −− section E. ABE CBE. AC and BD are perpendicular. AED and CDE are right angles; therefore, AED CDE. −− −− −− −− AE CE. ED ED. AED CED. CAD ACD. 9 mA mB mC 180 (6x 12) (8x 8) (3x 6) 180 17x 10 180 x 10 mA 72; mB 72; mC 36 The triangle is isosceles. 10 mB 180 2x 5-4 Working With Two Pairs of Congruent Triangles (pages 75–76) Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 1 1. ABC EFG −− −− 2. AC EG −− 3. BD is a median. 4. D is the midpoint (Definition of −− median) of AC. 1 5. DC _AC (Definition of mid2 point) −− 6. FH is a median. −− 7. H is the midpoint of EG. 1 EG 8. HG _ 2 9. AC EG 1 AC 10. HG _ 2 11. HG DC 5-4 Working With Two Pairs of Congruent Triangles 21 −−− −−− 12. HG DC −− −− 13. FG BC 14. BCD FGH 15. BCD FGH (SAS SAS) −− −− 16. BD FH −− −− 2 a 1. PR and QS bisect each other at V. −− −− 2. PV RV −−− −− 3. QV SV 4. PVQ RVS (Vertical angles are congruent.) 5. PQV RSV (SAS SAS) b A continuation of part a. 6. UPV TRV (Corresponding parts of congruent triangles are congruent.) 7. PVU RVT (Vertical angles are congruent.) 8. PUV RTV (ASA ASA) −− −−− 3 a 1. AC AD −− −− 2. BC BD −− −− 3. AB AB 4. ABC ABD (SSS SSS) 5. BAC BAD (Corresponding parts of congruent triangles are congruent.) −− −− 6. AE AE 7. ACD ADE (SAS SAS) b A continuation of part a. 8. 1 2 (Corresponding parts of congruent triangles are congruent.) −−− −− 4 1. CD CB 2. 1 2 −− −− 3. CF CF 4. CFB CFD (SAS SAS) 5. CBF CDF (Corresponding parts of congruent triangles are congruent.) 6. EDF ABF 7. BFA EFD −− −− 8. DF BF 9. BFA DFE (ASA ASA) 10. 3 4 (Corresponding parts of congruent triangles are congruent.) 22 Chapter 5: Congruence Based on Triangles 5 1. 2. 3. 4. 5. 6. 7. 8. −− −− AC BC −− −− AE BE −− −− EC EC ACE BCE ACE BCE −−− −−− CD CD ACD BCD −−− −− AD BD (SSS SSS) (SAS SAS) (Corresponding parts of congruent triangles are congruent.) −− −−− SP QR 1 2 −− −− AB bisects QS at C. −− −−− SC QC (Definition of bisector) −− −− 5. SQ SQ 6. SQP QSR (SAS SAS) 7. SQP QSR (Corresponding parts of congruent triangles are congruent.) 8. ACQ BCS 9. ACQ BCS (ASA ASA) b Continuation of part a. −− −− (Corresponding 10. AC BC parts of congruent triangles are congruent.) 6 a 1. 2. 3. 4. 5-5 Proving Overlapping Triangles Congruent (page 77) Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −− −− 1 1. AE BE 2. EAB EBA (If two sides of a triangle are congruent, the angles opposite these sides are congruent.) 3. 4. 5. 2 1. 2. 3. 4. 5. 6. 7. 3 1. 2. 3. 4. 5. 4 1. 2. 3. 4. 5. 6. 7. 8. 9. 5 1. 2. 3. 4. 5. 6 1. 2. 3. 4. 5. 6. 7. 8. −− −− AB AB −− −− AC BD ABC BAD (SAS SAS) 1 3 2 2 1 3 2 3 ADB EDC (Partition postulate) 4 5 −−− −− AD ED ADB EDC (ASA ASA) −− −− AE BE −− D is the midpoint of AB. −−− −− AD BD −− −− ED ED ADE BDE −− −− TP TQ TPQ TQP SPQ RQP SPT RQT STP RTQ (Vertical angles are congruent.) STP RTQ (SAA SAA) −− −− SP PQ −− −− PQ PQ SPQ RQP −− −− EF GF −− −− DF HF DFH DFH DFG EFH (SAS SAS) 1 2 (Corresponding parts of congruent triangles are congruent.) −− −−− AD and FC bisect each other at G. −−− −−− AG DG −− −− FG CG AGC DGF (Vertical angles are congruent.) AGC DGF (SAS SAS) GDE GAC (Corresponding parts of congruent triangles are congruent.) AGB DGE AGB DGE (ASA ASA) 5-6 Perpendicular Bisectors of a Line Segment (pages 79–80) −− 1 Given AB and P and Q, such that PA PB −− −− and QA QB, let AB intersect PQ at point M. APQ BPQ. APM BPM. −−− −−− APM BPM. AM BM. Therefore, AMP BMP. −− 2 a) Given AB and point P such that PA PB; if two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment. Select a second point that is equidistant from both A and B, for example −− −−− the midpoint of AB, M. Then PM is the per−− pendicular bisector of AB. b) Given point P on the perpendicular bisec−− −− tor of AB, let point M be the midpoint of AB. PMA PMB. −− 3 The perpendicular bisector of PQ Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −− −− 4 1. JL KL −− −−− 2. JM KM 3. JL KL (Congruent segments are equal in length.) 4. JM KM −−− −− 5. LM is the perpendicular bisector of JK. (Two points, each equidistant from the endpoints of a line segment, determine the perpendicular bisector of the line segment.) 6. N is the midpoint of JK. (A point on the perpendicular bisector of a line segment is equidistant from the endpoints of the line segment.) −− −−− (Definition of a midpoint) 7. JN KN 5-6 Perpendicular Bisectors of a Line Segment 23 −− −− 5 1. FG is the perpendicular bisector of HI. −− −− 2. JG is the perpendicular bisector of HI. (F, J, and G are collinear.) −− − 3. HJ IJ −− −− 4. GJ GJ 5. HJG and IJG are right angles. (Definition of perpendicular bisector) 6. HJG IJG (Right angles are congruent.) 7. HGJ IGJ (SAS SAS) 8. 1 2 (Corresponding parts of congruent triangles are congruent.) −−− −− 6 1. AB and CD bisect each other. −− −−− 2. AB CD −− −−− 3. AC AD −− −− 4. BC BD −−− −− 5. AD BD −− −− 6. AC BC 7. ACBD is an equilateral quadrilateral. (Definition of equilateral quadrilateral) 7 1. 1 2 2. 3 4 −− −− 3. XZ XZ 4. XVZ XWZ (ASA ASA) −−− −−− (Corresponding parts 5. XV XW of congruent triangles are congruent.) 6. XV XW −−− −−−− 7. XZY; VYW −− −−−− 8. XY is the perpendicular bisector of VYW. (A point equidistant from the endpoints of a line segment is on the perpendicular bisector of the line segment.) 8 BP CP 4y 4 y1 AP CP xy4 x14 x5 9 AP BP 3x y x y xy BP CP xy4 xx4 x2 y2 10 CG AG 10 24 Chapter 5: Congruence Based on Triangles 5-7 Constructions (page 83) 1 (4) BAC and GHI 2 A B 3 C V T R S 4 A B C D E F 5 1 2 3 6 If the legs of the compass form two sides of a triangle, then the line segment connecting the pencil to the point is the third side. If the angle between the legs does not change (and the legs remain the same length), then any line segment connecting the legs is congruent to the original because of SAS congruence. 7 A F G B H C E D L The first arc drawn in the construction creates an isosceles triangle, BGH. The rest of the construction copies the sides of BGH to a new triangle. ED BH, FD GH, and FE GB, since F and G lie on the intersections. Therefore, BGH EFD by SSS congruence and all corresponding angle measures, including B and E, are equal. 8 Draw a point. Draw an arc centering on the point. Draw two lines from the point intersecting the arc. Connect the points of intersection. This will form an isosceles triangle. 9 13 C E D A 10 P C D B A B A 14 C D B C D T A 11 B 15 B B M D C A 12 A C D B 16 A A C D B C 5-7 Constructions 25 Chapter Review (pages 84–85) Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −−− 1 1. MN bisects PNQ. 2. RND RNQ 3. RNP RQN −−− −−− 4. RN RN 5. RPN RQN (ASA ASA) −− −− 2 1. PQ PR −− 2. QT is a median. −− −− 3. PT RT −− 4. RS is a median. −− −− 5. PS QS 6. PQT PRS (SSS SSS) 3 1. 3 4 −− −− 2. DE DF −−− −−− 3. DC DC 4. DEC DFC (SAS SAS) 5. DCE DCF (Corresponding parts of congruent triangles are congruent.) 6. EDC FDC 7. mCAD m1 mEDC mDCE 180 8. mCBD m2 mFDC mDCF 180 9. mCAD mCBD 10. CAD CBD 11. ABC is isosceles. (Definition of isosceles triangle) 4 1. ABC is an equilateral triangle. ___ ___ (Definition of equi2. AB AC lateral triangle) 3. DCB DBC 4. DB ___ DC ___ 5. AD AD 6. ABD ACD (SSS SSS) 7. BAD CAD (Corresponding parts of congruent triangles are congruent.) −−− 8. AD bisects BAC. (Definition of an angle bisector) 26 Chapter 5: Congruence Based on Triangles 5 ____ ___ 1. TM TA 2. MTA is isosceles. ___ (Definition of isosceles triangle) ___ ___ (In an isosceles triangle, the bisector is the altitude.) ___ (SSS SSS) (Corresponding parts of congruent triangles are congruent) 3. TH ___ bisects MTA. 4. TH is an altitude of MTA. 6 1. 2. 3. 4. 5. 6. 7. 8. 9. AB AD ___ ___ CB CD ___ ___ AC AC ABC ADC BAE DAE ___ AE AE ABE ADE ___ ___ BE DE DAB is isoceles. ___ 10. AE is the median of DAB. ___ 11. AE is the altitude of DAB. 7 8 9 ___ ___ (ASA ASA) (Definition of isosceles triangle) (Definition of median) (In an isosceles triangle, the median is the altitude.) AB BD A D DBA CBE EBD EBD DBA EBD CBE EBD EBA CBD ABE DBC (ASA ASA) ADE BDC EDB EDB ADE EDB BDC EDB ADB EDC DAE DEC ___ ___ DA DE DAB DEC (ASA ASA) 1 2 3 ___ 4 ___ DE DF DEG____ DFH (ASA ASA) ___ GD HD (Corresponding parts of congruent triangles are congruent.) 6. EDF EDF 1. 2. 3. 4. 5. 6. 7. 1. 2. 3. 4. 5. 6. 7. 1. 2. 3. 4. 5. 1 EDF 2 EDF GDF EDH FGD EDH (ASA ASA) 5 6 (Corresponding parts of congruent triangles are congruent.) ___ ___ and BD bisect each other at G. 10 1. AC ___ ____ AG 2. CG ___ ___ 3. BG DG 4. AGB CGD (Vertical angles are congruent.) 5. AGB CGD (ASA ASA) ___ ___ 6. AB CD 7. GCD GAB (Corresponding parts of congruent triangles are congruent.) 8. 1 2 9. CED AFB (ASA ASA) ___ ___ (Corresponding 10. EC FA parts of congruent triangles are congruent.) 11 Assume that ___ ABC and _____ABC are congruent and that BD and BD are angle bisectors of B and B, respectively. ____ ___ AB _____ AB. A A. ABD ABD. ___ ABD ABC. BD BD. 12 1. ___ BIG___ is equilateral. ___ 2. IA BC GT 3. __ IA ___ BC ___ GT 4. IB BG GI 5. IB BG GI 6. IB IA AB; BG BC CG; GI GT TI 7. IA AB BC CG GT TI 8. IA AB IA BC CG BC GT TI GT 9. AB TI ___ __ ___ CG 10. AB CG TI 11. BIG IGB GBI 12. IAT BCA (ASA ASA) GTC ___ ___ ___ (Corresponding 13. AT CA TC parts of congruent triangles are congruent.) 14. CAT is (Definition of an equilateral. equilateral triangle) 7. 8. 9. 10. 15 16 17 18 19 ___ RU ___ ___ RT US RT US RT ST US ST RS ___ TU ___ RS TU R U VST WTS mVST mWTS VSR is the complement of VST. WTU is the complement of WTS. VSR WTU RVS UWT (ASA ASA) ____ ____ MQ NQ ___ ____ QP QO ___ ____ PQ MQ MQP____ is a right angle. ____ OQ NQ NQO is a right angle. MQP NQO (Right angles are congruent.) 8. PQO PQO 9. MQP PQO NQO PQO 10. MQO NQP 11. MQO NQP (SAS SAS) ___ ___ 1. AC BC 2. ACF BCG 3. DCF ECG 4. DCF ACF BCG ACF 5. DCF ACF BCG ECG 6. ACD BCE 7. CAF CBA 8. ___ CAD___ CBE (ASA ASA) 9. DC EC 10. DCE is isosceles. (Definition of an isosceles triangle) Draw a line longer than the sum of ___ the lengths of the two segments. Copy AB onto the new line. Place the compass vertex where the arc swing___ intersects the line and mark off the length of CD. The line segment from the original vertex to the final arc swing marks off the___ new segment. Bisect AB and then bisect each half of the original segment. Use angle bisector procedure. ___ Bisect side AB. Mark the point where the bisector intersects the line M. Draw a line from C to M. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14 1. 2. 3. 4. 5. 6. 7. 13 Chapter Review 27 20 Use constructing congruent angles procedure. 21 Use angle bisector procedure on each side of the triangle. The angle bisectors should meet at a common point, P. 22 Use constructing a perpendicular bisector procedure. 23 Copy one side and the angles at each vertex. Extend the rays until they meet. 24 Use the line bisector procedure. Mark the intersection of the line and its bisector as the midpoint. CHAPTER Transformations and the Coordinate Plane 6-1 Cartesian Coordinate System (pages 91–92) 1 (2) 5 2 (2) 6 3 (4) 10 4 (2) 6 2 5 (1) √ √ 226 6 (2) 7 a A(5, 2), B(3, 3) b A(3, 4), B(0, 5) c A(x, y), B(0, 0) ___ 8 The length of base AB is the difference between the x-coordinates of A and B; 6 2 4.___ The height is the vertical distance from C to AB or the difference between the y-coordinates; 4 1 3. Therefore, 1 (4)(3) 6. 1 bh _ A_ ___ 2 2 9 The length of the base AB is the difference between the x-coordinates of A and B; 1 (3) 4. The ___height is the vertical distance from C to AB or the difference between the y-coordinates; 8 4 4. Therefore, 1 (4)(4) 8. 1 bh _ A_ 2 2 28 6 −− 10 The length of the base AB is the difference in the y-coordinates of A and B; 6 2 4. The height is the horizontal distance from C −− to AB or the difference between the x-coordinates; 3 2 1. Therefore, 1 (4)(1) 2. 1 bh _ A_ ___ 2 2 11 The length of the base AB is the difference between the y-coordinates of A and B; 8 2 6. The height is the horizontal dis___ tance from C to AB or the difference between the x-coordinates; 4 (2) 6. Therefore, 1 (6)(6) 18. 1 bh _ A_ 2 2 12 D(x, y) (1, 2) 13 Draw a horizontal line from A___ to C. For ABC, the length of the base AC is the difference between the x-coordinates of A and C; 3 (7) 10. The ___ height is the vertical distance from B to AC or the difference between the y-coordinates; 5 1 4. The area of 1 (10)(4) 20. For 1 bh _ ABC is A _ 2 2 ADC, the base is 10 and the height is 4. The 1 bh _ 1 (10)(4) 20. area of ADC is A _ 2 2 The area of quadrilateral ABCD is 20 20 40. Chapter 6: Transformations and the Coordinate Plane 14 The distance from A to B is 2 2 [2 (7)] (5 1) √ 2 5 4 2 41 . The distance from B to C 2 is √ [3 (2)] (1 5)2 2 5 4 2 41 . The distance from C to D [(2) 3]2 [(3) 1]2 is √ 2 5 4 2 41 . The distance from D to A is √[(7) (2)] 2 [(1 (3)] 2 2 2 5 4 16 17 18 19 20 21 22 41 . The perimeter of ABCD 41 41 41___ 41 4 41 . The length of the base AC is the difference between the x-coordinates of A and C; 6 (4) 10. The ___ height is the vertical distance from B to AC or the difference between the y-coordinates; 6 (6) 12. Therefore, 1 (10)(12) 60. 1 bh _ A_ 2 2 Using the distance formula, AB BC 13 and CA 10. The perimeter of ABC 13 13 10 36. Isosceles because PA AT 5 53 , CB √ 74 , and Scalene because AB √ 29 AC √ 130 , AD √ 26 , Scalene because MA √ 104 and MD √ 26 , IT √ 125 , and Scalene because WI √ 109 WT √ All three sides measure 2, therefore the triangle must be equilateral. The radius is the distance from the center to any point on the circle. 2 2 1) (2 2) 5 r √(4 15 Diameter 2(5) 10 23 The two diagonals are congruent because AD CB 2. 2 2 2 24 AB [(1) 3] (2 2) 32 2 2 2 BC (3 1) (2 4) 8 2 2 2 AC [(1) 1] [(2) 4] 40. 2 2 2 32 8 40 or AB BC AC . 5 6 7 8 9 10 11 12 13 14 15 16 17 18 T 0, 2 T 3, 0 (4, 1) (3, 7) (2, 4) (4, 3) (4, 2) (2, 2) T 3, 2 T 2, 2 (3, 0) (7, 3) (4, 7) K(8, 5) → (2, 9), E(10, 3) → (4, 1), N(2, 2) → (8, 6) 19 y 10 9 8 7 D" 6 D E" E 5 4 W" 3 2 D' E' W 1 5 4 3 2 1 1 1 2 3 4 5 6 7 8 9 10 x W' 2 3 4 5 a D’(1, 2), E’(6, 3), W’(2, 1) b D(0, 6), E(7, 7), W(3, 3) c T 3, 1 20 y 10 9 8 7 6 S' S" 5 4 A' 3 2 W' 1 5 4 3 2 1 1 2 3 S(3, 4) H' A" A(0, 1) 1 2 3 4 W(1, 2) H" H(5, 1) 5 6 7 8 9 10 x W" 4 5 a W’(2, 0), A’(3, 3), S’(0, 6), H’(2, 3) b W(5, 1), A(4, 2), S(7, 5), H(9, 2) c T 4, 1 6-2 Translations (pages 94–95) 1 (1) (1, 2) 2 (2) (3, 2) 3 (4) T 8, 4 4 (4) 6 6-2 Translations 29 6-3 Line Reflections and Symmetry (pages 100–101) 1 (3) A 2 (3) V 3 (1) only vertical line symmetry 4 (2) 2 5 (3) (4, 7) 6 y-axis 7 both 8 x-axis 9 y-axis 10 P’(1, 8) 13 14 15 16 (5, 2) (2, 3) (2, 5) A’(2, 1), B’(3, 4), C’(4, 5) y 8 7 6 C(4, 5) 5 3 2 5 5 6 7 8 x B'(3, 4) 5 6 7 8 y 6 I'(–1, 4) 5 I(1, 4) 1 1 2 3 4 5 6 2 7 8 9 10 x 4 H'(–4, 3) 2 5 J'(5, 2) 1 6 5 4 3 21G 1 4 H(4, 3) 3 J(–5, 2) y 2 3 G'1 2 3 4 5 x 6 2 6 P'(1, 8) 18 A’(3, 3), B’(5, 8), C’(1, 5) 9 y 10 10 11 P’(3, 7) 9 B'(5, 8) 8 y 7 6 8 B(8, 5) C'(1,5 5) 7 4 6 3 A'(3, 5 2 4 6 5 4 3 2 1 1 1 2 3 4 5 6 7 8 x x y 1 8 7 6 5 4 3 2 1 1 2 3) A(3, 3) 1 3 2 1 2 3 4 C(5, 1) 5 6 7 8 9 10 x 2 3 4 y 2 3 19 Q’(2, 4), Z’(3, 4), D’(1, 2) 4 5 y 6 7 6 8 5 Q'(2, 4) Z'(3, 4) 4 Z(4, 3) 12 P’(4, 1) 3 2 y 1 8 7 6 5 4 3 2 1 1 8 7 Q(4, 2) 6 5 3 2 P'(4, 1) 1 8 7 6 5 4 3 2 1 1 2 3 4 5 1 2 3 4 5 6 7 8 x y 2 P(4, 5) 6 7 8 Chapter 6: Transformations and the Coordinate Plane D'(1, 2) 1 D (2, 1) 3 4 4 30 4 A'(2, 1) 4 C'(4, 5) 2 P'(3, 7) 3 3 3 P(3, 3) 2 2 P(2, 4) 4 8 1 17 G’(0, 0), H’(4, 3), I’(1, 4), J’(5, 2) 6 7 A(2, 1) 1 8 7 6 5 4 3 2 1 1 y 6 5 4 3 2 1 1 B(3, 4) 4 2 3 y 4 5 x 6 7 8 x 20 6-5 Rotations y 6 y 4 2 yx 5 x 3 2 1 6 5 4 3 2 1 1 2 3 1 2 3 4 5 6 x x y2 4 5 6 6-4 Point Reflection and Symmetry (pages 103–104) 1 (2) H 2 (1) only line symmetry 3 (3) both point and line symmetry 4 (3) (k, 2k) 5 line symmetry 6 both 7 point symmetry 8 both 9 (1, 3) 10 (3, 1) 11 (9, 0) 12 (2, 6) 13 (4, 4) 14 (8, 2) 15 (2, 4) 16 (2, 8) 17 (10, 10) 18 Point reflection is a transformation of a figure into another figure. (page 107) 1 (2) trapezoid 2 (1) regular hexagon 3 (2) regular pentagon 4 (2) rectangle 5 (1) equilateral triangle 6 (1) 7 Z 8 X 9 Y 10 Z 11 (1, 5) 12 (3, 3) 13 (2, 8) 14 (2, 3) 15 (2, 2) 16 (4, 4) 17 (6, 3) 18 (2, 6) 19 (5, 5) 20 W(3, 2), H(8, 2), Y(5, 10) 6-6 Dilations (page 110) 1 (3) (4, 10) 1 2 (2) _ 2 3 (4) (x, y) → (2x, 2y) 4 (4) 20 square inches 5 (6, 9) 9 6 3, _ or (3, 4.5) 2 7 (4, 6) ( ) For problems 8–11, see figure below. Point symmetry is a quality of a figure. A figure with point symmetry is not changed by a reflection through that point. (It is rotated in either direction 180 about a point.) 8 A(0, 8), B(2, 2), C(6, 4) 3 3 1 , 1_ 1 9 A(0, 3), B _, _ , C 2_ 2 4 4 4 10 A(0, 12), B(3, 3), C(9, 6) ( ) ( ) 6-6 Dilations 31 11 A(0, 8), B(2, 2), C(6, 4) 10 y 8 y 7 10 6 9 5 8 4 7 3 2 6 5 (8) 4 1 A 8 7 6 5 4 3 2 1 1 2 C B 2 1 2 3 4 5 6 7 8 9 10 x 5 Z'(3, 4) 6 7 x 8 M(5, 2) 7 8 (11) (10) 9 direct isometry 11 y 8 10 7 11 6 12 5 13 4 14 13 5 6 8 12 4 Z(3, 4) 5 (9) 6 7 3 4 3 4 2 3 1 10 9 8 7 6 5 4 3 2 1 1 1 K'(2, 2) K(2, 2) 2 M'(5, 2) 3 3 K'(1, 1) 2 (1, √3 ) 8 7 6 5 4 3 2 1 1 1 2 3 4 5 Z'(0, 1) 2 K(2, 2) ( √_22 , √_22 ) 14 (3a, 3b) 1 ; (2, 4) → (1, 2) 15 k _ 3 1 ; (2, 4) → (1, 2) 16 k _ 2 17 (3, 12) 18 G(1, 1), N(4, 1), A(4, 1), T(1, 1) M'(2, 1) 1 7 8 x M(5, 2) 3 4 6 Z(3, 4) 5 6 7 8 direct isometry 12 y 5 4 3 2 1 5 4 3 2 1 1 6-7 Properties Under Transformations (pages 111–113) 1 (2) dilation 2 (4) dilation 3 (2) translation 4 (2) r y x 5 (3) dilation D 1 6 r y-axis 7 T 5, 0 8 R 270 or R 90 9 T 4, 3 1 2 3 4 5 2 2) K(2, 5 7 8 9 10 11 12 x M(5, 2) 3 4 6 K'(4, 4) W'(10, 4) Z(3, 4) 6 7 8 Z'(6, 8) 9 10 not an isometry 13 y 10 9 8 A" R''' 7 6 5 C" A''' R" 4 3 C''' 2 1 10 9 8 7 6 5 4 3 2 1 1 C 1 2 3 4 5 2 3 4 5 6 7 8 9 10 R R' A C' 6 7 8 A' 9 10 a C(3, 5), A(4, 7), R(7, 4) 32 Chapter 6: Transformations and the Coordinate Plane x b C(3, 5), A(4, 7), R(7, 4) c C(5, 3), A(7, 4), R(4, 7) d c y 14 12 11 D 10 I 9 8 D" D' 7 6 K 5 D''' 4 3 2 I" I' 1 K" K' 11 10 9 8 7 6 5 4 3 2 1 1 I''' 1 2 3 4 5 6 7 8 9 10 11 x 13 A 14 R 90 15 Any combination of three rotations, the sum of whose angles is 100 16 Any combination of two rotations, the sum of whose angles is 180 17 Glide reflection 18 (6, 1) 19 (2, 3) 20 (2, 7) y 21 10 2 9 4 7 K''' 3 4 2 C 1 C' 10 9 8 7 6 5 4 3 2 1 1 1 2 3 4 5 6 7 8 9 10 x C"2 y 3 4 9 5 M" 8 7 6 7 A" 6 P" 5 4 2 J" J''' 1 J' 1 2 3 4 5 J 2 3 4 A' 5 8 9 10 R" 3 10 9 8 7 6 5 4 3 2 1 1 P 3 10 A''' yx M' 5 a K(6, 1), I(9, 2), D(10, 7) b K(6, 1), I(9, 2), D(10, 7) c K(1, 2), I(4, 1), D(5, 4) R''' M 6 5 15 P' 8 6 7 8 9 10 C(2, 1), M(7, 5), P(4, 8) C(2, 1), M(7, 5), P(4, 8) d r y x x A R' R 22 y 10 a J(1, 0), A(3, 1), R(2, 2) b J(0, 2), A(2, 6), R(4, 4) c J(2, 0), A(6, 2), R(4, 4) 9 8 Y' 7 Y 6 X" 5 X 4 X' 3 2 6-8 Composition of Transformations (pages 117–119) 1 (3) R 200 ___ 2 (2) AT 3 (1) (x, y) 4 (3) r y x D 3 5 (1) a direct isometry 6 (2) (x, y) 7 (4) (3, 8) 8 (4) D 9 (3) T 4, 0 10 (1) (x, y) 11 (3) orientation 12 N Z' 1 10 9 8 7 6 5 4 3 2 1 1 Z 1 2 3 4 5 2 6 7 8 9 10 x Y" 3 4 5 yx 6 7 8 Z" 9 10 X(5, 3), Y(2, 6), Z(7, 1) X(3, 5), Y(6, 2), Z(1, 7) d (1) rotation Chapter Review (pages 119–121) 1 (1) I 2 (1) WOW 3 (3) parallel to the y-axis 4 (3) (5, 2) Chapter Review 33 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 (3) (x, y) → (x, 2y) (1) translation (1) rotation (1) A (4) D (2) (5, 4) (3) reflection in the line y x (2) (x, y) → (4x, 2y) (4) y 0 (3) r x 1 r y-axis 1 (1) _ 2 (4) (3) (6, 1) 5 13 10 2 √ 13 10 3 √ (1, 1) (2, 6) (3, 0) (4, 1) r y x(1, 8) (8, 1) 28 (4, 9) 29 a (2, 2) b (4, 0) c (2, 2) d (2, 2) e (3, 2) 30 a and c y H 10 T 8 6 4 C 2 14 12 10 8 6 4 2 T'(9, 1) 2 4 6 2 4 C'(4, 3) 8 10 T"(8, 2) 12 H'(10, 3) 6 8 C"(2, 10) 10 H"(10, 10) b ry x CHAPTER Polygon Sides and Angles 7-1 Basic Inequality Postulates (pages 125–126) Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied 34 Chapter 7: Polygon Sides and Angles 14 7 in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 1 1. m1 m2 2. m2 m1 3. m2 m3 4. m3 m2 5. m3 m1 (Transitive postulate of inequality) 6. m1 m3 x 2 1. PQ ___ PS −− 2. PQ PS 3. 1 2 4. m1 m2 5. m1 m3 6. m2 m3 CE BD CF BF CE CF FE CF FE BD BD BF FD CF FE BF FD BF FE BF FD BF FE BF BF FD BF or ___FE ___FD 4 1. AE EB 2. ABE BAE 3 1. 2. 3. 4. 5. 6. 7. 8. 3. 4. 5. 6. 7. 8. 9. 10. 5 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. (Isosceles triangle theorem) (Substitution postulate of inequality) (Subtraction postulate of inequality) (Isosceles triangle theorem) mABE mBAE mDAB mCBA mDAB mCAD mBAE mCAD mBAE mCBA mCBA mABE mDBC mCAD mBAE mABE mDBC mCAD mABE mABE mDBC mCAD mABE (Subtraction mABE mABE postulate of mDBC mABE inequality) or mCAD mDBC −− C is the midpoint of AB. AC CB AB AC CB AB AC AC or AB 2AC AB AC _ 2 −− F is the midpoint of DE. DF DF DE DF FE DE DF DF or DE 2DF DE DF _ 2 AC DF AB DF _ 2 AB _ DE 13. _ 2 2 AB 2 _ DE 2 14. _ 2 2 or AB DE (Multiplication postulate of equality.) 6 1. RP 3RS 3RS RP _ 2. _ 3 3 RP _ RS or 3 3. RQ 3RT RQ 3RT 4. _ _ 3 3 RQ _ RT or 3 5. RS RT RP RT 6. _ 3 RQ RP _ _ 7. 3 3 RQ RP 3 _ 3 8. _ 3 3 or RP RQ 7 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 8 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. (Multiplication postulate of equality.) −− I is the midpoint of EH. EH EI IH EI IH EH EI EI EH 2EI −− J is the midpoint of EF. EF EJ JF EJ JF EF EJ EJ or EF 2EJ EH EF 2EI EF 2EI 2EJ EI EJ (Division postulate of equality) ___ BD bisects ABC. ABD DBC (Definition of bisector) mABD mDBC mABC mABD mDBC mABC mABD mABD or mABC 2mABD −− BD bisects ADC. ADB BDC mADB mBDC mADC mADB mBDC mADC mADB mADB or mADC 2mADB 7-1 Basic Inequality Postulates 35 11. mABC mADC 12. 2mABD 2mADB 13. mABD mADB (Division postulate of equality) 1 AC 9 1. AE _ 3 1 AC 3 or 3AE AC 2. AE 3 _ 3 (Multiplication postulate of equality) 1 AB 3. AD _ 3 1 AB 3 4. AD 3 _ 3 or 3AD AB 5. AC AB 6. 3AE 3AD (Substitution postulate) 7. AE AD (Division postulate of inequality) ___ ___ A to BC. 10 1. AE is the median from ___ 2. E is the midpoint of BC. (Definition of median) 3. BE EC (Definition of midpoint) 4. BC BE EC 5. BC BE BE or BC 2BE ___ ___ 6. CD is the median from ___C to AB. 7. D is the midpoint of AB. (Definition of median) 8. AD DB 9. AB AD DB 10. AB AD AD or AB 2AD 11. AB BC 12. 2AD 2BE 13. AD BE (Division postulate of inequality) 7-2 The Triangle Inequality Theorem (page 128) 1 Yes 2 No 3 Yes 4 No 5 Yes 6 No 36 Chapter 7: Polygon Sides and Angles Yes Yes Yes Yes 1 S 9 6 S 10 2.5 S 5.5 0 S 12 1 1 S 3_ 15 1_ 2 2 7 8 9 10 11 12 13 14 7-3 The Exterior Angles of a Triangle (pages 130–132) 1 (2) isosceles 2 (4) right 3 a CAD b ABC and ACB c mCAD mABC; mCAD mACB 4 a True b True c True d True 5 6 and 9 6 5, 9, 8, and 10 7 7 and 10 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 8 1. m1 m3 (Exterior angle theorem) 2. m3 m5 (Exterior angle theorem) (Transitive postu3. m1 m5 late of inequality) ___ ___ 9 1. AB CD 2. 1 is a right angle. 3. AED is a right angle. 4. AED 1 (Right angles are congruent.) 5. mAED m1 6. mAED m2 (Exterior angle theorem) 7. m1 m2 10 1. mDCB mCBA 11 12 13 14 (Exterior angle theorem) 2. mCBA mCBM mMBA 3. mCBA mMBA 4. mDCB mMBA (Transitive postulate of inequality) 12x 8 (4x 6) (7x 6) x8 12x 32 (5x 5) (5x 5) x 11 Let x be mB, then mA is 3x. 3x x 104 x 26 mA 3x 3(26) 78 If the vertex angle of the triangle measures 130, each base angle measures 25 and each exterior angle at the base measures 180 25 155. 7-4 Inequalities Involving Sides and Angles of Triangles (page 133) 1 ZXY 2 PRQ ___ 3 AC 4 GFH ___ 5 KL 6 ACB ___ 7 YZ 8 mDGE mDEF mGDE, so mDGE mGDE and DE EG. 9 PQS QSP, but QSP SQR, so PQS SQR 10 m4 m3 m1, so m4 m1 Chapter Review (pages 133–136) 1 (3) p m p n 2 (3) 28 3 (2) an obtuse angle 4 True. Transitive postulate of inequality 5 True. Additive postulate of inequality 6 True. Postulate of inequality 7 False Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 8 1. m2 m3 2. m1 m2 (Exterior angle theorem) 3. m1 m3 (Transitive postulate of inequality) 9 1. BEST is a parallelogram. 2. BT ___ ___ BE (Definition of a 3. BT ES parallelogram) 4. BT ES 5. ES BE (Substitution postulate of inequality) ___ ___ 10 1. GS GD 2. NPS RSP (Isosceles triangle theorem) 3. mNPS mRSP 4. mISR mIPN 5. mISR mRSP mIPN mRSP (Addition postulate of inequality) 6. mISP mRSP mIPN mNPS 7. mISP mIPS (Postulate of inequality) 11 1. mBCE mDBC 2. mACE mABD 3. mBCE mACE (Addition postulate mDBC mABD inequality) 4. mACB mABC (Postulate of inequality) ___ ___ 12 1. AD DC 2. ACD CAD (Isosceles triangle theorem) 3. mACD mCAD 4. mDAB mDCB 5. mDAB mACD (Subtraction mDCB mACD postulate of inequality) Chapter Review 37 6. mDAB mCAD mDCB mACD 7. mCAB mACB (Postulate of inequality) 13 1. PQRS is a parallelogram. 2. SP RQ (Definition of parallelogram) 3. ST TP RQ 4. ST TP QU RU or ST RU QU TP 5. TP QU 6. 0 QU TP (Subtraction postulate of inequality) 7. 0 ST RU (Addition 8. ST RU postulate of inequality) ___ 14 ST ___ 15 AB ___ 16 AB ___ 17 BC −− 18 DE 19 AD BD 20 mB 120 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) ___ 21 1. AD ___ bisects CAB. 2. BE bisects CBA. 3. mDAB mEBA 4. mDAB 2 mEBA 2 (Multiplication postulate of inequality) 5. mCAB 2(mDAB) (Definition of bisector) 6. mCBA 2(mEBA) (Definition of bisector) 7. mCAB mCBA 38 Chapter 7: Polygon Sides and Angles 22 1. CF AE 2. CF 2 AE 2 3. 4. 5. 6. 7. 8. 23 1. 2. 3. (Multiplication postlate ___of inequality) F is the midpoint of CD. CF 2 CD (Definition of midpoint) CD AE 2 ___ E is the midpoint of AB. AB AE 2 (Definition of midpoint) CD AB (Multiplication postulate of inequality) AC CB CB AC _ _ (Division postulate of 2 2 inequality) AC (Definition of AD _ 2 midpoint) CB 4. AD _ 2 CB 5. BE _ 2 24 25 26 27 (Definition of midpoint) 6. AD BE Let A be the acute angle and let B be its supplement. mB 180 mA. Since mA 90, mB 90 and is therefore obtuse by the subtraction postulate of inequality. Let C be the complement of A, and let D be the complement of B. mC 90 mA and mC mD by the subtraction postulate of inequality. m1 m3 because an exterior angle is greater than either nonadjacent interior angle and m1 m2. So m2 m3 by the substitution postulate of inequality. mABC mABD because a whole is greater than its parts. mBAC mABC because they are opposite congruent sides of a triangle. So mBAC mABD by the substitution postulate of inequality, and DB DA because the greater side lies opposite the greater angle. 28 mBDA mACB because a whole is greater than its parts. mACB mBCA because they are opposite congruent sides of a triangle. So mBDA mBCA by the substitution postulate of inequality, and AB AD because the greater side lies opposite the greater angle. −− 29 Draw diagonal AC, forming two triangles. In ABC, mBCA mBAC because they are opposite congruent sides. In ADC, mACD mCAD because the greater angle is opposite the greater side. mBCA mACD mBAC mCAD or mBCD mBAD by the addition postulate of inequality. 30 Look at the two right triangles formed with −− common side EG. mD mF by the subtraction postulate of inequality, so DE EF because the greater side lies opposite the greater angle. 31 m2 180 114 66 m1 180 50 66 64 32 x 45 105 x 60 33 y 180 110 70 x 70 130 x 60 34 True 35 False 36 True 37 False 38 False CHAPTER 8 8-1 The Slope of a Line (pages 141–144) 1 (2) (0, 4) 2 (2) It has an x-intercept of 2. 3 (4) y 3 4 (3) a y-intercept of 5 5 (3) 6 (2) x 1 5 7 m _; negative 3 8 m 2; positive 3 9 m _; positive 4 Slopes and Equations of Lines 10 Undefined 11 m 2; negative 3 12 m _; positive 2 13 m 0; zero 1 ; positive 14 m _ 2 15 m 3; positive 16 a y 5 b y1 c x 2 d x 10 e y 4.5 8-1 The Slope of a Line 39 ge 17 a _ fd b Undefined a c _ b d 1 3 18 m _ 2 19 y 3 20 y 14 21 y 4 22 x 0 23 x 15 24 x 8 25 x 4 26 a m 1 b m1 27 a–c Not collinear 28 m 4 29 15 feet 1 1 4 _ _ _ −− , m −−− , m −−− 30 a m AB 3 CD 3 AD 11 b Not collinear 2 , m −− is undefined, m −−− 0. The _ −− 31 m AB AD 3 BC −− triangle is a right triangle because BC is −− parallel to the x-axis and CA is parallel to the y-axis, making these sides perpendicular to each other. 8-2 The Equation of a Line (page 147) 1 a No b Yes c Yes 2 a y x 6; m 1, b 6 b y 3x 5; m 3, b 5 c y 3x; m 3, b 0 d y 2x 4; m 2, b 4 2 2 x 4; m _ e y _ ,b4 3 3 3 3 f y _x 3; m _, b 3 2 2 g y 3x 5; m 3, b 5 9 9 1x_ 1, b _ h y _ ; m _ 2 2 2 2 3 a x-intercept 7, y-intercept 7 b x-intercept 4, y-intercept 8 c x-intercept 3, y-intercept 6 40 Chapter 8: Slopes and Equations of Lines d e f 4 a b c d 5 a b c d e f 6 a b c d e f 7 a b c d e f x-intercept 2, y-intercept 4 x-intercept 0, y-intercept 0 x-intercept 6, y-intercept 4 y 2x 5 1x 2 y_ 2 y 3 y x 4 yx3 y2 1x 5 y_ 2 5 _ y x3 2 3 y _x 2 2 3 _ y x5 5 7x 5 y _ 2 y 2x 2 y 5x 10 3 y _x 3 2 y 4x 13 6 y _x 6 5 yx1 2x 2 y _ 3 y5 y 2x 1 y 2x 11 1 1x _ y _ 3 3 8-3 The Slopes of Parallel and Perpendicular Lines (page 150) 1 (4) Line l has a negative slope. 2 (1) 5 9 3 a Slope parallel _ 7 7 Slope perpendicular _ 9 b Slope parallel 0 No slope for perpendicular line 7 c Slope parallel _ 8 8 Slope perpendicular _ 7 4 5 6 7 2 d Slope parallel _ 5 5 Slope perpendicular _ 2 9 e Slope parallel _ 10 10 Slope perpendicular _ 9 f Slope parallel 13 1 Slope perpendicular _ 13 a Perpendicular; slopes are negative reciprocals b Perpendicular; slopes are negative reciprocals c Neither; slopes are neither equal nor negative reciprocals d Neither; slopes are neither equal nor negative reciprocals e Perpendicular; lines with a slope of 0 are perpendicular to lines with no slope f Parallel; slopes are the same a Yes b Yes c No 17 1x _ a y _ 4 4 b y 2x 7 1x 4 c y_ 5 a Right triangle b Not a right triangle 10 7 6 (pages 153–154) 1 (2) (1.5, 1) 2 (4) (2.5, 2) 3 (2) (2, 0.1) 4 (1) (8, 0) 5 (4) (17, 20) 6 (4) (13, 13) 7 (5a, 5r) 8 (3x, 5y) 9 (10, 11) D 5 4 3 G 2 A 1 8 7 6 5 4 3 2 1 1 1 2 3 4 5 6 7 8 x 2 3 4 L 5 6 7 8 11 12 13 A rectangle is formed. M GL (0, 1), M LA (2, 1), M DA (2, 3), M GD (0, 3) (0, 2) −− −− a BD is parallel to EF because their slopes 3 are _. 4 1 BD b EF _ 2 1 (10) 5_ 2 55 5 a m_ 2 2 b m _ 5 5 c M 4, _ 2 35 5 d y _x _ 6 6 y 4x 9 1x _ 4 y_ 5 5 ( 14 8-4 The Midpoint of a Line Segment y 8 15 ) 8-5 Coordinate Proof (pages 163–165) 1 The triangle is an isosceles triangle because 40 . WI WN √ −−− −−− 2 Slopes of NA and AQ are negative reciprocals proving these two lines form a right angle by being perpendicular; therefore, NAQ is a right triangle. 26 ; the midpoint of the 3 AM CM BM √ hypotenuse is equidistant from all three vertices. 4 The slopes are negative reciprocals so the median is also the altitude. 8-5 Coordinate Proof 41 5 BIG is isosceles because it has two con50 ; and BIG is gruent sides, BI IG √ −− a right triangle because BI is perpendicular −− to IG. −−− −− 6 Two sides (QR and PS) have the same slope −− −− 2 and the other two sides (OP of _ and RS) 3 have the same slope of 3. PQRS is a parallelogram. 7 SAND is a parallelogram because the diagonals bisect each other at the point (1.5, 0.5). 8 LEAP is a parallelogram because all of the 50 and are congruent. sides measure √ 9 ABCD is a parallelogram because the diagonals bisect each other at the point (2.5, 0.5). 5 and 2 √ 13 10 a 6 √ b The diagonals meet at their midpoint (1, 1). c BETH is a parallelogram because the diagonals bisect each other. d BETH is not a rectangle because the diagonals are not congruent. 11 a NICK is a parallelogram; both pairs of opposite sides are parallel. b NICK is not a rhombus; the diagonals are not perpendicular. 12 The figure KATE is a square. KATE is a rectangle because the diagonals bisect each other and are the same length. KATE is a square because two adjacent sides are congruent. 13 ABCD is a rhombus. ABCD is a parallelogram because the diagonals bisect each other. ABCD is a rhombus because two adjacent sides are congruent. −− −− 14 a Slopes of SU and UE are negative reciprocals proving these two lines form a right angle by being perpendicular; therefore, SUE is a right triangle. b SU 5 and UE 10 15 a NORA is a parallelogram because the diagonals bisect each other. NORA is a rhombus because two adjacent sides are congruent. b NORA is not a square because it does not contain a right angle. 16 a JACK is a trapezoid because it has only one pair of opposite sides parallel; −− −− slope of JA slope of CR 1. b JACK is not an isosceles trapezoid because the legs are not congruent. 17 MARY is a trapezoid because it has only one pair of opposite sides parallel; slope of −−− −− MA slope of RY 0. MARY is an isosceles trapezoid because the legs are congruent; AR YM 5. 18 C(a, 0) 19 P(b a, c) 20 E(a, 0), F(a, 2a), G(a, 2a) 21 C(a, b r) 22 M(a c, b) 23 T(a, 2a) 24 R(a b, c) 25 Yes −− −− 26 Slope of AC is 1 and the slope of BD is 1. Since the slopes are negative reciprocals, −− −− AC BD. −− 27 a Midpoint of BC is (a, b). 2 b MA MB MC √a b2 28 a y E(2x, 2y) T(0, 0) 29 30 31 32 Chapter 8: Slopes and Equations of Lines x b A(x, y), B(3x, y), C(2x, 0) −− 2 2 c The length of median TB is √ 9x y . −− The length of median EC is 2y. The length −−− of median QA is √ 9x 2 y 2 . d TEQ is an isosceles triangle. c 2 d 2 TA TR _ 2 −− −− JA and NE are parallel to the x-axis and each slope is 0; thus they are parallel to each −−− −− c ; thus other. Slope of AN slope of EJ _ b they are parallel. Therefore, JANE is a parallelogram. −− Midpoint of AC midpoint of −− a r, _ s BD _ 2 2 (a b) 2 c 2 TA EM √ √( ) ( 42 Q(4x, 0) ) 8-6 Concurrence of the Altitudes of a Triangle (the Orthocenter) (page 168) 1 (4) obtuse 2 (4) at one of the vertices of the triangle Exercises 3–9: Check students’ graphs. 3 (2.2, 1.2) 5 4 _, 4 3 5 (1, 0) 6 (6, 6.5) 22 7 2, _ 3 8 (1, 1) 9 (1, 0.5) −− −− 4 3 10 a Slope of AB is _ . Slope of BC is _. Since 3 4 the slopes are negative reciprocals, the segments are perpendicular and B is the right angle. −− −− b AB and BC can each be altitudes. B is the endpoint of the altitude drawn from B to AC. ( ) ( ) Chapter Review (pages 168–171) 3 1 (2) y _x 4 4 2 (2) (0, 3) 3 (4) (1, 9) 4 (2) y 5x 1 5 (1) y 3x 3 6 No. Sub-in: 3(5) 4(2) 0 7 x-intercept is (2, 0) and y-intercept is (0, 3). 13 . Distance is √ 8 x-intercept is (15, 0) and y-intercept is (0, 3). Distance is 15.3. 9 a 10 10 B(10, 10) ___ 1 . Slope of ___ AC is 2. Their 11 Slope of OB is _ 2 1 (2) 1. product is _ 2 12 a (0, 3) b (4, 0) c (8, 3) 13 a m 3 b x-intercept is (4, 0) c y-intercept is (0, 12) d (3, 3) is not on the line. () 14 a neither b vertical c vertical d neither e horizontal f vertical 3 1 b 2 c _ 15 a _ 2 4 a 2 2 e _ f _ d _ 3 3 d 16 a no slope b 0 c 1 d b k 7 _ _ d e ca f _ 2 h 17 a 7 b 2 c 14 d 2 e 6 f 5 g 4 18 a not collinear b collinear c collinear d collinear e collinear f collinear 19 a y 3 9 3 b y _x _ 2 2 5 1 _ c y x_ 3 3 2 _ d y x 3 19 3 e y _x _ 2 2 √ 20 a AB AC 2___ 10 b Midpoint of BC is (2, ___ 1). Slope of the median from ___ A to BC slope of altitude from A to BC 1. 21 a (3, 2) b (2, 3) c (6.5, 8.5) ____ ___ 1 22 Slope of DR slope of AW _. Slope of 3 ___ ____ RA ___ is not ____equal to the slope of DW. RA DW 5 2 and OP 2 √ 5 23 a NO 5 √ ___ ___ 1 _ b Slope of EO is . Slope of PN is 3. 3 Slopes of perpendicular lines are negative reciprocals of each other. 24 A(0, 3), B(4, 2), C(5, 4), and D(1, 3). ABCD is a parallelogram because both pairs ___ of opposite sides are parallel. Slope of AB slope ___ ___ 1 . Slope of ___ of CD _ BC slope of DA 6. 4 25 QRST is a parallelogram because both pairs of opposite sides are parallel. Slope of ___ ___ b . Slope of ___ QR slope of ST _ RS slope a ___ of TQ 0. 26 a–b Check students’ graphs. c (2, 10) Chapter Review 43 27 a M ___ (5, 1), M ___ (4, 3), M ___ TK (2, 1) KA AT ___ ___ 2 b m KA 1, m AT 0, m ___ TK ___ c Slope of line perpendicular to KA is 1. There ___ is no slope for the line perpendicular to AT. Slope of the line perpendicular ___ 1. to TK is _ 2 ___ d Perpendicular bisector of KA ___ : y x 6 Perpendicular bisector of AT: x 4 ___ 1x Perpendicular bisector of TK: y _ 2 e (4, 2) 28 a y 2x 5 b (6, 8) 1x 5 c y_ 2 CHAPTER Parallel Lines 9-2 Proving Lines Parallel (pages 177–179) 1 none 2 a b, c d 3 none 4 bc 5 ab 6 ac 7 bc 8 lm 9 a and b 10 c and d 11 a and b Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 44 Chapter 9: Parallel Lines 9 12 1. 1 3 2. 2 4 3. 1 2 3 4 (Addition postulate of equality) 4. ABG DEG −− −− 5. ED BA (If two lines are cut by a transversal forming a pair of congruent alternate interior angles, ___ the two lines are parallel.) ___ CD 13 1. AF ___ −− 2. FC FC ___ ___ −− −− 3. AF ___ FC ___ CD FC DF 4. AC ___ −− 5. BC ___ EF ___ 6. BC AD 7. BCF is a right angle. (Definition of right angle) ___ ___ 8. EF AD 9. EFD is a right angle. 10. BCF EFD (Right angles are congruent) 11. ABC DEF (SAS SAS) 12. ___ BAC EDF (CPCTC) ___ 13. ED AD (Alternate interior angles are congruent) ___ 14 1. BE bisects ABC. 2. 1 EBC 3. ___ m1 mEBC 4. CE bisects DCB. 5. 2 ECB 6. m2 mECB 7. m1 m2 90 8. mEBC mECB 90 (Substitution postulate) 9. m1 m2 mEBC mECB 10. m1 mEBC 90 11. m2 mECB 90 12. ABC___ and DCB are right angles. BC 13. BA ___ BC 14. CD (Two lines perpendicular to the CD 15. BA same line are parallel.) 15 1. 1 3 2. m1 m3 3. 2 4 4. m2 m4 5. m1 m2 m3 m4 360 6. m1 m1 m2 m2 360 or 2(m1) 2(m2) 360 7. m1 m2 180 (Division postulate) −−− −− 8. AD BC (Interior angles on the same side of the transversal are supplementary.) 9. m1 m1 + m4 m4 360 or 2(m1) 2(m4) 360 10. ___ m1___ m4 180 11. AB CD ___ ___ CF and of BE. 16 1. D is the midpoint of ___ ___ DF 2. CD ___ ___ DE 3. BD ___ ___ 4. FC FC 5. CBD FDE (Vertical angles are congruent.) 6. CBD FDE (SAS SAS) 7. ___ EFD (CPCTC) ___ DCB 8. AC FE (Alternate interior angles are congruent.) 17 1. 1 2 2. AFC DCF (Supplementary angles of congruent angles are congruent.) ___ ___ 3. EF ___ CB ___ FC ___ ___ 4. FC ___ ___ 5. EF ___ FC ___ = BC FC BF 6. EC ___ ___ 7. AF CD 8. AFB DCE (SAS SAS) 9. ABF DEC (CPCTC) ___ ___ (Alternate interior an10. AB ED gles are congruent.) ___ ___ FC 18 1. AE ___ ___ EF 2. EF ___ ___ ___ ___ EF FC EF 3. AE ___ ___ 4. AF ___ ___ CE 5. DE AC 6. ___ DEA___ is a right angle. 7. BF AC 8. BFC is a right angle. 9. DEA BFC (Right angles are congruent.) ___ ___ 10. DE BF 11. ______ AFB CED (SAS SAS) 12. AEFC 13. BAF DCE (CPCTC) ___ ___ 14. AB DC 9-3 Properties of Parallel Lines (pages 182–185) 1 (1) same-side exterior angles 2 (2) 2, 3, 6, 7, 10, 11, 14, 15 3 m1 m4 m6 m7 60; m2 m3 m5 120 4 m1 m4 m5 m8 135; m2 m3 m6 m7 45 5 ma md mg 65; mb mc me mf 125 6 w y 70; x z 110 7 a 8x 6x 30 180 x 15 b 2x 10 5x 47 x 19 m3 2(19) 10 48 9-3 Properties of Parallel Lines 45 106 mA 75, mC 67 57 60 1. Perpendicular lines form right angles. 2. Right angles measure 90. 3. When two parallel lines are cut by a transversal, corresponding angles are congruent. 4. Congruent angles have equal measure. 5. Transitive property of congruence (3, 5) 6. If an angle measures 90, it is a right angle. 7. If two lines intersect to form a right angle, they are perpendicular. Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) ___ 13 1. AB 2. 2 is supplement of 1. 3. 2 is supplement of 3. 4. 1 3 (If two angles are supplements of the same angle, then they are congruent.) ___ ___ 5. BC AD (When two lines are cut by a transversal creating corresponding angles, the lines are congruent.) 14 1. ABC ___ ___ 2. AB BC 3. BAC BCA (Isosceles triangle theorem) _____ ___ 4. FHD BC 5. FDA BCA (Corresponding exterior angles are congruent.) _____ ___ 6. EHG AB 7. BAC GEC (Corresponding exterior angles are congruent.) 8. ___ GEC____ FDA 9. HE HD 10. EHD is isosceles. (Definition of isosceles triangle) 8 9 10 11 12 46 Chapter 9: Parallel Lines ___ ___ 15 1. DE AC 2. 1 2 3. 3 4 4. 2 3 5. 1 4 ___ (Corresponding exterior angles are congruent.) (Transitive property of congruence) ___ BD at E. 16 1. AC ___ intersects ___ 2. AE ED 3. A D ___ ___ 4. AD BC 5. A C (Alternate interior angles are congruent.) 6. B D (Alternate interior angles are congruent.) 7. B C (Transitive property of congruence) ___ ___ (Isosceles triangle 8. AE CE theorem) ___ ___ 9. BE ___ DE ___ (Addition postulate) 10. AC BD 17 1. n m 2. ABE and BAD are supplementary. (Two interior angles on the same side of the transversal are supplementary.) 3. mABE mBAD 180 1 mBAD 90 1 mABE _ 4. _ 2 2 (Division postulate) ___ 5. BC bisects ABE. 1 mABE 6. mABC _ 2 (Definition of angle bisector) ___ 7. AC bisects BAD. 1 mBAD 8. mACB _ 2 (Definition of angle bisector) 9. mABC mACB 90 10. mBCA mABC mACB 180 11. mBCA 90 12. BCA is a right angle. ___ ___ (Perpendicular lines form 13. BC AC right angles.) 18 1. r m and a b 2. 4 and 3 are supplementary. 3. 2 and 3 are supplementary. 4. (a) 4 2 (Supplements of the same angle are congruent.) 5. 4 and 5 are supplementary. 19 20 21 22 6. 2 and 1 are supplementary. 7. (b) 5 1 (Supplements of the same angle are congruent.) a (x 12) (3x) 180 x 42 m1 m3 m5 54 m2 m4 m6 126 b (2x) (2x 20) 180 x 50 m1 m3 m5 100 m2 m4 m6 80 c (7x 65) (5x 5) x 30 m2 m5 145 m1 m3 m4 m6 35 a 5 b 7 c 6 and 8 d m5 60 e m4 90 Since the corresponding angles are congruent, the halves of each are congruent. They form new congruent corresponding angles cut by the same transversal. The bisectors are therefore parallel. Draw a diagonal line, forming two congruent triangles by SAS. The other two sides of the quadrilateral and remaining angles are congruent by CPCTC. Therefore, since the diagonal is a transversal for these sides, the sides are parallel. 9-4 Parallel Lines in the Coordinate Plane (pages 187–188) 1 a 4 7 b _ 3 2 c _ 5 x _ d a 2 a 1 3 b _ 5 2 c _ 5 1 3 a _ 4 b 7 c 1 3 d _ 2 4 a perpendicular b parallel c perpendicular d parallel e perpendicular f neither 5 a 1 5 b _ 2 3 c _ 10 7 d _ 9 4 _ e 3 3 _ f 2 6 a 7 b 1 c 11 d no slope e 0 1 f _ 2 ___ 1 , ___ 7, slope ___ _ AB CD 7 a Slope ___ AB CD 7 ___ ___ _ ___ ___ b Slope AB 2, slope CD 1 , AB CD 2 8 y 5x 3 1x 3 9 y=_ 5 3 7 10 y _x _ 2 2 1 _ 11 y x 3 3 12 y 4 13 y 2x 6 2x 9 14 y _ 3 15 k = 1 16 (7, 5) 3 4 , slope ___ _ 17 Slope ___ _ . The slopes are AB CD 3 4 −−− −− negative reciprocals, therefore AB and CD are perpendicular and ABC is a right triangle. 18 Opposite sides have the same slope. 1 and m ___ m ___ 5 m ___ _ m ___ PQ RS PS QR 3 19 H(5, 6) 20 a D(4, 3) 9-4 Parallel Lines in the Coordinate Plane 47 9-5 The Sum of the Measures of the Angles of a Triangle (pages 193–195) 1 (4) scalene 2 (3) obtuse 3 (3) 120 4 (2) right 5 (3) 112 6 a base angles are 30, vertex angle is 120 b base angles are 60, vertex angle is 60 c base angles are 52, vertex angle is 76 d base angles are 36, vertex angle is 108 e base angles are 20, vertex angle is 140 7 a base angles are 50, exterior angle is 130 b base angles are 40, exterior angle is 140 c base angles are 54, exterior angle is 126 d base angles are 60, exterior angle is 120 e base angles are 80, exterior angle is 100 8 base angles are 74, vertex angle is 32 9 mP 27, mQ 45, mR 108 10 base angles are 28, vertex angle is 124. 11 exterior angle is 30, base angles are 15, vertex angle is 150 12 18, 54, 108 13 99, 45, 36 14 mc 35 15 mB 78 16 mx 150 17 mx 30 18 m1 150 19 mD 20 20 md 125 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 21 1. ___ A C 2. BD bisects ABC. 3. ABD CBD ___ ___ 4. BD BD 5. ABD CBD (AAS AAS) 48 Chapter 9: Parallel Lines ADB CDB (CPCTC) ADB is supplementary to CDB. ADB and CDB are right angles. −− −− BD AC (Segments forming right angles are perpendicular.) 22 1. A C ___ ___ 2. BD AC 3. BDA and BDC are right angles. 4. BDA BDC ___ ___ 5. BD BD 6. BDA DBC (AAS AAS) 7. ABD CBD (CPCTC) ___ (Definition of angle 8. BD bisects ABC. bisector) 23 Both ABC and DEC share C. Since the remaining angles are the same as well, 1 and B are___ congruent corresponding angles. ___ Therefore, BA ED. 24 Compare ABC and DEC. Both are right triangles with one pair of congruent acute angles. Therefore, the other pair of acute angles are also congruent, B CED. But CED 1, vertical pairs. Then B 1 by the transitive postulate of congruence. 6. 7. 8. 9. 9-6 Proving Triangles Congruent by Angle, Angle, Side (pages 198–200) 1 b and f 2 a Not sufficient. If the third side is congruent, SSS. If included angles are congruent, SAS. b Sufficient, SSS c Sufficient, hypotenuse-angle d Not sufficient. If either pair of corresponding angles are congruent, AAS. e Sufficient, AAS f Not sufficient. If any pair of corresponding sides are congruent, AAS. g Sufficient, hypotenuse-leg h Sufficient, SAS i Sufficient, SSS j Sufficient, AAS Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) ___ ___ 3 1. AB EF 2. ABC FED ___ ___ 3. BC DE 4. 1 2 5. I II (ASA ASA) 4 1. 1 4 2. ___ B ___ D 3. AC AC 4. I II (AAS AAS) ___ ___ 5 1. AB DE 2. DEC BAC (Alternate interior angles are congruent.) 3. ABC EDC (Alternate interior angles ___ are congruent.) BD. 4. C is the midpoint of ___ ___ (Definition of 5. BC CD midpoint) 6. ABC EDC (AAS AAS) 6 1. B D 2. BEC DEA ___ ___ 3. BC AD 4. AED CEB (AAS AAS) ___ ___ (CPCTC) 5. AE CE 7 1. A E ___ ___ DC 2. BC ___ ___ CE 3. AC ___ ___ BC (Subtraction 4. AC ___ ___ CE___ DC___ postulate) AB DE or ___ ___ 5. BG AE 6. BGA is a right (Definition of angle. right angle) ___ ___ AE 7. DF ____ 8. DFE is a right angle. 9. BGA DFE (Right angles are congruent.) 10. BGA DFE (AAS AAS) ___ ___ (CPCTC) 11. BG DF 8 1. 2. 3. 4. 5. 9 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11 1. 2. 3. 4. 5. 6. 12 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. ___ ___ AB EF ABC FED A F ___ ___ AC DF I II (AAS AAS) ___ ___ AB CD ___ ___ CD AB CDA is a right angle. ___ ___ AE CB AEC is a right angle. BAC BCA (Isosceles triangle theorem) ___ ___ AC AC CDA___ AEC (AAS AAS) ___ CD AE (CPCTC) ___ AC. E is the midpoint of ___ ___ AE CE ___ ___ AF BD AFE is a right angle. ___ ___ CD BD CDE is a right angle. AFE CDE CED AEF (Vertical angles are congruent.) AFE CDE (AAS AAS) ___ ___ AF CD (CPCTC) ___ ___ QR SR RQS RSQ (Isosceles triangle theorem) 1 2 ___ ___ QS QS QTS (AAS AAS) ____ ___SMQ QM ST (CPCTC) ___ ___ BA CD ___ ___ BA AD BAQ is a right angle. ___ ___ CD AD (Two lines perpendicular to the same line are parallel.) CDP is a right angle. BAQ CDP B C ___ ____ AP ___ QD ___ PQ ___ PQ ___ AD PD (Addition postulate) BAQ CDP (AAS AAS) ___ ___ BA CD (CPCTC) 9-6 Proving Triangles Congruent by Angle, Angle, Side 49 9-7 The Converse of the Isosceles Triangle Theorem (pages 202–203) 1 2(3x 4) 2x 4 180 x 22 mA mC 70 mB 40 2 2(3x 3) 4x 16 180 x 19 mA mB mC 60 3 2x 180 82 82 x8 4 (2x 14) (3x 2) (5x 8) 180 x 16 2x 14 2(16) 14 46 3x 2 3(16) 2 46 5x 8 5(16) 8 88 Since two angles are equal in measure, ADC is isosceles. 5 mA mB 49 mx 180 49 49 my 49 mz 180 82 98 6 mQ 58 mx mz 64 my 116 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 7 1. A C ___ ___ AB 2. BC ___ ___ 3. AD EC 4. ABD CBE (SAS SAS) ___ ___ 8 1. BD AC 2. 1 ACB 3. 2 BAC 4. 1 2 5. ACB BAC (Substitution postulate) 6. ABC is isosceles. (Definition of isosceles triangle) 50 Chapter 9: Parallel Lines 9 1. 1 2 2. 3 4 ___ ___ 3. AB BC 4. ABC is isosceles. ___ ___ 10 1. BD ___ AE ___ 2. AC CE 3. A E 4. A B (Supplements of congruent angles are congruent.) (Definition of isosceles triangle) (Two parallel lines are cut by a transversal then the corresponding angles are congruent.) 5. E D (Two parallel lines are cut by a transversal then the corresponding angles are congruent.) 6. B D (Transitive postulate of congruence) ___ ___ 7. BC DC (Congruent angles imply congruent sides.) ___ ___ 11 1. BC BD 2. BCD BDC 3. BDA BDE (Linear pairs of congruent angles) 4. BAC BED (ASA ASA) ___ ___ BE (CPCTC) 5. AB ___ ___ 12 1. AB EB 2. BAE BEA ___ ___ 3. BD AE 4. BAE 1 5. BAE 2 6. 1 2 (Substitution postulate) ___ ___ SR 13 1. PQ ___ ___ 2. PQ SR 3. QPR SRP (Alternate interior angles are congruent.) ___ 4. PR bisects QPS. 5. QRP RPS (Definition of bisector) 6. SRP RPS (Transitive postulate of congruence) ___ ___ (Converse of the 7. PS SR isosceles triangle theorem) 14 1. 2. 3. 4. 5. 6. 7. 15 1. 2. 3. 4. 5. 6. 7. 8. ___ ___ DF FE 2 3 1 4 ADF CEF (Linear pairs of congruent angles) ADF CEF (ASA ASA) A C (CPCTC) ABC is an isosceles triangle. (Definition of an isosceles triangle) ______ AEDC 1 2 ___ ___ BE BD AEB CDB ___ ___ AE DC ABE CDB (ASA ASA) A C (CPCTC) ABC is an isosceles triangle. (Definition of an isosceles triangle) 9-8 Proving Right Triangles Congruent by HypotenuseLeg; Concurrence of Angle Bisectors of a Triangle (pages 207–208) 1 (a) 2 (d) 3 (c) 4 (b) 5 (a) 6 m1 24, m2 52, m3 104, m4 52, m5 14, m6 114, m7 14, m8 24, m9 142 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) ___ ___ PR 7 1. SQ ___ ___ 2. SA PB 3. SQP and SAP are right angles. 4. SQP and SAP are right triangles. ___ ___ 5. SP SP 6. 7. 8. 9. ___ ___ SQ SA SQP SAP SPQ APS ___ PT bisects RPB. ___ ___ (HL HL) (CPCTC) (Definition of angle bisector) 8 1. AE ___ BC ___ 2. CD AB 3. CDA and CEA are right angles. 4. CDA and CEA are right triangles. ___ ___ AC 5. AC ___ ___ 6. AE CD 7. ACE CAD (HL HL) ___ ___ 9 1. AE ___ BC ___ 2. CD AB 3. AEB and CDB are right angles. 4. AEB and CDB are right triangles. ___ ___ 5. DB EB 6. DBE DBE 7. ___ AEB___ CDB (Leg–acute angle) CD (CPCTC) 8. AE ___ ___ AC 10 1. BD ___ ___ 2. QS PR 3. BDA and QSP are right angles. 4. BDA and QSP are right triangles. 5. ABC PQR ___ ___ (CPCTC) 6. AB PQ 7. A P (CPCTC) 8. BDA QSP (Hypotenuse– acute angle) 9. ABD PQS (CPCTC) 10. mABD mPQS ___ 11. BD bisects ABC. 1 mABC mABD mPQS 12. _ 2 1 mPQR (Division _ 13. 1 mABC _ 2 2 postulate) 1 14. mPQS _mPQR (Transitive 2 postulate) ___ (Definition of 15. QS bisects PQR. angle bisector) ___ ___ ___ ___ 11 1. AB CF, DE CF 2. ABF and CED are right angles. 3. ABF and CED are right triangles. ___ ___ AF 4. CD ___ ___ BE 5. BE ___ ___ 6. CE FB 7. ABF CED (HL HL) ___ ___ 8. AB DE 9-8 Proving Right Triangles Congruent by Hypotenuse-Leg; Concurrence of Angle Bisectors of a Triangle 51 ___ ___ ___ ___ CE BA, BD AC CEA and BDA are right angles. CEA and BDA are right triangles. ___ ___ AB AC ABC ABC CEA BDA (Hypotenuse–acute angle) ___ ___ (CPCTC) 7. CE BD 12 1. 2. 3. 4. 5. 6. 13 1. 2. 3. 4. 5. 6. 7. ___ ___ ___ ___ AC BF, ED BF ACB and FDE are right angles. ACB and FDE are right triangles. ___ ___ AC ED ___ ___ BA EF ACB FDE (HL HL) A E (CPCTC) 9-9 Interior and Exterior Angles of Polygons (pages 210–212) 1 Polygon 52 Number of Sides Number of Triangles Sum of Interior Angles 180(n 2) Measure of Each Exterior 180(n 2) Angle n Measure of Each Interior Angle _ Triangle 3 1 180(1) 180 180 _ 60 3 360 _ 120 3 Quadrilateral 4 2 180(2) 360 360 _ 90 4 360 _ 90 4 Pentagon 5 3 180(3) 540 540 _ 108 5 360 _ 72 5 Hexagon 6 4 180(4) 720 720 _ 120 6 360 _ 60 6 Heptagon 7 5 180(5) 900 900 _ 128.57 7 360 _ 51.43 7 Octagon 8 6 180(6) 1,080 1,080 _ 135 8 360 _ 45 8 Nonagon 9 7 180(7) 1,260 1,260 _ 140 9 360 _ 40 9 Decagon 10 8 180(8) 1,440 1,440 _ 144 10 360 _ 36 10 24-gon 24 22 180(22) 3,960 3,960 _ 165 24 360 _ 15 24 n-gon n n2 180(n 2) 180(n 2) _ n 360 _ n Chapter 9: Parallel Lines 2 18,000 180(n 2) n 2 100 n 102 3 Diagonals n 3 a 0 b 1 c 2 d 4 e 8 4 Triangles n 2 a 7 b 10 c 15 d 98 5 180n a 720 b 900 c 1,160 d 3,420 sum _ 6 180 a 1,000 sides b 100 sides 360 _ 7 n a 40 b 36 c 10 d 5 360 8 __ # of degrees a 12 b 10 c 6 d 8 180(n 2) 9 # of degrees _ n a 18 b 360 c 8 d 6 10 sum 180(n 2) a 12 b 24 c 50 d 1,000 360 11 Each exterior angle is 30. _ 12 sides 30 12 180(n 2) 720 180n 360 720 180n 1,080 n6 180(n 2) 8 360 _ 13 _ n n 180(n 2) 2,880 n 2 16 n 18 14 m1 70, m2 75, m3 105, m4 145 15 m1 90, m2 110, m3 70, m4 35, m5 95, m6 85 16 m1 m2 m6 120 m3 m4 m5 60 17 180(n 2) 5(360) n 12 18 (5x 10) (6x 25) (6x 25) (5x 10) (3x 5) 180(3) 25x 65 540 x 19 Interior angles: 105, 139, 139, 105, 52 19 x + 3x + 4x + 4x + 6x 360 18x 360 x 20 Exterior angles: 20, 60, 80, 80, 120 Interior angles: 160, 120, 100, 100, 60 20 (3x 4) (7x 7) (6x 5) (5x 8) (3x 2) 5x 360 29x 348 x 12 Exterior angles: 40, 91, 67, 68, 34, 60 Interior angles: 140, 89, 113, 112, 146, 120 Chapter Review (pages 212–216) 1 a no slope 1 b _ 4 15 c _ 7 5 d _ 2 3 e _ 5 f 0 2 a 2 b 1 c no slope d 0 7 e _ 5 f no slope 3 5 3 m ___ _, m ___ _. Slopes are negative AC 5 BC 3 reciprocals. m ___ m ___ m ___ 1. Opposite sides 4 m ___ LA SL BC PS are parallel (slopes are equal) and slopes of consecutive sides are negative reciprocals (sides are perpendicular). ___ ___ ___ 5 m ___ PL m AN 1 and m PS m LA 4. Opposite sides are parallel (slopes are equal). 6 For parallel lines m and n cut by transversal a, m6 m10. So 2y 3y 10, and y 38. m1 m6 m9 m14 76 m2 m5 m10 m13 104 For parallel lines m and n cut by transversal b, m4 m12. So 2y 15 3y 7, and y 22. m4 m7 m12 m15 59 m3 m8 m11 m16 121 7 m1 42, m2 48, m3 42, m4 42 8 mx 85 9 mx 60 10 mx 68 11 130 Chapter Review 53 mx 45, my 45 mx 98, my 82 mx 60, my 70 mx 65, my 52 mx 67, my 78 mx 15, my 55 mx 55, my 62.5 5 b 20 c 35 n 3 _ d 594 e 2 14 102 sides 15 8 sides 16 14 sides 17 6 sides 18 a mx 85 b my 55 19 mD 45 20 m1 m2 25 21 m1 140 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 22 1. 6 4 2. x y 3. x z 4. y z (Transitive postulate) 23 1. 2 4 2. 1 3 3. 1 2 3 4 (Corresponding angles are 4. m r congruent.) ___ ___ 24 1. QR PS 2. QPS QRS 3. RPS QRP 4. QPS RPS QRS QRP or QRP ___ ___ SRP (Corresponding angles are 5. QP RS congruent.) ___ 25 1. BC bisects ABD. 2. 2 3 3. 1 2 4. 1 3 (Transitive postulate) 5. 5 1 12 a b c d e f g 13 a 54 Chapter 9: Parallel Lines 26 27 28 29 30 31 6. 3 5 7. a b (Alternate interior angles are congruent.) ___ 1. AC bisects BCD. 2. 3 4 ___ ___ 3. AB BC 4. ABC is a right angle. ___ ___ 5. CD AD 6. CDA is a right angle. ___ ___ 7. AC AC 8. I II (Hypotenuse–acute angle) 1. ABCD ___ ___ 2. CF DE 3. ACF BDE ___ ___ DE 4. CF ___ ___ CD 5. AB ___ ___ ___ ___ BC 6. AB ___ ___ CD BC 7. AC BD 8. ACF BDE (SAS SAS) ___ ___ 9. AF ___ BE 1. BE bisects ABC. 2. DBE EBA ___ ___ 3. DE BA 4. DEB EBA (Alternate interior angles are congruent.) 5. DEB DBE (Transitive postulate) ___ ___ 6. DB DE 7. BDE is isosceles. (Definition of isosceles triangle) ___ 1. AF ___ ___ 2. AB ___ CD ___ EF 3. AC ___ ___ ___ ___ CE EF CE 4. AC ___ ___ CF 5. AE ___ ___ 6. AB CD 7. BAE DCF (Corresponding angles are congruent.) 8. BAE DCF (SAS SAS) ___ ___ DF (CPCTC) 9. BE ___ ___ 1. AB CB 2. A C 3. B B 4. ABE CBD (ASA ASA) ___ ___ 5. AE ___ CD ___ 1. BA AE 2. BAE is a right angle. 3. 1 2 4. B D 5. 6. 7. 8. 9. 32 1. 2. 3. 4. 5. 33 1. 2. 3. 4. 5. 6. 7. 8. 34 1. 2. 3. 4. 5. 6. 7. 35 1. 2. ___ ___ AE AE ABE EDA (AAS AAS) BAE DEA (CPCTC) DAE is a right angle. ___ ___ DE AE (Perpendicular lines intersect to form right angles.) ___ ___ BA DE CAE CEA ___ ___ AE AE ABE EDA (SAS SAS) BEA DAE (CPCTC) 1 2 3 4 5 6 (Linear pairs of congruent angles are congruent.) ___ ___ AD EC ___ ___ DE DE ___ ___ AE DC (Addition postulate) ABE CBD (ASA ASA) ABE CBD (CPCTC) ___ ___ ___ RS ST TR 1 ___ 2 ___ RP TP (Converse of isosceles triangle theorem) ___ ___ SP SP RPS TPS (SSS SSS) RSP TSP (CPCTC) ___ SP bisects RST. (Definition of an ___ ___ ___ angle bisector) ___ PT QR, RS PQ RSP and PTR (Definition of right are right angles. angles) RSP and PTR are right triangles. ___ ___ PT RS ___ ___ PR PR RSP PTR (HL HL) SRP TRP (CPCTC) PQR is isosceles. (Converse of isosceles triangle theorem) ___ ___ ___ ___ 1. AS PR, BT PR 2. ASP and BTR are right angles. 3. PAS and RBT are right triangles. 4. 1 2 5. PAS RBT (Linear pairs of congruent angles are congruent.) ___ ___ 6. AS BT 7. PAS RBT (Leg–acute angle) 8. APS BRT (CPCTC) 9. PQR is isosceles. (Definition of an isosceles triangle) True True False True True False True False True False False True True True 3. 4. 5. 6. 7. 8. 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Chapter Review 55 CHAPTER Quadrilaterals 10-2 The Parallelogram (pages 221–223) 1 mP 48, mQ mS 132 2 mR 30, mQ mS 150 3 mQ mR 180 (3x 5) (4x 25) 180 7x 210 x 30 mS mQ 85 mP mR 95 4 mP mQ 180 (7x 12) (2x 3) 180 9x 189 x 21 mR 135 mS 45 1 (5x 3) 5 3x 5 _ 2 6x 10 5x 3 x7 PR 5(7) 3 32 QS 3(7) 5 16 6 PRS PRQ; QPS SRP; QMP SMR; QMR SMP 7 x (5x 6) 180 x 31 The angles are 31 and 149. 8 (3x 24) x 180 x 39 The angles are 39 and 141. 9 x3 10 4 11 mETS mBTE mTEB 42 56 Chapter 10: Quadrilaterals 10 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 12 1. ABCD ___ and DAPQ are parallelograms. −−− (Opposite sides of 2. AD BC a parallelogram are congruent.) −−− −− (Opposite sides of 3. AD PQ a parallelogram are congruent.) −− −− (Transitive property) 4. BC PQ 13 1. ABCD is a parallelogram. −− −−− 2. BC AD 3. GBF GDE 4. BGF DGE −−− (Diagonals of a par5. BG GD allelogram bisect each other.) 6. BFG DEG (ASA ASA) −− −− 7. FG GE 8. G is the midpoint (Definition of −− midpoint) of FE. −−− −− 14 1. MX QS −− −−− 2. TX QM 3. QMXT is a parallelogram. −−− 4. M is the midpoint of QR. −−− 5. QM MR −− (Opposite sides of 6. QM TX a parallelogram are congruent.) −−− −− 7. MR TX 8. XTS MXT 9. MXT MQT 10. MQT RMX 11. XTS RMX 12. XST RXM (Transitive postulate) (Alternate interior angles) (Opposite angles) (Corresponding angles) (Transitive postulate) (Corresponding angles are congruent) (AAS AAS) 13. MRX TXS 15 1. Parallelogram ABCD −− 2. X is the midpoint of BC. 1 BC 3. XC _ (Definition of a 2 midpoint) −−− 4. Y is the midpoint of AD. 1 AD 5. AY _ 2 −−− −− 6. AD BC 7. AD BC 1 BC 1 AD _ 8. _ 2 2 9. AY XC −− −− 10. AY XC 11. AMY CMX (Vertical angles are congruent.) 12. MAY MCX (Alternate interior angles are congruent.) 13. MAY MCX (AAS AAS) −−− −−− (CPCTC) 14. XM YM 15. a M is the midpoint (Definition of −− midpoint) of XY. −−− −−− (CPCTC) 16. AM MC 17. b M is the midpoint (Definition of −− midpoint) of AC. −− 16 1. AC is a diagonal in parallelogram ABCD. −− −− 2. AF CE −− −− 3. FE FE −− −− −− −− 4. AF FE CE FE −− −− 5. AE CF 6. DAC BCD (Definition of a parallelogram) 7. BAF DCE (Alternate interior angles are congruent.) 8. EAD FBC 9. 10. 11. 12. 17 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 18 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 19 1. 2. 3. 4. 5. 6. (Subtraction postulate) −−− −− AD BC (Opposite sides of a parallelogram are congruent.) ADE BCF (SAS SAS) AED CFB (CPCTC) −− −− DE BF (Alternate interior angles are congruent.) ABCD is a parallelogram. −− −− DE AF −− −− CF AF DAE and CFB are right angles. DAE CFB −− −−− CA AD −−−− AEBF −− −− EB EB −− −− AE BF (Subtraction postulate) DEA CFB (SAS SAS) −− −− DE CF (CPCTC) Parallelogram ABCD −− H is the midpoint of AB. 1 AB AH _ 2 −−− F is the midpoint of DC. 1 DC FC _ 2 −− −−− AB DC (Opposite sides of a parallelogram are congruent.) AB DC 1 DC 1 AB _ _ 2 2 AH FC −−− −− AH FC −−− −− −− −− HG AC, FE AC HGA and FEC are right angles. HFA FEC −− −−− AB DC CAB ACD GAH ECF (AAS AAS) −−− −− HG FE Parallelogram ABCD −− −−− AR CM −− −−− −−− −−− AR MR CM MR −−− −− AM CR −−− −− AD BC −−− −− AD BC 10-2 The Parralallogram 57 20 7. 8. 9. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 21 1. 2. 3. 4. 5. 6. 7. 22 1. 2. 3. 4. 5. 6. 7. 58 MAD RCB MAD RCB (SAS SAS) −− −−− BR DM (CPCTC) Parallelogram ABCD −−− QD bisects D. 1 mCDP mCDQ _ 2 −− PB bisects B. 1 mABQ mABP _ 2 CDP ABQ mCDP mABQ 1 mCDP _ 1 mABQ _ 2 2 mCDQ mABP CDQ ABP DCQ BAP ABP CDQ (SAS SAS) −− −−− (CPCTC) a AP CQ −− −−− BC AD −− −− (Subtraction b BQ PD postulate) Parallelogram PQRS PS PQ mx mQSP −−− −− QR PS y QSP (Alternate interior angles are congruent.) my mQSP mx > my (Substitution postulate) Parallelogram MARC AR MA mAMR mARM ARM CMR (Alternate interior angles are congruent.) mARM mCMR mAMR mCMR (Substitution postulate) AMR is not congruent to CMR. Chapter 10: Quadrilaterals 10-3 Proving That a Quadrilateral Is a Parallelogram (pages 224–225) 1 Check students’ answers. The following is one possible solution. −−− −− Slope of AB slope of CD 3. Slope of −− −−− AD slope of BC 0. Both pairs of opposite sides are parallel. 2 a PR 5 7 , 8 or (3.5, 8) b _ 2 3 a Check students’ answers. The following is one possible solution. −−− DR AB 10. Slope of DR slope −− of AB 0. One pair of opposite sides are both congruent and parallel. −−− b The length of the altitude from B to DR is 5. Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 4 1. Parallelogram LOVE 2. AOV BEL 3. OAV EBL −−− −− 4. OV EL 5. OAV EBL (AAS AAS) 1 5 BF _CE (segment connecting midpoints of 2 1 CE (definsides of a triangle) and CD _ 2 1 AC and tion of midpoint). BF CD. DF _ 2 1 AC. BF CD and DF BC. Opposite BC _ 2 sides of a parallelogram have equal measure so they are congruent. −− −− 6 1. KL and EU bisect each other at M. −−− −−− 2. KM LM −−− −−− 3. EM MU 4. EMK UML 5. EMK UML (SAS SAS) −− −− (CPCTC) 6. EK LU −− 7. K is the midpoint of JE. ( ) 1 EJ 8. EK _ 2 −−− 9. L is the midpoint of UN. 1 UN 10. UL _ 2 11. EK UL 1 EJ _ 1 UN 12. _ 2 2 13. EJ UN −− −−− 14. EJ UN −− −− 15. EU EU 16. LUM KEM 17. EUN UEJ −− −− 18. EN UJ 19. JUNE is a (Both pairs parallelogram. of opposite sides are congruent.) 7 1. ABCD is a parallelogram. −− 2. RC −−− −− −− −− 3. DQ RC, AR RC 4. DQC and ARB are right angles. 5. DQC ARB 6. RBA QCD (Corresponding angles are congruent.) −− −−− (Opposite sides of 7. AB DC a parallelogram are congruent.) 8. AB DC 9. ARB DQC (AAS AAS) −− −−− (CPCTC) 10. AR DQ 11. AR DQ −− −−− Segments perpen12. AR DQ dicular to the same segment are parallel.) 13. ARQD is a (One pair of opparallelogram. posite sides is both congruent and parallel.) −− −− 8 1. PE bisects HL at M. −−− −−− 2. HM ML 3. EPL PEH 4. HME LMP 5. HME LMP (AAS AAS) −− −− 6. HE LP −− −− 7. HE LP 8. HELP is a (One pair of opparallelogram. posite sides is both congruent and parallel.) 9 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 10 1. 2. 3. 4. 5. 6. 7. 8. 9. 11 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 12 1. 2. 3. 4. Parallelogram ABCD −−− −− AD BC −−− −−− AM NC −−− M is midpoint of AD. 1 AD AM _ 2 −− N is the midpoint of BC. 1 BC NC _ 2 −−− −− AD BC AD BC 1 BC 1 AD _ _ 2 2 AM NC −−− −−− AM NC ANCM is a (One pair of opparallelogram. posite sides is both congruent and parallel.) BR and DM 2 3 −− −−− BC AD 1 4 BAD DCB (Supplements of congruent angles are congruent.) −− −− BD BD BCD DAB (AAS AAS) −− −−− BC AD ABCD is a (One pair of opparallelogram. posite sides is both congruent and parallel.) −− −− QS bisects PR (at M). −−− −−− QM MS 1 2 QMR SMP QMR SMP (AAS AAS) −−− −− QR SP −− −− PR PR PSR RQP (SAS SAS) −− −− QP SP PQRS is a (One pair of opparallelogram. posite sides is both congruent and parallel.) −−− −− QV bisects RT. −− −− RS ST −−− −− QR PV RQS TVS (Alternate interior angles) 10-3 Proving That a Quadrilateral Is a Paralallogram 59 5. 6. 7. 8. 9. 10. 11. RSQ TSV QRS VTS −−− −− QR PT −− −− PT TV −−− −− QR PT −−− −− QR PT PQRT is a parallelogram. (Vertical angles) (AAS AAS) (CPCTC) (One pair of opposite sides is both congruent and parallel.) 13 1. Parallelogram ABCD −−− −− (Opposite sides of 2. DC AB a parallelogram are congruent.) −− −− 3. DF BE −−−− 4. DFEB 5. CDF ABE (Alternate interior angles are congruent.) 6. DFC BEA (SAS SAS) −− −− (CPCTC) 7. CF AE −−− −− (Opposite sides of 8. AD CB a parallelogram are congruent.) 9. ADF EBA 10. AFD CEB (SAS SAS) −− −− (CPCTC) 11. AF CE 12. AECF is a (Both pairs parallelogram. of opposite sides are congruent.) −− 14 1. KJ is a diagonal in parallelogram KBJD. −− −− 2. KA JC 3. BJC AKD −− −−− 4. BJ KD 5. BJC KAD (SAS SAS) −− −−− (CPCTC) 6. BC AD −− −− 7. BK JD 8. CJD BKA 9. ABK CDJ (SAS SAS) −− −−− (CPCTC) 10. AB CD 11. ABCD is a (Both pairs parallelogram. of opposite sides are congruent.) −− 15 1. BD is a diagonal in parallelogram ABCD. 2. BEC AFD 3. EBC ADF (Alternate interior angles are congruent.) −− (Opposites sides of 4. BC AD a parallelogram are congruent.) 60 Chapter 10: Quadrilaterals 5. 6. 7. 8. 9. 10. 11. CBE AFD −− −− AF EC −− −− DF EB ABE FDC −− −−− AB DC ABE CDE −− −− EA CF (AAS AAS) (CPCTC) (CPCTC) (SAS SAS) (CPCTC) 10-4 Rectangles (pages 227–228) 1 (1) are congruent 2 AR DR 3(4x 3) 10x 1 x5 AR CR DR BR 51 3 AR BR 2(x 6) 3x 20 x8 AR CR 4 BD 8 4 DR CR 4(3x 10) 3(x 2) 12 46 x_ 9 46 AR _ 9 128 AC BD _ 9 5 AC BD 1 2 (12x 3) 5x 3(2x 5) _(4x 4) _ 3 4 x2 AC 24 DR 12 6 2x 30 36 x3 7 PR QS 4x 3 6x 7 x5 PR 4(5) 3 23 QS 6(5) 7 23 PR QS √ 23 23 23 √ 8 mADB mDAC 90 49 41 9 (5, 6) Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 10 1. Rectangle ABCD 2. ABCD is a parallelogram. −− −−− 3. BC AD −− −−− (Opposite sides of 4. AB CD a parallelogram are congruent.) −− −− (Diagonals of a 5. BD CA rectangle are congruent.) 6. CDA BAD (SSS SSS) 7. CAD BDA (CPCTC) 11 1. Rectangle PQRS 2. PQRS is a parallelogram. −− −− 3. PQ PQ −−− −− 4. QR PS 5. SQP PQR (SSS SSS) 6. 1 2 (CPCTC) 12 1. Rectangle ABCD 2. ABP and NCD are right angles. 3. ABP and NCD are right triangles. −− −−− 4. AP DN −−− −− (Opposite sides of 5. DC AB a parallelogram are congruent.) 6. ABP DCN (HL HL) 7. DNC APB (CPCTC) 8. PAD APB (Alternate interior angles are congruent.) 9. NDA APB 10. PAD NDA (Transitive postulate) −− −− (Definition of an 11. AE DE isosceles triangle) −− 13 By the addition postulate of inequality, AY −− is not congruent to TX, and AMY is not −−− congruent to THX. Therefore, MY is not −−− congruent to HX. 14 1. Rectangle ABCD −−− 2. N is the midpoint of CD. 3. CN DN −−− −−− 4. CN DN 5. BCN NDA (A rectangle is equiangular.) −− −−− (Opposite sides of 6. BC AD a parallelogram are congruent.) −− −−− (CPCTC) 7. BN AN −− −−− 1. AB CD 2. BJH and AJH are linear angles. 3. mBJH mAJH 180 1 mAJH 90 1 mBJH _ 4. _ 2 2 (Division postulate) −− 5. JG bisects BJH. 1 mBJH 6. mHJG _ 2 −− 7. EJ bisects AJH. 1 mAJH 8. mEJH _ 2 9. mHJG mEJH 90 (Substitution) 10. mEJG 90 11. EJG is a right angle. 12. CHJ BJH 13. EHJ GJH 14. AJH DHJ 15. HJE JHG −− −− 16. HJ HJ 17. EJH GHJ −− −−− 18. EJ GH −− −− 19. EH JG 20. EJGH is a parallelogram. 21. EJGH is a rectangle. (Definition of a rectangle) 16 1. Rectangle PQRS −− −− 2. PA CS −− −− −− −− 3. PA AC CS AC −− −− 4. PC AS −− −− 5. QP RS 6. QPS RSP 7. QPC RSA (SAS SAS) 8. a 1 2 (CPCTC) 9. PQR and SRQ are right angles. 10. 3 is complementary to 1. 11. 4 is complementary to 2. 12. b 3 4 (Complements of congruent angles are congruent.) −− −− (Definition of an 13. c QB RB isosceles triangle) 15 10-5 Rhombuses (pages 229–231) 1 mB mD 105; mC 75. AB CD 7 2 mADB 66 10-5 Rhombuses 61 3 4x 2 3x 3 x5 RS 18 4 Perimeter of MABC 8 5 mADC 110 6 2x 2 x 8 x 10 CD 18 7 mADC 94 −− −− 8 a Midpoint of AC midpoint of BD (6, 5) −− −− b Slope of AC 1, slope of BD 1. Slopes are negative reciprocals, diagonals are perpendicular. 9 a (9, 5) 2 b PQ PS 5 √ −− −− _ 1 c Slope of PR , slope of QS 3. 3 Slopes are negative reciprocals, diagonals are perpendicular. −− k −−− 10 a Slope of AD slope of BC _ x. AD BC √ x 2 k 2 . ABCD is a parallelogram. (One pair of opposite sides have the same length and are parallel.) −− −− _ k , slope of BD k , b Slope of CA _ xk xk not negative reciprocals. The diagonals are not perpendicular. Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −−− −− −− −− −−− −− 11 QR AC, PS AC, QR PS. Likewise, −− −− −− −− −− −− QP BD, RS BD, and QP RS. Thus, PQRS −− −− −−− is a parallelogram. Since AC BD, QR −− −− −− and PS are perpendicular to QP and RS. Thus, PQRS has at least one right angle, and PQRS is a rectangle. 12 1. Rhombus ABCD −− 2. ED −− −− 3. BF AC −− −− 4. EF AC −− 5. F bisects AC. 6. EA EC 62 Chapter 10: Quadrilaterals −− −− 7. EA EC 8. ACE is isosceles. 13 14 15 16 17 18 (Definition of an isosceles triangle) −− −− −− −− Prove that DF AB and DE BF by using the midpoints and alternate interior angles. Thus, EBFD is a parallelogram. By the divi−− −− sion postulate, EB FB. EBFD is a rhombus because a rhombus is a parallelogram with two congruent consecutive sides. 1. Parallelogram ABCD −− −−− 2. AB CD 3. BAD 2 (Corresponding angles are congruent.) 4. mBAD m2 5. BAD 1 CAD 6. mBAD m1 (Whole is greater than a part.) 7. m2 m1 (Substitution postulate) 1. Rhombus PQRS −− −− 2. QS PR −−− −− 3. QC CS −−− 4. B is the midpoint of QC. −− 5. D is the midpoint of CS. −− −−− 6. BC ___ CD ___ 7. AC AC 8. ABC ADC (SAS SAS) −− −−− (CPCTC) 9. AB AD 10. BAD is isosceles. (Definition of an isosceles triangle) If a quadrilateral is equilateral, it is a parallelogram with a pair of consecutive congruent sides. The diagonal creates two congruent triangles (ASA ASA) and corresponding congruent sides are consecutive. −−− 1. CD BE −−− −− 2. CD BA −−− 3. AD BF −−− −− 4. AD BC 5. ABCD is a (Definition of a parallelogram. parallelogram) 6. BAD BCD (Opposite angles are congruent.) bisects FBE. 7. BG 8. CBD ABD −− 9. ABCD is a rhombus. (Diagonal BD bisects opposite angles.) 10-6 Squares (pages 232–233) 1 (4) A rectangle is a square. 2 (3) sides and angles are congruent 3 (4) A trapezoid is a parallelogram. −− −−− 4 (3) AC DC 5 (1) congruent and bisect the angles to which they are drawn 2 6 (2) x √ −−− −− 3 7 Slope of DA slope of VE _. Slope of 4 −− −− 4 . A parallelogram ED slope of AV _ 3 has two pairs of opposite sides that are parallel. A rectangle is a parallelogram in which consecutive sides have slopes that are nega−−− 1 . Slope tive reciprocals. Slope of DV _ 7 −− of AE 7. Slopes of the diagonals are negative reciprocals, thus they are perpendicular. Therefore, DAVE is a square. −−− −− 4 . Slope of 8 Slope of MA slope of TH _ 3 −− 3 −−− HM slope of AT _. A parallelogram has 4 two pairs of opposite sides that are parallel. A rectangle is a parallelogram in which consecutive sides have slopes that are negative −−− 1 . Slope of reciprocals. Slope of MT _ 7 −−− AH 7. Slopes of the diagonals are negative reciprocals, thus they are perpendicular. Therefore, MATH is a square. 9 The diagonals bisect the vertex angles, creating four congruent isosceles triangles. The bisected angles (the base angles of the triangles) measure 45. The vertex angles thus measure 90. The diagonals cross, forming 90 angles. −− −− −−− −− 10 a Slope of PQ slope of RS 0. QR and SP have no slope. A parallelogram has two pairs of opposite sides that are parallel. A rectangle is a parallelogram in which consecutive sides are perpendicular. b PQ RS 20. QR PS 7. Since all sides are not congruent to each other, PQRS is not a square. Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 11 a Given ABCD is a square. BFE is a right −− −− angle because EF BD. mEBF 45 −− because diagonal BD bisects right angle −− −− ABC. mFEB 45. Therefore, BF EF. b 1. ABCD is a square. −− −− 2. EF BD 3. EFD is a right angle. 4. DAE is a right (Definition of angle. a square) 5. DAE and DFE are right triangles. −− −− −− 6. (Draw DE). DE DE 7. DAE DFE (HL HL) −− −− (CPCTC) 8. EF EA −− 12 1. Rhombus ABCD with diagonals AC −− and BD intersecting at E. 2. 1 2 −− −− 3. BE CE −− −− 4. E bisects AC and BD. −− −− 5. BD AC 6. ABCD is a rectangle. 7. ABCD has all right angles. 8. ABCD is a square. (A square is a rhombus in which all angles are right angles.) 10-7 Trapezoids (pages 236–238) 1 (2) They are congruent. 2 (3) ADC ABC 3 a x z 70; y 110 b x 73; y z 107 c x 40; y 108; z 32 d x 70; y 44; z 66 e x 130; y 20; z 30 f x 82; y z 41 −−− −− 1 . These legs 4 Slope of MA slope of TH _ 2 −− 3 −−− are parallel. Slope of AT _. HM has 4 no slope. These legs are not parallel. AT HM 10. Therefore, MATH is an isosceles trapezoid. 10-7 Trapezoids 63 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 5 1. ABCD is an isosceles trapezoid. 2. BAD CDA −− −−− 3. BC AD 4. NBC BAD (Corresponding angles are congruent.) 5. NCB CDA 6. NBC NCB (Substitution postulate) 7. NBC is isosceles. (Base angles of an isosceles triangle are congruent.) 6 1. Isosceles trapezoid ABCD −− −−− 2. AB DC 3. BAD CDA (Base angles of an isosceles trapezoid are congruent.) −−− −−− 4. AD AD 5. ADB DAC (SAS SAS) 7 1. Trapezoid ABCE −− −−− 2. BD AD −− −− 3. AB CE 4. A B 5. CED A 6. ECD B (Corresponding angles are congruent.) 7. CED ECD (Substitution postulate) −−− −− 8. CD ED −− −−− (Subtraction 9. BD CD −−− −− AD ED postulate) −− −− 10. BC AE 11. ABCE is an (Nonparallel sides isosceles trapezoid. of an isosceles trapezoid are congruent.) 8 1. Quadrilateral PQRS −−− −−− −−− 2. QAB, RAS, and PSB −− −− 3. QB bisects RS. −− −− 4. RA AS 64 Chapter 10: Quadrilaterals −−− −−− 5. PSB QR 6. RAQ BAS 7. QRA BSA 8. 9. 9 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 10 11 12 1. 2. 3. 4. 5. 6. 7. 8. 9. 1. 2. 3. 4. 5. 6. 7. 8. 9. 1. 2. 3. 4. 5. (Vertical angles are congruent.) (Alternate interior angles are congruent.) (ASA ASA) QRA BSA −−− −− QA AB Isosceles trapezoid ABCD A D −−− −− AD BC −− −−− AB DC −−− E is the midpoint of AD. −− −− AE ED ABE DCE (SAS SAS) −− −− (CPCTC) a BE CE BE CE BCE is isosceles. (Definition of isosceles triangle) −− −− (The median from b EH BC the vertex angle of an isosceles triangle is perpendicular to the base.) Isosceles trapezoid PQRS −− −− PQ RS −−− −− QR PS P S −− −− PS PS PQS SRP (SAS SAS) PQS SRP (CPCTC) QAP BAS (Vertical angles are congruent.) PAQ SAR (AAS AAS) Trapezoid ABCD −− −−− BR AD −−− −−− CM AD ARB and DMC are right angles. ARB and DMC are right triangles. −− −−− AB CD BAR CDM ARB DMC (Hypotenuse–acute angle) 1 2 (CPCTC) Trapezoid PQRS Q R −− −−− −− −−− PA QR, SE QR PAQ and SER are right angles. PAQ and SER are right triangles. 6. AP ES −− −− 7. AP ES 8. PAQ SER −− −− 9. QP RS (Distances between parallel lines are equal.) (Leg–Acute Angle) (CPCTC) 10-8 Kites (page 239) 1 A kite is a quadrilateral with only two pairs of adjacent congruent sides. A rhombus has four congruent sides. 2 True 3 True 4 False 5 True 6 True 7 False 8 False 9 True −− 10 a AC b 1 8; 2 7; 3 6; 4 5 11 a 90 b 45 c 17 d m4 25 and mKLM 50 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −− −− 12 1. QS is the perpendicular bisector of PR. −− −− 2. PT TR 3. QTP QTR −− −− 4. QT QT 5. QPT QRT (SAS SAS) 6. STP STR −− −− 7. TS TS 8. STP STR (SAS SAS) −− −−− (CPCTC) 9. QP QR −− −− (CPCTC) 10. SP SR −− −− 11. QT ST 12. PQRS is not (Definition of a rhombus. a rhombus) 13. PQRS is a kite. (Definition of a kite) 10-9 Areas of Polygons (pages 243–245) 1 4 2 16 3 9, 12, 15 4 Length is 4 ft and width is 5 ft. 5 a h 10 x b A x(10 x) 2 b 9x 2 c 6 a x 2 2 d x 4x 4 e 4x 4x 1 25 7 a _ b 18 c 2 d 1 e 9 8 11 9 a 12 b 45 c 45 3 e _ d 18 √ 2 √ √ 10 36 144 6 5 11 a BD 10 b 120 √ 3 12 72 13 s 6 14 a 13 b 120 c x 2 4x 4 49 _ 2 20 120 _ 13 c 4x 12 15 a 21x b 10x 2 2 d 18x 12 e 2x 6x 16 a x 4 b 4(4) 4 12 17 8 3 18 27 √ √ 2 19 30 20 10 and 20 21 a 96 b 260 c 60 22 72 23 100 24 a 6 b SMP and AML are similar. Let x be the −− perpendicular drawn from M to AP. 20 x _ Then _ , and x 5. 6x 4 10 x _ b x4 25 a _ 20 12 x c Area of RCS 20. Area of QCT 80. 4(8 11) 9 26 a A _ 38 b 5, _ 2 2 −− −− _ 3 _ 27 a Slope of JE , slope of EN 4 . The 3 4 slopes of the two legs are negative reciprocals, therefore perpendicular, forming a right angle. 1 (JE)(EN) 25 b A_ 2 ( ) 10-9 Areas of Polygons 65 28 Enclose PAT in a large rectangle and subtract the excess areas. a Area of PAT 120 60 24 6 12 18 b AT 10 c 3.6 52 ; sides are congruent. 29 a SA SM √ −− _ −−− 3 2; Slope of SA , slope of SM _ 2 3 slopes are negative reciprocals, therefore perpendicular and forming a right angle. 1 (SA)(SM) 26 b A_ 2 c √ 26 30 Subdivide pentagon SIMON into two triangles and a trapezoid and take the sum of all the areas. Area 72 31 Divide pentagon JANET into two triangles and take the sum of the two areas. Area 64 32 Enclose quadrilateral RYAN in a large rectangle and subtract the excess areas. 13 Area 72 _ 6 12 12 35.5 2 −− −− 33 NI and CK have zero slope, the two sides are −−− −− 10 . Therefore, NICK parallel. NK IC m √ is an isosceles trapezoid. −− −−− 1 ; the sides are 34 Slope of JO slope of HN _ 3 −− parallel. Slope of NJ 3. The slopes are negative reciprocals, therefore perpendicu−−− lar, forming a right angle. Slope of OH is not −− equal to the slope of NJ, therefore, only one pair of sides are parallel. JOHN is a right trapezoid. 35 a 38 b 28 c 28 d (5, 4) Chapter Review (pages 245–248) 1 (4) The diagonals of a parallelogram bisect each other. 2 (4) mB mC 360 3 (1) rhombus 4 (1) a rhombus 5 (3) A rhombus is a square. 6 (4) parallelogram 7 (3) 3 −− −− 8 (4) QS and PR bisect each other. 9 (1) the diagonals are congruent 10 a square b rhombus c parallelogram d rectangle 66 Chapter 10: Quadrilaterals 11 3x 40 x 50 x5 12 5 13 5x 90 x 18 14 (3x 20) (7x 40) 180 x 20 15 2 16 mDAE 90 75 15 17 (4, 3) 18 3x 15 7x 55 x 10 19 a mACD mCAB 30 b rectangle 180 75 20 mAEK _ 52.5 2 21 5 ft 22 (1, 1) 2 23 3 √ 24 mA 45 25 h 8 26 Rhombus. Since the triangles are congruent, the opposite angles that are originally vertex angles are congruent. By the addition postulate, the opposite angles formed by joining the triangles at their bases are congruent. Thus the quadrilateral is a parallelogram. All sides are congruent. The parallelogram is a rhombus. 27 PS QR 2x 3 x 2 x5 PS QR SR PQ 7 28 a (2x 8) 3(x 34) 180 x 14 mABC mCDA 144 mDAB mBCD 36 b AE EC 4y 6y 36 y 18 BE ED 3x 1 x 13 x7 AC 144, BD 40 29 mABP mBAP 90 (5x 10) (2x 4) 90 x 12 mDCB 140, mAPB 90 30 a mR 150 b mRAD 150 c mGAD 135 d mD 30 20 √ 3 35 20 200 √3 346.4 2 . ABC is isosceles beAB BC 5 √ cause there is one pair of congruent sides. −− −− b Slope of AB 1. Slope of BC 1. Slopes −− −− are negative reciprocals, thus AB BC. c Area of ABC 25 −− −− −− 33 DE AV because the slope of DE slope −− 1 −−− −− of AV _ . DA is not parallel to VE because 3 −−− the slope of DA is not equal to the slope −− of VE. DA VE 5; nonparallel sides are congruent. Therefore, DAVE is an isosceles trapezoid. −− −− 3 34 Slope of PQ slope of RS _. Slope of 4 −− −−− −− 4 . Slope of PR 1. PS slope of RQ _ _ 3 7 −− Slope of QS 7. PQRS is a parallelogram because opposite sides have the same slope, and a rhombus because the diagonals are perpendicular—slopes are negative reciprocals, and a square because consecutive sides are perpendicular—slopes are negative reciprocals. −−− −− 35 Slope of AB slope of CD 0. Slope of −− −− −−− 4 . Slope of AC BC slope of AD _ 2. 3 −− 1 . ABCD is a parallelogram Slope of BD _ 2 because opposite sides have the same slope, and a rhombus because the diagonals are perpendicular—slopes are negative reciprocals. −− −− 4 . Slope 36 Slope of LE slope of AF _ 5 −− −− −− 4 of EA slope of FL _ . Slope of LA is −− 5 udefined. Slope of EF 0. LEAF is a parallelogram because opposite sides have the same slope, and a rhombus because the diagonals are perpendicular. Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 31 a b c 32 a 37 1. 2. 3. 4. 5. 6. 7. 8. Quadrilateral TRIP −− −− TR RI m1 m2 1 2 −− −− RA RA a RAT RAI −− −− TA AI TAP IAP (SAS SAS) (CPCTC) (Supplements of congruent angles are congruent.) −− −− AP AP TAP IAP (SAS SAS) b 3 4 (CPCTC) Parallelogram PQRS −− Diagonal QS bisects PQR. PQS RQS RQS QSP (Alternate interior angles are congruent.) 5. PQS QSR (Alternate interior angles are congruent.) 6. RQS QSR (Transitive postulate) 7. QSP QSR −− −− 8. PS PQ 9. PS PQ 10. PQRS is a rhombus. (A rhombus is a parallelogram with two congruent consecutive sides.) 39 1. ABED is a rhombus. −−−− −− 2. BD intersects AOEC at O. 3. BOE and DOE are right angles. 4. BOE and DOE are right triangles. −− −− 5. OE OE −− −− 6. BE DE 7. BOE DOE (HL HL) −− −−− (CPCTC) 8. BO OD 9. BOC and DOC are right triangles. −−− −−− 10. OC OC 11. BOC DOC (Leg-leg) −− −−− (CPCTC) 12. a BC DC −− −−− 13. AB AD −− −− 14. AC AC 15. ABC ADC (SSS SSS) 16. b ABC ADC (CPCTC) 40 Mark the point of intersection of the diagonals E. The diagonals bisect each other, thus AE DE. By the triangle inequality theorem, mADE mDCE and mADC mDCB by the multiplication postulate of inequality. 9. 10. 11. 38 1. 2. 3. 4. Chapter Review 67 CHAPTER Geometry of Three Dimensions 11-1 Points, Lines, and Planes (pages 251–252) 1 False 2 False 3 True 4 True 5 True 6 True 7 True 8 False 9 True 10 False 11 False 12 H 13 E 14 C 15 G 16 C 17 G 18 E 19 D −− −− −− −−− −− −− 20 a AB FG, AB HC, AB ED, −−− −−− −−− −− −−− −− −− −− AH DG, AH CB, AH EF, FA GB, −− −−− −− −− FA DC, FA EH b There are many possible answers. The following are just a few. −− −− −− −−− −− −− −− AB & EF, AB & DG, BC & EF, BC & −− −−− −−− −−− −− ED, DC & AH, DC & EF. 21 Infinitely many 22 Infinitely many 23 Infinitely many 24 One 25 One 26 Infinitely many 68 Chapter 11: Geometry of Three Dimensions 11 27 28 29 30 31 32 33 34 35 Infinitely many One Infinitely many Infinitely many One One One None One 11-2 Perpendicular Lines, Planes, and Dihedral Angles (pages 257–258) 1 False 2 True 3 True 4 False 5 True 6 True 7 False 8 True 9 One 10 Infinitely many 11 Four 12 Infinitely many 13 One 14 One 15 a ABCD b GFE 16 a BCDE and GCDE b PRS, SRT, ADE, and EDF 17 The plane that bisects the dihedral angle −−− −− −−− −−− −− −− 18 QR, QS, QU, WP, TP, VP Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 19 1. 2. 3. 4. 5. 6. 7. plane M BC BCA and BCD are right angles. BCA BCD −− −− AB DB −− −− BC BC BCA BCD (HL HL) BAC BDC (CPCTC) 11-3 Parallel Lines and Planes (pages 259–260) 1 Parallel 2 Perpendicular 3 Perpendicular 4 Parallel 5 Parallel 6 Parallel 7 Parallel 8 Infinitely many 9 None 10 One 11 Line a is parallel to plane M. Line b is contained in plane R. Line b is in plane M. Lines a and b are coplanar and do not intersect. Therefore, a b. 12 True 13 True 14 True 15 False 16 False 17 True 18 False 19 False 20 True 21 True 22 True 23 True 24 True 25 26 27 28 29 30 True False False False False True 11-4 Surface Area of a Prism (pages 264–265) 1 (3) 80 2 a 104 b 108 c 224 3 4 faces, 7 edges, 4 vertices 4 5 faces 5 Right prism 2 2 b 223 ft 6 a 404 cm 7 208 8 242 square inches 9 588 2 10 435 in. 2 11 Lateral area: 255 cm ; total area: 25 √ 3 255 _ cm 2 2 12 Lateral area: 45 cm 2; total area: 9 √ 3 45 _ cm 2 2 13 144 32 √ 3 cm 2 14 3,880 in. 2 3 15 Lateral area: 72; total area: 72 12 √ 16 Lateral area: 510 square inches; total area: 3 square inches 510 75 √ 17 3 √3 in. 2 2 2 18 d 2 h 2 (AB) , but (AB) l 2 w . By sub2 stitution, d 2 h 2 l 2 w 11-5 Symmetry Planes (pages 267–268) 1 (1) A 2 (2) E 3 (1) F Exercises 4–7: Check students’ sketches. 8 7 symmetry planes 9 Check students’ sketches. 10 112.5 √3 11-5 Symmetry Planes 69 11-6 Volume of a Prism (pages 269–270) 1 288 3 2 15,625 m 3 2 3 a Volume: 105 cm ; surface area: 142 m 3 2 b Volume: 3,600 in. ; surface area: 1,500 in. 3 2 c Volume: 48 ft ; surface area: 88 ft 4 147 5 a 2 b 6 c 4 d 5 e 3 6 14 1 7 _ 2 8 Volume: 1,920 ft 3; surface area: 992 ft 2 9 1 3 3 ft 10 Volume: 2,197 ft ; diagonal: 13 √ 11 30 inches 3 12 480 cm 13 9 14 2.7 ft 3 15 x √ √ 3 16 81 17 Volume: 343; surface area: 294 3 in. 3 18 24 √ 19 Volume is ten times as large. 3 cubic 20 a Each solid has a volume of 360 √ units. b triangular prism 360 square units, 6 hexagonal prism 120 √ 2 2 2 21 Diagonal, d √l w h , so d 2 2 l 2 w h 2. Multiply by four, so that 2 2 4d 4l 2 4w 4h 2. 11-7 Cylinders (pages 273–274) 1 (3) 6 inches 2 h 11 3 V_ 9 1 _ 4 r 4 1 5 h_ 6 7 8 9 10 70 a a a a a 112 cm 2 2 96 in. 2 24 in. 2 6 m r3 b b b b b 136.5 cm 168 in. 2 26 in. 2 14 m 2 r4 2 c c c c 196 cm 288 in. 2 12 in. 2 6 in. 2 3 Chapter 11: Geometry of Three Dimensions 11 A cylinder with a radius of 8 and a height of 12 has a volume of 768. A cylinder with a radius of 12 and a height of 8 has a volume of 1,152. Their difference is 384. 12 38 pounds 13 r 3.4 14 h 29.4 15 a 64 b The volume is twice as large; 128 c 32 d The lateral area stays the same; 32. 1 of the 16 a Doubled b Volume is _ 4 volume original. 17 Yes. The volume of the glass with a diameter of 2.4 is 1.44. Double the volume is 2.88. The volume of the second glass is 2.89, which is greater than twice the volume of the first glass. 18 False 19 True 2 20 Lateral surface area: 960 in. ; volume: 3 3,840 in. 6,930 385 21 h _ _ 36 2 22 16 : 64 or : 4 23 a V 256 b V 512 c V 256 512 11-8 Pyramids (pages 278–279) 2 1 234 cm 2 2 240 cm 2 3 1,152 in. 3 4 4,848 cm 3 5 57.2 in. 3 6 7.5 ft 3 7 48 √ 8 h 300 ft 9 h6 2 10 279 ft 3 11 864 in. 12 120 3 b 340 13 a 180 25 √ 3 c 360 150 √ 14 9 √3 15 Volume of pyramid is 80. Volume of prism is 240. 16 240 3 cm 2 17 144 √ 3 ft 3 18 432 √ 11-9 Cones (pages 283–284) 1 Lateral area: 60; total area: 96; volume: 96 2 Lateral area: 20; total area: 36; volume: 16 3 Lateral area: 136; total area: 200; volume: 320 4 Lateral area: 80; total area: 105 2 3 5 Lateral area: 135 cm ; volume: 324 cm 2 2 6 Lateral area: 260 in. ; total area: 360 in. ; 3 volume: 800 in. 7 r3 8 V 36 9 V 16 10 V 12 11 r 10 1 base height 12 _ 3 13 equal 14 three times 15 one-third 16 No, because the slant height is always greater than the radius. 17 4 : 1 18 h 8 3 19 28.5 in. 3,056 20 Volume: _ 1,018.67 in. 3; lateral area: 3 300 in. 2 3 21 144 √ 22 60 11-10 Spheres (pages 287–288) 1 a r 10.5 ft 2 a r 5 mi c r 4 mi 3 3 a 288 in. b r 6.2 ft c r 2.9 ft √ b r 2 5 4.5 mi d r 1.6 mi b 36 ft 3 3 1 yd 3 d _ c 972,000 mi 6 4 m 3 e _ 81 4 r 7 in. 500 2 3 5 Surface area: 100 cm ; volume: _ cm 3 6 Surface area: 192 in. 2; volume: 256 √3 in. 3 7 Volume: 1,679,616,000 cubic miles; surface area: 4,665,600 square miles 8 63,361,600 square miles 9 1,000,000 times 10 r 5 11 r 3 9 12 9 : 25 or _ 25 4 2 b 4 : 25 or _ 13 a 2 : 5 or _ 5 25 2 4 14 a _ b _ 3 9 15 a m 2 : n 2 b m3 : n3 64 16 _ 27 17 108 ounces 18 1,458 lb. 2 4 b r_ 19 a r _ √ √ 24 _ 20 r 1 ft 3 4 ft 3 21 a _ b _ c 4.5 ft 3 6 3 22 64 23 h 4r 24 : 6 or _ 6 2 _ 25 2 : 3 or 3 26 They are equal. 2 Lateral area of cylinder 16 in. 2 Surface area of sphere 16 in. √ Chapter Review (pages 289–290) 2 1 (4) 216 in. 2 False 3 False 4 True 5 True 6 False 7 False 8 False 9 False 10 True 11 True 12 a 8 vertices, 6 faces, 12 edges b 6 vertices, 5 faces, 9 edges c 8 vertices, 6 faces, 12 edges d 10 vertices, 7 faces, 15 edges 13 a Surface area: 216, volume: 432 b Surface area: 90, volume: 100 2 3 c Surface area: 52 ft , volume: 24 ft Chapter Review 71 14 15 16 17 18 19 20 21 22 23 47.5 in. 2 h 4 in. 3 25 cm The cylinder formed by spinning the rectan3 gle about the shorter side is 96 cm greater in volume. 3 r 6 and h 8, V 288 cm 3 r 8 and h 6, V 384 cm 2 3 in. 36 9 √ 3 in. 3 36 √ 3 cm 3 144 √ 480 2 156 in. √ 65 in. 24 25 26 27 28 29 30 31 32 3 100 ft 2 60 ft 2 3 Lateral area: 15 in. ; volume: 12 in. 3 r 3 in.; volume: 36 in. 2 Surface area: 40,000 cm 4,000,000 1 cm 3 Volume: _ cm 3 or 1,333,333_ 3 3 2 3 4 _ Surface area: 4 ft ; volume: ft 3 8 r _ meters √ 5 _ inches or 2.5 inches 2 40.57 41 scoops CHAPTER Ratios, Proportion, and Similarity 12-1 Ratio and Proportion (pages 294–295) 1 No 2 Yes 3 No 4 Yes 3 5 a _ 2 _ b 2 5 3 c _ 5 5 d _ 3 6 77 7 15 8 9 9 15 10 5 72 Chapter 12: Ratios, Proportion, and Similiarity 12 11 44 12 15 13 39 mt 14 _ a 2ma _ 15 r 4ar 16 _ t 17 9 18 8 19 15 20 4 √3 11 21 3 √ 22 5 √3 23 6 √2 24 4 √6 1 25 _ 10 √ 3 1 or _ 26 _ 9 3 √3 27 28 29 30 31 32 33 34 40, 50 70, 110 20, 35 16, 20, 24 length 28, width 49 45, 135 a 8 b 6 a 20 b 9 23 a 1. 2. 3. 4. 5. 6. 7. 8. 9. c 10 12-2 Proportions Involving Line Segments (pages 298–300) 1 a–d are all true 4 Proportions are not true, lines not 2 _ 2 _ 5 8 parallel. 3 6 4 28 5 5.4 6 6 7 24 8 5 ac 9 _ b np _ 10 m 11 6 12 20 c(a b) 13 _ a 14 No 15 No 16 No 17 No 18 Yes 19 12 1 20 1 : 3 or _ 3 21 11 cm 22 x 6, AB 47, DE 17, AC 34, EC 9, BE 9, AD 23.5, DB 23.5 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) b 24 1. 2. ABC −− P is the midpoint of AB. −− Q is the midpoint of BC. −− −− PQ AC (A line joining the midpoints of two sides of a triangle is parallel to the third side.) −− ___ PQ AR −− R is the midpoint of AC. −− −−− AB RQ −− −−− AP RQ PQRA is a parallelogram. (Definition of a parallelogram) 30 ABC, ADC −− −− AC wx −− −− AC zy −− zy −− wx BAD, BCD −− −− BD wz −− −− BD xy −− xy −− wz wxyz is a parallelogram. 25 48 √3 3. 4. 5. 6. 7. 8. 9. (A line dividing two sides of a triangle proportionally is parallel to the third side.) (Definition of a parallelogram) 12-3 Similar Polygons (pages 302–303) 1 1:1 2 4, 4.5, 5 3 1:4 4 45, 55 5 a 2:3 b 4.5, 7.5 c 18 6 9, 13.5, 18 7 5a, 5b, 5c 8 All corresponding angles are congruent, all corresponding sides are in proportion, 1 : 2. y _ x, _ w, _ , z 9 _ 3 3 3 3 10 32, 40 11 mR mQ 110 12 mI 40; mK 160 13 12 12-3 Similar Polygons 73 b QR 10, RS 20, ST 12, PT 22 15 a True b False c True d True e False f True g False h False 16 If the vertex angles are congruent, then the base angles must also be congruent. Similarly, if the base angles are congruent, then the vertex angles are congruent. Triangles are similar by (AA) or (AAA). 14 a 3 : 2 12-4 Proving Triangles Similar (pages 307–308) 1 (2) similar 2 Vertical angles are congruent. (AA) 3 Not similar Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −− −− 4 a 1. AE BC 2. BCD DAE (Alternate interior angles) −− −− 3. AC and BE intersect at D. 4. BDC ADE (Vertical angles) 5. ADE CDB (AA) b AD 30 5 All corresponding sides are in BE _ AB _ AE _ 4 . Triangles proportion; _ 5 DE CD CE are similar. 5 4 and _ 6 _ , corresponding sides are not in 6 3 proportion. Triangles are not similar. 6 4 , corresponding sides are not in 7 _ and _ 7 5 proportion. Triangles are not similar. −−− −− 8 1. CD AB 2. CDA and CDB are right angles. 3. CDA CDB 4. CAD CBD 5. CAD CBD (AA) 74 Chapter 12: Ratios, Proportion, and Similiarity −− −− 9 1. AB DE 2. EDF BAC −− −− BC EF EFD BCA ABC DEF (AA) −−− −− −− −− BE AC and CD AB HEC and HDB are right angles. HEC HDB DHB CHE (Vertical angles are congruent.) 5. EHC DHB (AA) −− −− −− −− 1. DE BC and DF AB 2. DEC and DFA are right angles. 3. DEC DFA 4. Parallelogram ABCD 5. FAD ECD 6. AFD CED (AA) 1. DBE ADF 2. BAC BDE (Corresponding angles) 3. B B 4. DBE ABC (AA) 1. Parallelogram ABCE −−− 2. AED −− −− 3. BC AE −− −−− 4. BC AD 5. CBF ADB (Alternate interior angles) 6. BAD FCB 7. BAD FCB (AA) 1. ABC PQR 2. ABC PQR −− 3. BD bisects ABC. −− 4. QS bisects PQR. 5. ABD PQS 6. BAD QPS 7. ABD PQS (AA) 1. Parallelogram ABCD −−− −− 2. AD BC 3. GAE BCG (Alternate interior angles are congruent.) −− 4. AC bisects HAE. 5. HAG GAE 6. HAG BCG −−− −− 7. AH AE 3. 4. 5. 10 1. 2. 3. 4. 11 12 13 14 15 (Corresponding angles are congruent.) 8. 9. 10. 11. GEA GHA GEA GBC GHA GBC HAG BCG 12-6 Proving Proportional Relationships Among Segments Related to Triangles 12-5 Dilations (pages 310–311) 80 10 b _ c 9 1 a _ 3 3 2 a BC 10, PA 2.2, BA 11, PS 9.6 10 5 b __ 8 4 3 (21, 9) 4 (6, 12) 5 (9, 0) 6 (27, 3) 7 (12, 12) 8 (10, 36) 9 (20, 12) 10 (2.5, 1) 11 (4, 2) 12 (3, 0) 13 (3, 3.5) 2 , 2.5) 14 (2 √ 15 (16, 8) 16 (10, 2) 17 (15, 0) 18 (3, 15) 19 (3, 2) 20 (4, 8) 21 D 2 r x-axis 22 D 3 r y-axis 23 r x-axis D _1 2 24 r x-axis D _1 4 25 a A(0, 0), B(12, 0), C(15, 6), D(3, 6) b Slope AB 0; slope BC 2; slope CD 0; slope DA 2 c Midpoint M (2.5, 1) and midpoint M (7.5, 3). Yes, M is the image result of D 3 operating on point M. Midpoints are preserved under dilation. (pages 315–316) 1 3 : 4 and 3 : 4 2 5 cm 3 4:9 4 11 in. 5 AD 6, DC 8 6 18 7 15 8 4:7 9 AC 23, PR 8 10 RQ 6, BC 12.2, AC 20.6 11 13 19 12 _ or 4.75 4 13 22 14 12 15 a 4 : 7 b 16 and 20 48 4 c 84 and 48 d __ 7 84 16 a BQ 8, QR 5, PR 8 b 36 12-7 Using Similar Triangles to Prove Proportions or to Prove a Product (pages 319–321) 1 128 2 40 3 108 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −− −− −− −− 4 a 1. ED AC and AB CB 2. EDA and ABC are right angles. 3. EDA ABC 12-7 Using Similar Triangles to Prove Proportions or to Prove a Product 75 5 6 7 8 76 4. A A 5. ADE ABC (AA) AC _ AB _ b AE AD c 32 a Each smaller triangle is similar to the larger triangle so they are similar to each other. b Corresponding sides of similar triangles are in proportion. c Product of means product of extremes. d BD 6 1. MAH is a right angle. −− −−− 2. UT AH 3. UTH is a right angle. 4. MAH UTH 5. H H 6. MAH UTH (AA) MA AH (Corresponding 7. _ _ UT TH sides of similar triangles are in proportion.) −− −− 1. AB DE 2. CAB CED (Alternate interior angles are congruent.) 3. ACB ECD (Vertical angles are congruent.) 4. ACB ECD (AA) AB ED _ _ 5. AC EC AC AB _ _ (Corresponding 6. ED EC sides of similar triangles are in proportion.) 1. EBA CDA 2. A A 3. EBA CDA 4. BCF DEF −− −− 5. AC AE 6. DEC BCE (Isosceles triangle theorem) 7. DEC DEF (Subtraction BCE BCF postulate) 8. FEC FCE 9. CBD EDB 10. CBD EDB (AA) BC BE 11. _ _ DE DC Chapter 12: Ratios, Proportion, and Similiarity 12. BC DC BE DE 9 1. EBC CDE 2. BFC DFE 3. DEF BCF FB _ FD 4. _ FE FC FC FB _ _ 5. FE FD 10 1. 2. 3. 4. 5. 6. 7. 11 1. 2. 3. 4. 5. 6. 7. 12 1. 2. 3. 4. 5. 6. 7. (The product of the means equals the product of the extremes.) (Vertical angles are congruent.) (AA) (Corresponding sides of similar triangles are in proportion.) −− −− AE BC ADC and EDC are right angles. ADC EDD DAC DEC DEC DAC (AA) AC EC _ _ ED AD EC AD (The product of the AC ED means equals the product of the extremes.) −−− −−− Isosceles triangle WXZ with WX WZ X Z YTW YRW XTY ZRY (Supplements of congruent angles are congruent.) XTY ZRY (AA) YR YT _ _ XY ZY XY YT _ _ (Corresponding sides YR ZY of similar triangles are in proportion.) Rectangle DEFG DGB and EFC are right angles. DGB EFC Isosceles triangle ADE BDG CEF (Supplements of congruent angles are congruent.) BDG CEF (AA) BG DG _ _ (Corresponding sides EF CF of similar triangles are in proportion.) 6 8 2 240 cm 32 in. 5 , 2x 20 √ 5 x 10 √ 24 5 19.2 in. Perimeter is 56. Area is 192. 3 10 3 √3 2 2 BD a b 2 2 2 BD d 2 c 2 2 a b2 d2 c 2 2 a d2 b2 c 2 2 2 25 AB BC AC 2 2 2 AC AD CD 2 2 2 2 AB BC AD CD 12 13 14 15 16 17 18 19 20 21 22 23 24 12-8 Proportions in a Right Triangle (pages 323–324) 1 (1) 10 21 2 (3) √ 3 (3) 4 77 4 (1) √ 5 a False b True c False d False e True f True g True h False 15 6 2 √ 7 5 8 6 55 9 √ 5 , y 4 √5, z 4 10 x 2 √ 5 , y 4 √5, z 8 11 x 2 √ √ 12 x 3 6 , y 6 √3, z 6 3 , y 8 √3, z 12 13 x 4 √ 6 , z 40 √ 6 14 x 20, y 8 √ 15 x 9, y 6.75, z 18.75 5 , y 10 16 x 4 √ 10 , y 3 √6, z 3 √ 15 17 x 3 √ √ √ 18 x 12, y 4 5 , z 6 5 3 19 x 9, y 6, z 6 √ 20 x 25, y 17.5, z 29.2 21 x 2.4, y 3.2, z 1.8 2 22 a x 21 b (10) x(x 21) c 4 23 13 26 1, LM 4 √ 26 24 a LR 2 √ b 9 and 10 Special Triangles 12-9 Pythagorean Theorem and Special Triangles (pages 327–329) 1 (2) 10 √2 2 (3) 25 feet 13 3 (1) √ 4 (1) 50 miles 41 5 (4) √ 6 (1) 24 feet 7 (2) 13 8 (4) {10, 24, 26} 9 14 10 b, c, d, e, f, g 2 11 a 3 √ d 9 b √ 3 5 e _ 12 c √ 5 f √ 14 (pages 332–334) 2 ft 1 (1) 6 √2 ft and 6 √ √ 2 (3) 1 : 3 : 2 3 (4) 6 √2 4 (1) 1 5 (2) 6 √3 6 (1) 30 7 (3) 3.75 8 (2) 2 √6 9 (3) 60 10 a BC 7, AC 7 √3 3 b BC 2.5, AC 2.5 √ √ c BC 3 3 , AC 9 d AC 9 √3, AB 18 3 e AC 12, AB 8 √ f BC 7, AB 14 3 , AC 8.25 g BC 2.75 √ h BC 4 √3, AB 8 √3 2 11 a AC 4, AB 4 √ 2 b BC 10, AB 10 √ c AC 8, BC 8 d AC 1.5, BC 1.5 e AC 6 √2, BC 6 √2 f BC 3 √2, AB 6 g BC 10 √2, AB 20 2 , BC 7.5 √2 h AC 7.5 √ √ i BC 4 3 , AB 4 √6 j AC 4 √6, BC 4 √6 12-9 Pythagorean Theorem and Special Triangles 77 12 13 14 15 16 17 18 19 20 21 7 √ 2 4 3 5 √ 3 , y 12, z 6 √ 2 x 6 √ a 6 by 6 √ 3 b 12 12 √3 3 c 36 √ 2 9 √ 46 3 , AC 24, DC 8, DB 16 AB 16 √ 2.28 a 8 b 2 12-10 Perimeters, Areas, and Volumes of Similar Figures (Polygons and Solids) (pages 336–337) 1 (1)1 : 3 2 (3) 3 : 5 3 (1) 28 4 a 9:1 b d 25 : 1 e 5 a 2:3 b d 2:5 e 6 1:9 7 343 8 13.5 9 25 : 1 2 10 539 ft 11 1 : 25 12 27 : 64 13 48 3 14 27 ft 15 27.7 in. 16 506.25 kg 17 a 1 : 9 18 a Yes b 2:3 16 : 81 2:3 6:5 9 : 11 c 81 : 4 c 1:7 b 1:3 c 4:9 d 8 : 27 312 _ 4 _ e Ratio of surface areas: 702 9 8 360 _ _ Ratio of volumes: 27 1,215 19 72 20 a 2.4 b 27 : 125 78 Chapter 12: Ratios, Proportion, and Similiarity Chapter Review (pages 338–341) 1 (1) 4 : 25 2 (4) 5 : 14 3 (3) 15 a 4 (2) x _ a 5 (3) √3 6 (4) 64 pounds an 7 a 6 b 6 c _ m 8 a 5:6 b 2:9 c n:m d c : (a b) e p : (m n) f (m n) : (a b) g (n b) : (a m) 10 9 a 6 b 8a c _ 3 10 30 11 11.25 12 6 : 35 13 6 14 6 15 14 ac bc 16 _ a 8 5 20 8 _ 17 _ , EC _, AC _ 3 8x 3 x4 18 a 3 b Expansion 19 54 20 102 3 21 Yes, the ratio of similitude is _. 2 1. 22 Yes, the ratio of similitude is _ 2 6 _ 23 mQ 125, PQ , QR 2 5 24 2 25 13 13 , BC 6 √ 13 26 BD 12, AB 4 √ 27 20, 21, 29 b 50 28 a 5 √2 29 34 3 , y 80 30 x 40 √ 3, y 8 31 x 8 √ 3, y 9 32 x 6 √ 2 33 √ 34 126 2 35 75 ft 8 36 _ 27 37 80 in. 3 38 81 39 45 40 18 2 3 in. 41 72 √ 42 Area 60, perimeter 24 10 √2 Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −− −− 43 1. TS QP 2. QRP TSR (Corresponding angles are congruent.) 3. PQR STR 4. R R 5. PQR STR (AAA AAA) −− −− −− −− 44 1. AE BC, BD AC 2. FEB and BDC are right angles. 3. FEB BDC 4. FBE FBE 5. BFE BCD (AA) −− −−− 45 1. AB CD 2. ABE DCE (Alternate interior angles are congruent.) 3. BAE CDE 4. AEB CED 5. a ABE DCE EC BE _ 6. _ ED AE BE _ AE 7. b _ ED EC 8. c BE ED EC AE (Vertical angles are congruent.) (AAA AAA) (Corresponding sides of similar triangles are in proportion.) (The product of means equals the product of extremes.) −− −− BC AF ECA and FDA are right angles. ECA FDA A A ADF ACB BA = _ AF 6. _ BC DF 7. BA DF (The product of AF DC means equals the product of extremes.) 46 1. 2. 3. 4. 5. CHAPTER Geometry of the Circle 13-1 Arcs and Angles (pages 345–346) 1 a 36 b 51 d 180 e 2r 2 a BC, CD, AD, BD, AB , ADC b ABC , BDA , ADB c ABD c 90 13 75, AS 105, AR 75 105, BS RB a 155 b 25 c 155 d 335 a 296 b 244 c 296 d 244 a 73 b 132 c 155 d 180 e 155 f 228 g 253 h 205 i 287 7 a 20 b 60 c 120 d 110 e 7 f 60 g 70 h 120 i 110 j 230 k 240 l 290 3 4 5 6 13-1 Arcs and Angles 79 13-2 Arcs and Chords (pages 350–351) m 1 a 16 b 7.5 c _ 2 6 e 5 √ 2 d 3 √ 2 a True b True c False 3 109 4 a 90 b 72 c 60 d 36 5 a 24 b 22.5 c 20 d 18 e 15 2 c 5 6 a 10 b 5 √ 2 c 12 d 6 √2 7 a 6 b 6 √ 8 6 in. 9 17 10 a 15 b 7.5 11 6. 2 12 8 √ 2 13 x √ 14 5 15 25 16 a 30 b 60 c 60 d 60 e 60 f 300 g 150 17 a 30 b 60 c 10 3 d 5 e 5 √ Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −−− −−− 18 1. Radii OK and OM 2. AOL DOL (Central angles are congruent.) −− −−− −− −−− 3. LA OK, LD OM 4. LAO and LDO are right angles. 5. LAO and LDO are right triangles. 6. LAO LDO −− −− 7. LO LO 8. LAO LDO (Leg-angle) 19 1. ABD CBD −−− −− (All radii of the 2. OA OB same circle are congruent.) 3. BAO ABD −−− −− 4. OC OB 5. BCO CBD 80 Chapter 13: Geometry of the Circle 6. 7. 8. 20 1. 2. 3. 4. AOB COB AOD COD CD AD (Congruent inscribed angles intercept congruent arcs.) −−− −− −−− OA, OB, and OC are radii of circle O. BAC BCA BC AB (Congruent inscribed angles intercept congruent arcs.) AOB COB (Central angles are equal to their arcs.) 13-3 Inscribed Angles and Their Measure (pages 354–357) 1 a 25 b 55 c 125 d 60.4 e 4 2 a 52 b 124 c 15 d 180 e (16x) 3 mx 70, my 35; 70 35 105 4 a 50 b 130 c 56 d 25 e 62 f 25 5 53 6 70 7 80 8 35 9 140 10 mC 75, mM 82 11 85.5 12 m1 120, m2 45, m3 75, m4 60 13 m1 57, m2 33, m3 90, m4 114 14 m1 74, m2 74, m3 37, m4 106 15 a 140 b 80 c 40 d 70 e 70 16 m1 57, m2 48, m3 57, m4 48 17 a 80 b 188 c 46 d 94 18 a 64 b 108 c 140 d 70 e 24 f 54 g 94 19 a 86 b 102 c 88 d 56 e 116 20 a 44 b 88 c 44 d 44 e 92 f 88 21 a 60 b 60 c 60 d 30 e 60 f 120 g 30 h 60 i 120 j 60 k 90 l 30 22 Label arcs with degree measures that sum to 360, as with x, 2x, 3x, 4x. Then find the values of the inscribed angles. 23 Parallel lines cut congruent arcs x and y. Therefore, 2x 2y 360 x y 180 Each inscribed angle is equal to one-half the intercepted arc: 90. 13-4 Tangents and Secants (pages 363–365) 1 a b c d 2 a 12 b 3 3 a 3.25 b 1 4 a disjoint b externally tangent c intersecting twice d internally tangent e disjoint internally f concentric 5 a Sketch of two circles externally disjoint b Sketch of two intersecting circles, not tangent c Sketch of externally tangent circles d Sketch of internally tangent circles 6 a 1 b 3 c 2 d 4 7 a 160 b 130 c 90 d 40 e 180 x 8 a 60 b 45 c 35 d 51 e 67.5 180 n n b 90 2n c _ 9 a _ 2 2 180 m n d 45 n e __ 2 10 16 11 80 12 48 13 234 14 a 18 b 72 c 29.25 15 a 11, 17, 18 b no sides equal 16 a 12, 16, 20 b Satisfies Pythagorean theorem: 2 2 2 12 16 20 17 a 10, 15, 15 b two sides are equal 18 BE 4, EC 5, CF 5, AF 6, AC 11, AB 10 19 AB 20, CB 20, RB 10 20 AB 24, DR 14, DB 32 21 OB 26, DB 36, RB 16 91 , CB 2 √ 91 22 OB 20, AB 2 √ 23 OB 22, CB 8 √6, AB 8 √6 24 AB 20, CB 20, OC 15, OB 25 25 Tangent segments from the same point are congruent. Base angles are equal, each measuring 60. Therefore, APB is equilateral. Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −− 26 1. Let E be the point of intersection of AB −−− and CD. −− −− 2. EB and ED are tangent segments to circle P. −− −− (Two tangent segments 3. EB ED drawn from an external point are congruent.) −− −− 4. EA and EC are tangent segments to circle O. −− −− 5. EA EC −− −− (Addition postulate) 6. EA EB −− −−− EC CD −− −−− or AB CD 13-4 Tangents and Secants 81 −−− −−− Common external tangents, AG and CM −−− −−− Radii OA and OC OAG and OCM are right angles. OAG and OCM are right triangles. −−− −−− OA OC −−− −−− Tangents OM and OG of circle P −−− −−− OM OG (Two tangent segments drawn from an external point are congruent.) 8. OAG OCM (HL HL) −−− −−− (CPCTC) 9. AG CM 28 1. Circles A and B are congruent with . tangent FG −− −− 2. AF is the radius of A. BG is the radius of B. −− −− 3. AF BG 4. AFC and BGC are right angles. 5. FCA GCB 6. FCA GCB (AAS AAS) −− −− 7. AC BC 1 (CD BE ). 2(mA) 29 mA _ 2 mCOD mBOE. But mBOE mA. 2(mA) mCOD mA. Therefore, 3(mA) mCOD. −−− −− 30 1. AB is tangent to O at E; CD is tangent to O at F. −− 2. OE OF −−− −−− −− −−− 3. OA OC; OB OD −− −− 4. OE AB 5. OEA and OEB are right angles. −− −−− 6. OF CD 7. OFC and OFD are right angles. 8. OEA OFC (HL HL) OEB OFD −− −− (Addition 9. AE EB −− −− CF FD postulate) −− −−− or AB CD 27 1. 2. 3. 4. 5. 6. 7. 13-5 Angles Formed by Tangents, Chords, and Secants (pages 368–371) 1 (2) 96 2 (1) 24 3 (2) 144 82 Chapter 13: Geometry of the Circle (4) 173 (1) 38 (1) 90 a 40 b 62 c 225 d 45 e 75 f 100 g 80 h 30 8 a 79 b 38 c x 50, y 60, z 120 9 a 66 b 80 c 73 d 130 e 33 f 73 10 a 90 b 120 c 90 d 60 e 30 f 60 g 30 h 90 11 a 120 b 40 c 80 d 40 e 140 f 100 CD because in a regular hexagon, 12 BC congruent chords subtend congruent arcs. Chords and a tangent that intercept con−− BD gruent arcs are parallel. Therefore, PC Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) −− −− 13 1. FB EC 2. BEC EBF (Alternate interior angles are congruent.) 3. EHJ BHG 4. EHJ BHG BH _ HE (Corresponding sides 5. _ HJ HG of similar triangles are in proportion.) 6. BH JH (The product of means HE HG equals the product of extremes.) −− −−− 14 1. BA DA −−− 2. Diameter DC 3. BAD and DBC are right angles. −− 4. AB is tangent to O at B. 1 mBD 5. mDBA _ 2 1 mBD 6. mDCB _ 2 7. mDBA mDCB 8. DBA DCB 9. DBA DCB 4 5 6 7 DC BD _ 10. _ DB DA BD _ DA 11. _ BD DC 2 12. (BD) DC DA (Corresponding sides of similar triangles are in proportion.) 1 (The product of means equals the product of extremes.) 13-6 Measures of Tangent Segments, Chords and Secant Segments (pages 374–376) 1 (4) 20 2 (2) 46 3 (1) 17 4 (3) 24 5 (2) 10 6 (1) 3 7 (3) 25 cz 8 (3) _ a 9 (3) 17 10 (3) 33 11 (1) 9 12 24 13 19 14 40 15 44 16 12 17 16 18 NQ 6, QP 16 19 x 4, GH 16 −−− −−− 2 2 20 AOB WZ . Radius 12.5; 10 (7.5) 2 (12.5) 2 3 4 13-7 Circle Proofs (pages 381–382) Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 5 ; 1. A is the midpoint of CD −−− AOB is a diameter. AD 2. CA 1 mCA 1 mAD _ 3. _ 2 2 1 mAD 4. mABC _ 2 1 mCA mABC _ 2 5. mABC mABD 6. ACB and ADB are right angles. 7. mACB mADB −− −− 8. AB AB 9. ACB ADB (AAS AAS) −− −− (CPCTC) 10. CB DB −−− −− 1. AD CB CB 2. AD 1 mAD 3. mABD _ 2 1 mCB mCDB _ 2 4. mABD mCDB −− −− 5. DB DB 1 mDB 6. mDAB _ 2 1 mDB mDCB _ 2 7. mDAB mDCB 8. ADB CBD (AAS AAS) −− −−− (CPCTC) 9. AB CD −−−− −−−− 1. Circle O with diameters MOT and AOH 2. MAT and HTA are right angles. −−−− −−−− 3. MOT AOH −− −− 4. AT AT 5. MAT HTA (HL HL) −−− −− (CPCTC) 6. MA HT 1. Circle O, tangents PR , PV ; PV 2. PR OR OV 3. ORP and OVP are right angles. 4. mORP mOVP −−− −−− 5. OR OV −− −− 6. PR PV 7. RPO VPO (SAS SAS) 8. RPO VPO (CPCTC) 1. Circle O with T as the midpoint of CH 2. mCT mTH 180 3. mCTH 90 4. mCT 5. mCOT mCT 6. COT and CTH are right angles. 7. mCOT mCTH 13-7 Circle Proofs 83 1 mTH 8. mHCT _ 2 1 mCT mCHT _ 2 9. mHCT mCHT 10. CTO HTC CT TO 11. _ _ (Corresponding TH CH sides of similar triangles are in proportion.) −−−− GI 6 1. Circle O with diameter DOG; DR 1 _ ) TG 2. mTEG m(DR 2 1 m(GI TG ) 3. mTRI _ 2 GI TGI 4. TG 1 5. mTEG _mTGI 2 1 mTGI 6. mTRI _ 2 7. mTEG mTRI 8. mRTI mRTI 9. TEX TRI (AA) EX TX 10. _ _ TI RI 11. TX RI EX TI (The product of means equals the product of extremes.) 7 1. CT CH −−− −− 2. HA TD 3. CTD CHA; TDH HAT 4. TDH is supplementary to TDC; HAT is supplementary to HAC. 5. TDC HAC 6. TDC HAC (AAS AAS) −− −−− (CPCTC) 7. CT CH 8. HA TD (Contradiction) −− −−− 8 1. BC QR −− −− 2. BA QP RQ (Congruent chords 3. BC subtend congruent arcs.) 4. BA QP 5. A D (Inscribed angles intercepting congruent arcs are congruent.) 6. B Q 7. RQP CBA (SAS SAS) 8. AC PR 84 Chapter 13: Geometry of the Circle 9 a 1. BEA CED 2. BDC CAB (Vertical angles are congruent.) (Congruent inscribed angles) (AA) 3. BEA CED 16 b (i) 4 : 5 (ii) _ 25 −− −−− 10 1. BA PGD 2. DBA BDP 1 mBGD 3. mPBD _ 2 1 mBGD 4. mBAD _ 2 5. mPBD mDAB 6. PBD DAB 7. PBD DAB BD PD _ (Corresponding 8. _ BD BA sides of similar triangles are in proportion.) −− 11 1. AB is the diameter. 2. Line l is tangent to circle O. 3. ABC and AEB are right angles. 4. ABC AEB 5. CAB BAE 6. ABC AEB (AA) AC AB (Corresponding 7. _ _ AB AE sides of similar triangles are in proportion.) −− 12 1. AC is the diameter. −− 2. BD intersects the diameter at point E. −− −− 3. AB BE 4. BAE BEA 5. BEA CED 6. BAE CED 7. ABE and DCE are right angles. 8. ABE DCE 9. ABE ECD (AA) AC _ ED _ 10. AB EC 11. AC EC (The product of ED AB means equals the product of extremes.) 13 1. DCG CBD 1 m(ABC DF ) 2. mE _ 2 1 m(CB ) _ 2 1 m(CB ) 3. mCDB _ 2 4. mE mCDB 5. CBD FCE EC EF _ 6. _ DB CD 14 1. AZT MZH 2. HMT TAH 3. MZH AZT MZ AZ 4. _ _ ZH ZT 15 a 1. BAC BDC 2. ABD ACD 3. ACD ABE b BE 6 (AA) 7 (Corresponding sides of similar triangles are in proportion.) (AA) 8 (Corresponding sides of similar triangles are in proportion.) (Congruent inscribed angles) 10 (AA) 11 9 12 13-8 Circles in the Coordinate Plane 13 (pages 386–387) 2 2 1 (2) (x 4) (y 3) 36 2 (4) (13, 1) 2 2 3 a (x 1) (y 4) 25 2 2 b (x 3) (y 2) 49 2 2 c (x 4) (y 1) 121 2 2 d (x 2) (y 5) 64 2 2 e (x 2) y 121 2 2 f x y 144 2 2 4 a x y 49 2 2 b (x 4) y 25 2 2 c x y 41 2 2 d (x 2) y 36 2 2 e (x 3) (y 2) 9 2 2 f (x 1) (y 3) 25 5 For parts a–d, check students’ graphs. The center and radius of each circle are listed below. a C(2, 4), r 3 b C(3, 4), r 5 c C(0, 2), r 4 d C(6, 0), r 2 6 a C(0, 0), r 9 b C(0, 0), r 11 c C(0, 0), r 20 d C(0, 0), r 1 e C(0, 0), r √2 Note: Students’ answers may vary for the two other points on the circle. a C(5, 3), r 8; (5, 11), (5, 5) b C(4, 0), r 10; (6, 0), (14, 0) c C(2, 5), r 4; (2, 5), (6, 5) d C(3, 7), r 2 √3; (3, 7 2 √3), (3, 7 2 √3) 2 2 a x y 25 2 2 b x y 4 2 2 c x y 34 2 2 d x y 37 2 2 e x y 68 2 2 a (x 2) (y 3) 2 2 2 b (x 3) (y 7) 80 2 2 a x (y 0.5) 12.25 2 2 b (x 3) (y 3) 5 a Circumference 8, area 16 b Circumference 12, area 36 13 , area 13 c Circumference 2 √ a (3, 0) 2 2 b (x 3) (y 1) 1 c a C(1, 3), r 2 √2 b Area 8, circumference 4 √2 13-9 Tangents, Secants, and the Circle in the Coordinate Plane (pages 392–393) 1 (1) 0 2 a (3, 11), (3, 1) b (3) x 1 3 y0 4 y 0.5x 2.5 5 yx4 6 y 3 7 y4 8 y 2x 10 9 a (4, 3), (3, 4) 10 a (0, 4), (4, 0) 11 a (2, 3), (2, 5) 12 a (0, 3) 13 a (0, 2), (2, 0) 14 a (5, 5) 15 a (1, 4), (1, 4) b b b b b b b secant secant secant tangent secant tangent secant 13-9 Tangents, Secants, and the Circle in the Coordinate Plane 85 16 17 18 19 20 21 22 a a a a a a a b 23 a b c 24 a d (2 √ 2 , 2 √ 2 ), (2 √ 2 , 2 √ 2 ) b secant (10, 0) and (10, 0) b secant (7, 7) and (7, 7) b secant (6, 0) b tangent (8, 6) and (6, 4) b secant (0, 5) and (4, 3) b secant 2 Sub-in the given points in (x 3) 2 (y 2) 25. −−− Midpoint of ME is (3.5, 2.5); 0.5 . distance √ y x 8 yx8 (0, 8) y 3x 10 b x 10 c P(10, 20) √ √ √ PA 160 4 10 , PB 1000 10 , PM 20 10 √ 2 PA PB PM 10 10 √ 10 20 2 4 √ 400 400 Chapter Review (page 393–397) 1 (1) All chords in a circle are congruent. 2 2 2 (2) (x 4) (y 2) 9 3 (2) 1 4 (3) 8 5 (3) 6 and 15 6 (2) 12 7 (1) 12 8 (2) 6 9 (2) 68 10 (3) 122 11 (1) 43 12 (3) 86 13 (2) 2 : 1 14 (1) 41 15 (3) 10 inches 16 (3) 125 17 (1) 40 18 (4) 100 19 (2) 230 20 16 21 90 22 104 23 20 24 48 25 12 26 a 40 b 40 c 40 d 40 e 110 f 110 g 35 86 Chapter 13: Geometry of the Circle 40 b 70 c 20 45 e 25 f 65 30 b 45 c 15 135 e 90 C(0, 0), r 20 Answers may vary. (0, 20), (0, 20) C(7, 11), r 9 Answers may vary. (7, 20), (7, 2) C(3, 13), r 6 Answers may vary. (3, 19), (3, 7) 5 C(3, 5), r 2 √ 5 ), Answers may vary. (3, 5 2 √ √ (3, 5 2 5 ) 2 2 33 (x 3) (y 2) 16, C(3, 2), r 4 2 2 34 a (x 4) (y 2) 49 2 2 b (x 5) y 2 2 2 c x y 169 2 2 d C(3, 0), r 5, (x 3) y 25 2 10 , (x 4) (y 4) 2 10 e C(4, 4) r √ 2 2 2 35 6 6 (6 √2) 1 x 10 36 y _ 3 37 a (4, 3) and (4, 3) b secant 38 a (0, 3) and (3, 0) b secant 39 a (8, 2) and (2, 8) b secant 40 a (0, 5) and (3, 4) b secant Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 41 1. AOB COB −− −− 2. AB CB CB 3. AB −−− 42 1. DA is the diameter of circle O. −−− −−− −−− 2. Radii OA, OC, and OD −−− −−− −−− (Radii are equal.) 3. OA OC OD −−− −−− 4. AD CD 5. AOD COD (SSS SSS) 6. AD CD 7. ABD CBD (Congruent arcs are intercepted by congruent inscribed angles.) 27 a d 28 a d 29 a b 30 a b 31 a b 32 a b 43 1. 2. 3. 4. 5. 6. 7. −−− −−− −−− −− MA OK, MD OL OAM and ODM are right angles. −−− −−− AM DM −−− −−− OM OM OAM ODM (HL HL) AOM LOM (CPCTC) MK ML (Congruent angles have congruent arcs.) −− −− −− 44 1. OE AB, OD AC 2. AEO and ADO are right angles. −− −− 3. AB AC −− −− 4. OE bisects AB. −−− −− 5. OD bisects AC. −− −−− 6. AE AD −−− −−− 7. OA OA 8. AEO ADO (HL HL) 45 Use equal arcs are subtended by equal chords. CHAPTER Locus and Constructions 14-1 Basic Constructions (page 404) Note: For exercises 1–20, check students’ constructions; procedures may vary. 1 ab Use construction of congruent angles procedure. 2 Use construction of a perpendicular to a line through a given point on the line procedure. 3 ac Use construction of an angle bisector procedure. 4 Use construction of a line perpendicular to a line through a given point not on the line procedure. 5 Use construction of a perpendicular bisector of a line segment procedure. 6 Use construction of a parallel line through a given point not on the line procedure. 7 Use construction of the median of a triangle procedure. 8 Use construction of line perpendicular to a line through a given point not on the line procedure. 14 9 Use construction of a perpendicular bisector procedure. 10 Use construction of an angle bisector procedure. 11 a Use construction of a perpendicular line procedure. b Use construction of a perpendicular line procedure and then construct an angle bisector. 12 a Use construction of an equilateral triangle procedure. b Using angle constructed in a, construct an angle bisector. c Construct angle bisector of b. 13 Using the vertex of the angle as the center of a circle and one of the legs as its radius, construct a circle. Construct the diameter by extending the radius. 14 Using the vertex of the angle as the center of a circle and one of the legs as its radius, construct a circle. Construct the diameter by extending the radius. Construct a perpendicular through the center of the circle. 14-1 Basic Constructions 87 15 a Use constructing a congruent angle procedure. b Use constructing a congruent angle procedure. c Construct a perpendicular bisector of a segment to form a 90 angle. Then use constructing a congruent angle procedure to construct the sum. d Use constructing an angle bisector procedure. e Use constructing an angle bisector procedure for A. Then use constructing a congruent angle procedure to construct the sum. 16 Use construction of an equilateral triangle procedure. 17 Use the construction of an equilateral triangle procedure using the length of the longer segment to construct the leg of the isosceles triangle. 18 Using the compass, measure the radius and construct a circle passing through the point on the tangent line. 19 Use construction of a line tangent to a given circle through a given point outside the circle procedure. 20 Use procedures for constructing parallel lines and perpendicular lines. 14-2 Concurrent Lines and Points of Concurrency (pages 407–408) 1 (4) obtuse 2 (3) at one of the vertices of the triangle 3 Check students’ constructions. 4 All at the same point 5 10 2 6 6_ 3 7 36 8 16.5 9 4.5 10 SP 4, SC 6 11 PB 15, PR 30 12 PA 15, QA 45 1 13 4_ 3 14 9 88 Chapter 14: Locus and Constructions 15 16 17 18 19 20 21 22 23 24 25 3 √3 QE 4, DE 6 a, b, c: all interior a, b, c: all interior a interior b on a side c exterior a interior b at a vertex c exterior √ √ Incenter: 3 , circumcenter: 2 3 3 , circumcenter: 4 √3 Incenter: 2 √ √ 3 , circumcenter: 6 √3 Incenter: 3 Incenter: 6, circumcenter: 12 Since AG GC, the base is the same and the altitude is the same. Therefore, the area is the same. 14-4 Six Fundamental Loci and the Coordinate Plane (pages 413–414) 1 (3) One concentric circle of radius 11 inches 2 (2) one line 3 (2) a point 4 (1) a circle of radius 4 with center at P 5 (4) a circle Note: For exercises 6–17 check students’ sketches. 6 Circle with given point as center and radius 3 7 Two parallel lines, one on each side of the given line 8 One line parallel to and midway between the given parallel lines 9 The perpendicular bisector of the segment joining R and S 10 The perpendicular bisector of the segment AB 11 A circle with radius 2.5 inches and the given point as the center 12 The line that is the bisector of ABC 13 Two lines each the bisector of the vertical angles 14 Two concentric circles with radii 5 and 9 15 One concentric circle midway between the given circles 16 All the points in the interior of a circle with the given point as the center and with a radius 2 inches 17 A circle with the given point as the center and a radius of 3 inches and all the points in the exterior of that circle 18 Two lines, one on each side, parallel to the given line and at a distance r from the given line 19 One line parallel to the two parallel lines and midway between them 20 The diameter of the circle that is the perpendicular bisector of the given chord 21 The diameter of the circle that is the perpendicular bisector of the given chord 22 A ray that is the angle bisector 23 A line perpendicular to the point on the given line 24 The perpendicular bisector of the chord that joins the two given points Locus in the Coordinate Plane (Pages 418–419) 1 (4) y 2 or y 2 2 (1) x 2 3 (2) x 5 4 (2) y 1 2 2 5 (4) (x 1) (y 3) 25 6 (3) y 3 and x 0 7 (4) y x 1 and y x 5 8 y1 9 y 2, y 10 10 y 4 11 y x 3 and y x 3 12 y x 3 13 x 1 and y 0 2 2 14 a (x 1) (y 4) 16 2 2 b x y 36 2 2 c x (y 1) 9 2 2 d (x 1) y 1 2 2 e x (y 2) 6.25 2 2 f (x 1) (y 3) 30.25 15 The showers could be placed anywhere on the perpendicular bisector of the line segment joining the diving boards, which is 35 ft from each of the diving boards. 14-5 Compound Locus and the Coordinate Plane (pages 421–422) 1 (3) a pair of points 2 (4) the empty set 3 (1) 0 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 (3) a pair of points (3) 2 (2) 1 a 4 b 2 c 0 Find the point of concurrency (circumcenter) of the perpendicular bisectors of the sides of the triangle formed. 3 2 Parallel lines 2 inches from the given line, one on each side a Circle with radius 3, center R. 1 point b Circle with radius 6, center R. 2 points c Circle with radius 7, center R. 3 points d Circle with radius 9, center R. 3 points For all parts, check students’ sketches. a Two intersecting circles, but not tangent. 2 points b Two tangent circles. 1 point c Two disjoint circles. 0 points Sketch circle and two lines; 4 points. Sketch a line and a circle; 2 points. Sketch two lines bisecting vertical angles, and two parallel lines; 4 points. Sketch two concentric circles and two lines bisecting vertical angles; 8 points. The intersection of the line parallel to m and k and midway between them, and a circle with A as the center and d as the radius. 1 f (0 points) Case I d _ 2 1 f (1 point) Case II d_ 2 1 Case III d _ f (2 points) 2 The intersection of the perpendicular bisector of segment AB and a circle about center A with radius x. 1 d (0 points) Case I Radius x _ 2 1 d (1 point) Case II Radius x _ 2 1 d (2 points) Case III Radius x _ 2 Compound Loci and the Coordinate Plane (pages 423–424) 1 (4) 4 2 (3) 2 3 (4) 4 4 a 2 b 1 c 0 14-5 Compound Locus and the Coorinate Plane 89 5 Two horizontal lines y 11 and y 1, and vertical line x 4. Locus: 2 points. 6 Locus 2 points: (0, 8) and (8, 0) 7 Locus 2 points: (2, 7) and (2, 1) 8 Locus 2 points: (3, 1) and (2, 2) 9 a y2 b x 4 c (4, 2) 2 2 e one d (x 2) (y 2) 16 10 a Circle with radius, d b Two lines: x 1 and x 1 c (i) 1 point (ii) 3 points (iii) 4 points 14-6 Locus of Points Equidistant From a Point and a Line (page 427) Note: For exercises 1–10, check students’ sketches. 1 b (0, 2), (2, 3), (2, 3) 2 x 2 c y _ 4 2 b (2, 1), (2, 1), (6, 1) x2 _ x _ 1 c y _ 2 2 8 3 b (2, 0), (2, 2), (6, 2) x2 _ 1x _ 1 c y _ 8 2 2 4 b (3, 0.5), (0, 1), (6, 1) 2 x x1 c y _ 6 5 b (1, 0), (3, 4), (3, 4) y2 _ c x 1 8 6 b (3, 1.5), (6, 0), (0, 0) x2 x c y _ 6 7 b (2, 2), (6, 4), (2, 4) 5 x2 _ 1x _ c y _ 2 2 8 8 b (1, 4), (4, 10), (4, 2) y 2 2y 7 c x _ _ _ 3 3 12 9 b (0, 0), (2, 4), (2, 4) y2 c x _ 8 10 b (3, 1.5), (1, 2), (6, 2) x2 x 2 c y _ 6 90 Chapter 14: Locus and Constructions 14-7 Solving Other LinearQuadratic and QuadraticQuadratic Systems (page 430) Note: For exercises 1–20, check students’ sketches. 1 (2, 0), (2, 0) 2 , 0), (2 √2, 0) 2 (2 √ 3 (0, 5), (4, 3) 4 (3, 2) 5 (2, 3), (2, 5) 6 (2, 1), (1, 2) 7 (1, 1), (1, 1) 8 (5, 27), (1, 5) 5 9 _, 6 , (2, 5) 3 10 (4, 0), (4, 0) 11 ( √2, 0), ( √2, 0) 12 (3, 2), (6, 1) 2 , √2), (2 √ 2 , √2), ( √ 2 , 2 √ 2 ), 13 (2 √ 2 , 2 √2) ( √ 14 (0.5, 1.25), (4, 3) 15 (1, 0) 16 (1, 3), (1, 3) 2 , √5), (2 √2, √5), (2 √ 2 , √5), 17 (2 √ (2 √2, √5) 18 (2, 1), (2, 1) 19 (3, 0) 20 (2, 4), (2, 0) ( ) Chapter Review (pages 430–433) 1 (2) acute triangles −− 2 (4) median to side AC 3 (2) AAS 4 (2) two circles 5 (2) X lies on the locus of points equidistant from R and S. 6 (2) y 2x 3 2 2 7 (3) (x 3) (y 4) 36 8 (1) x 3 9 (3) x 5 10 (1) (0, 0) and (0, 8) 11 (4) (0, 2) and (4, 2) 12 (3) perpendicular bisectors of the sides of the triangle 13 (3) 4 14 (3) 3 15 (3) 2 16 (4) 4 17 (1) 90 x 18 8 19 x 3, PD 6, RP 12, RD 18 20 x 2, y 4, BE 9, AD 12 Note: For exercises 21–24, check students’ sketches. 21 Two concentric circles. Locus: radius 1 and radius 7. 2 2 22 Circle, radius 3, x y 9. Two lines, x 4, x 4. Locus: 0 points 2 2 2 2 23 Circle: x y 4. Circle: x (y 4) 9. Locus: 1 point, (0, 2) 24 Two lines parallel to line m on opposite sides and 4 centimeters from m. Circle with center H and radius 2. Locus: 1 point 2 2 25 (x 4) (y 2) 1 26 y x 4 27 Two lines: y x 2 and y x 2 Note: For exercises 28–37, check students’ sketches. 28 b (2, 0.5), (5, 1), (1, 1) 2 2x _ x _ 1 c y _ 6 3 6 29 b (5, 7), (1, 3), (3, 1) y2 _ 3y _ _ c x 1 8 8 4 30 b (2, 0), (2, 2), (6, 2) 31 32 33 34 35 36 37 38 x2 _ x _ 1 c y _ 8 2 2 b (1, 0), (1, 4), (1, 4) y2 c x _ 1 8 (0, 5), (3, 4), (3, 4) (0, 5), (3, 8) (0, 3), (3, 9) 7 , 1 √ 7 ), (2 √ 7 , 1 √ 7) (2 √ (3, 10), (1, 6) 7 , 3), ( √ 7 , 3) (0, 4), (√ The intersection of the perpendicular bisector of AB, and circle with A as center 1 f (0 points) Case I x _ 2 1 f (1 point) Case II x _ 2 1 f (2 points) Case III x _ 2 39 The intersection of one circle with radius of 2.5, concentric with the given circles, and two lines parallel to the given line, one on each side at a distance x from the given line. Case I x 2.5 (4 points) Case II x 2.5 (2 points) Case III x 2.5 (0 points) 40 The intersection of circle with center P and radius d and circle with center Q and radius d. Case I x 2d (2 points) Case II x 2d (1 point) Case III x 2d (0 points) 41 The intersection of two lines parallel to line m, one on each side at a distance d from line m and the circle with center A and radius r. Case I r d (0 points) Case II r d (2 points) Case III r d (4 points) 42 Locus: A line parallel to the north side and south side of the courtyard and midway between them. Since the width of the garden is 60 ft, every point on this line will be equidistant from the north and south walls and at least 30 feet from the North Entrance. 43 Check students’ constructions of the orthocenter of an acute triangle. 44 Check students’ constructions of the perpendicular bisector of the three sides of a right triangle. 45 Check students’ constructions of a circle that passes through the three vertices of an obtuse triangle. Chapter Review 91 !33%33-%.4 Cumulative Reviews Each review has a total of 58 possible points. Use the following table, adapted from the Regents Examinations, to convert the student’s raw score to a scaled score. Raw Score 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 Scaled Score 100 99 98 97 96 93 92 91 90 89 87 86 85 84 82 80 79 78 77 76 Raw Score 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 Scaled Score 75 74 72 70 68 67 66 65 64 63 61 60 59 58 57 56 54 53 51 Raw Score 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Scaled Score 49 47 46 45 44 43 42 41 39 37 36 34 32 29 26 21 16 10 5 Chapters 1–2 (pages 434–437) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. −− 3 (3) p q 1 (4) a → r 2 (2) AB 92 4 (4) 2BC AC 5 (1) If I win a scholarship, then I will play soccer. 2 6 (3) _ 3 7 (3) The triangle may be acute. 8 (4) If two angles are not congruent, then they are not right angles. 9 (2) right 10 (1) isosceles Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 12 13 14 Score Explanation 2 AB CD EF, and an appropriate explanation is given. 1 AB CD EF, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 3, and an appropriate explanation is given. 1 3, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 55, and an appropriate explanation is given. 1 55, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 9, and an appropriate explanation is given. 1 9, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–2 93 Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 16 17 94 Score Explanation 4 a (i) scalene (ii) right triangle b (i) isosceles (ii) acute c (i) equilateral (ii) equiangular d (i) isosceles (ii) obtuse e (i) isosceles (ii) right triangle f (i) scalene (ii) obtuse 3 Answered all but two parts correctly. 2 Answered all but three parts correctly. 1 Answered two parts correctly. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 3 : 1, and an appropriate explanation is given. 3 2 AB is given instead. An appropriate method is shown, but _ BD An appropriate method is shown, but an incorrect answer is given. 1 3 : 1, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 x 77, and an appropriate method is used. 3 An appropriate method is shown, but y is given instead. 2 An appropriate method is shown, but an incorrect answer is given. 1 x 77, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 Score 6 Explanation a distributive property b associative property of addition c additive identity d commutative property of multiplication 19 20 5 Student answers all but one part correctly. 3 Student answers only two parts correctly. 1 Student answers only one part correctly. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 6 34, and an appropriate method is given. 5 An appropriate method is used, but a single arithmetical error is made. 4 An appropriate method is used, but two arithmetical errors are made. 3 An appropriate method is used, but student found m⬔1 or m⬔2. 2 An appropriate method is shown, but multiple computational mistakes are made. 1 34, but no appropriate explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 6 Explanation a {6, 7, 8, 9, 10} b {0, 2, 4, 6, 8, 10} c {1, 3, 5} d {0, 1, 2, 3, 4, 5, 7, 9} 5 One part is missing elements from the solution set. 4 Elements are missing from two solution sets. 3 Answered two parts correctly. 2 Answered all parts, but solution sets are missing elements. 1 Only one part is answered correctly. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–2 95 Chapters 1–3 (pages 438–441) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (3) 80 2 (2) 54 3 (1) Subtraction postulate 4 (4) If Cecilia is not a senior, then she does not take AP Calculus. 5 (4) 22 6 (3) If a triangle is not equilateral, then it is not isosceles. 7 (1) 20 3 1 9 (4) _ 10 (2) b c 8 (3) _ 2 2 Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 12 Score Explanation 2 70, and an appropriate explanation is given. 1 70, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 2 Explanation Affirming the conclusion does not affirm the premise. a The quadrilateral could be rhombus or any other figure. b x could be 7 or any other positive number less than 7, so that x is not greater than 8. 13 14 96 1 Student answers only one part correctly. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 m⬔r 20, and an appropriate explanation is given. 1 m⬔r 20, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 RB 2, and an appropriate explanation is given. 1 RB 2, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 16 Score Explanation 4 m⬔A 82, m⬔B 82, and m⬔C 16, and an appropriate explanation is given. 3 An appropriate method is used, but a single arithmetical error is made. 2 An appropriate method is used to find x, but the measures of the angles are not given. 1 m⬔A 82, m⬔B 82, and m⬔C 16, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 4 Explanation a ⬔ABD and ⬔DBE b ⬔ABE and ⬔EBC c ⬔DBC d ⬔ABC 17 3 Student answers all but one part correctly. 2 Student answers only two parts correctly. 1 Student answers only one part correctly. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 m⬔4 50, m⬔5 20, m⬔ABD 160, and an appropriate method is used. 3 An appropriate method is used, but a single arithmetical error is made. 2 m⬔4 50, m⬔5 20, but student did not give the measure of ⬔ABD. 1 m⬔4 50, m⬔5 20, m⬔ABD 160, but no appropriate method is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–3 97 Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 Score 6 Explanation a If the sum of the measures of two acute angles is 90, the two angles are complementary. b If two acute angles are not complementary, then their sum is not 90. c If the sum of the measures of two acute angles is not 90, then the two acute angles are not complementary. d Two acute angles are complementary if and only if the sum of their measures is 90. 19 20 5 Student answers all but one part correctly. 3 Student answers only two parts correctly. 1 Student answers only one part correctly. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 1 (2) Reflexive postulate 2 (3) Addition postulate 1 (4) Partition postulate 2 (5) Substitution postulate 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 1 (2) Definition of bisector 1 (3) Definition of midpoint 2 (4) Doubles of equals are equal. 2 (5) Substitution postulate 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–4 (pages 442–445) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (3) 220 5 (2) 120 2 (2) a median 6 (1) ⬔3 and ⬔2 are nonadjacent complementary angles. 3 (4) 120 7 (2) b 4 (3) If the diagonals of a quadrilateral 8 (3) ⬔BAR and ⬔RAI are not congruent, then the 9 (1) m⬔x m⬔y quadrilateral is not a rectangle. 10 (3) a b 4 98 Cumulative Reviews Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 12 13 14 Score Explanation 2 a True 1 Student answers only two parts correctly. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score b False c False d True Explanation 2 11, and an appropriate explanation is given. 1 11, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 68 and 112, and an appropriate explanation is given. 1 68 and 112, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 57.5, and an appropriate explanation is given. 1 57.5, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 Score Explanation 4 AC 12, AB 16, BC 16, and an appropriate explanation is given. 3 An appropriate method is used, but a single arithmetical error is made. 2 An appropriate method is used, but multiple arithmetical errors are made. 1 AC 12, AB 16, BC 16, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–4 99 16 17 Score Explanation 4 a 91, b 35, c 54, d 91, and an appropriate explanation is given. 3 An appropriate method is used, but a single arithmetical error is made. 2 An appropriate method is used, but multiple arithmetical errors are made. 1 a 91, b 35, c 54, d 91, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 m⬔PTC 50, and an appropriate explanation is given. 3 An appropriate method is used, but a single arithmetical error is made. 2 An appropriate method is used, but multiple arithmetical errors are made. 1 m⬔PTC 50, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 Score 3 −− −− −−− −− a Given ⬔A ⬔F and AB EF. AD CF and by the reflexive property of −−− −−− −−−− −−− congruence DC DC. By the addition postulate, ADC FCD. Therefore, by SAS, 䉭ABC 䉭DEF. 2 a An appropriate proof with correct conclusions is shown, but one reason is faulty and/or one statement is missing. 1 a A correct conclusion is reached and a reason is given, but no appropriate method is shown. −− −−− −−− b Given ⬔D ⬔A. C is the midpoint of AD, then AC DC. ⬔ACB ⬔DCE because vertical angles are congruent. Therefore, by ASA, 䉭ABC 䉭DEC. 3 100 Explanation 2 b An appropriate proof with correct conclusions is shown, but one reason is faulty and/or one statement is missing. 1 b A correct conclusion is reached and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 19 Score 6 20 Explanation The following proof or another appropriate proof is given. Statements Reasons 1. ⬔B ⬔D −− −−− 2. BC DC 1. Given. 2. Given. 3. ⬔BCA ⬔DCE 3. Vertical angles are congruent. 4. 䉭ACB 䉭ECD 4. ASA ASA. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but more than two reasons are faulty or more than two steps are missing or have errors. 1 䉭ACB 䉭ECD and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 6 Explanation The following proof or another appropriate proof is given. Statements −− −− 1. BD is the median to AC. −− 2. D is the midpoint of AC. −−− −−− 3. AD DC −− −− 4. AB BC −− −− 5. BD BD 6. 䉭I 䉭II Reasons 1. Given. 2. Definition of a median of an isosceles triangle. 3. Definition of a midpoint. 4. Given. 5. Reflexive Property of Congruence. 6. SSS SSS. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or three steps are missing. 2 An appropriate proof with correct conclusion is used, but more than three reasons are faulty or more than three steps are missing or have errors. 1 䉭I 䉭II and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–4 101 Chapters 1–5 (pages 446–449) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct____ answer instead of the numeral 1, 2, 3, or 4. −− −−− 6 (1) isosceles 1 (4) QB, QA, QM −−− −− 2 (1) q → p 7 (3) CD bisects AB. 3 (3) 70 8 (2) ⬔3 ⬔4 4 (2) 9 (3) 28 5 (4) ⬔ADB ⬔WDB 10 (3) 78.5 Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 12 13 14 102 Score Explanation 2 6, and an appropriate explanation is given. 1 6, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 x 8, A 81, B 99, and an appropriate explanation is given. 1 x 8, A 81, B 99, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 256, and an appropriate explanation is given. 1 256, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 1 a 45, and an appropriate explanation is given. 1 b 90, and an appropriate explanation is given. 1 (a) 45 and (b) 90, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 Score 2 2 0 16 Score 4 Explanation a ⬔BCA ⬔ACD −−− −− b AD AB A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Explanation The following proof or another appropriate proof is given. Statements −− −− −−− −−− 1. DB ⬜ AC, AD ⬜ DC 17 Reasons 1. Given. 2. ⬔DBA and ⬔ADC are right angles. 2. Definition of perpendicular lines. 3. ⬔1 is complementary to ⬔x. 3. Given. 4. m⬔1 m⬔x 90 4. Definition of complementary angles. 5. m⬔x m⬔2 m⬔ADC 5. Addition postulate. 6. m⬔x m⬔2 90 6. Substitution postulate. 7. m⬔1 m⬔x m⬔x m⬔2 7. Substitution postulate. 8. m⬔1 m⬔2 8. Subtraction postulate. 9. ⬔1 ⬔2 9. Angles with equal measures are congruent. 3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or multiple steps are missing or have errors. 1 ⬔1 ⬔2 and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 2 1 2 Explanation −−− −− −− −− a Since CD bisects AB at E, AE BE. Given ⬔1 ⬔2, and because vertical angles are congruent, ⬔AEC ⬔BED. Hence, by ASA, 䉭ACE 䉭BDE. a A correct conclusion is reached and reason is given, but no appropriate method is shown. −− −−− b Given ⬔1 ⬔2 and LN ⬜ MO. ⬔LNM and ⬔LNO are right angles. Since right −− −− angles are congruent, ⬔LNM ⬔LNO. LN LN by the reflexive property of congruence. Therefore, by ASA, 䉭LMN 䉭LON. 1 b A correct conclusion is reached and reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–5 103 Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 Score 6 104 Explanation The following proof or another appropriate proof is given. Statements Reasons 1. 䉭PET −−− −− −−− −− 2. PCB ⬜ ET; TCA ⬜ EP 1. Given. 2. Given. 3. ⬔CBT and ⬔CAP are right angles. 3. Definition of perpendicular lines. 4. ⬔CBT ⬔CAP 4. Right angles are congruent. 5. ⬔ACP ⬔BCP −− −− 6. PA TB 5. Vertical angles are congruent. 6. Given. 7. 䉭ACP 䉭BCT −− −− 8. CP CT 7. AAS AAS. 8. CPCTC. 9. 䉭PCT is isosceles. 9. Definition of isosceles triangle. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or three steps are missing. 2 An appropriate proof with correct conclusion is used, but more than three reasons are faulty or more than three steps are missing or have errors. 1 䉭PCT is isosceles and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 19 Score 6 Explanation The following proof or another appropriate proof is given. Statements −− −− 1. AC BD −− −− −− −− 2. AC BC BD BC −− −−− 3. AB CD −− −−− −− −−− 4. GB ⬜ AD; EC ⬜ AD Reasons 1. Given. 2. Subtraction postulate. 3. Partition postulate. 4. Given. 5. ⬔GBA and ⬔ECD are right angles. 5. Definition of perpendicular lines. 6. ⬔GBA ⬔ECD 6. Right angles are congruent. 7. ⬔AGB ⬔DEC 7. Given. 8. 䉭AGB 䉭DEC 8. AAS AAS. 9. ⬔A ⬔D −− −− 10. AF DF 9. CPCTC. 10. Converse of isosceles triangle theorem. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or three steps are missing. 2 An appropriate proof with correct conclusion is used, but more than three reasons are faulty or more than three steps are missing or have errors. −− −− AF DF and a reason is given, but no appropriate method is shown. 1 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–5 105 20 Score 6 Explanation The following proof or another appropriate proof is given. Statements −− 1. BD is the median of 䉭ABC. −− −− 2. BA BC 3. 䉭ABC is isosceles. −− −− 4. BD ⬜ AC −− −− 5. BD is an altitude to AC. Reasons 1. Given. 2. Given. 3. Definition of isosceles triangle. 4. The median from the vertex angle of an isosceles triangle is perpendicular to the base. 5. Definition of an altitude. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or multiple steps are missing or have errors. −− −− BD is an altitude to AC and a reason is given, but no appropriate method is shown. 1 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–6 (pages 450–453) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 6 (3) 4z 1 (3) a ab b 2 2 (1) (x 9, y 5) 7 (2) −− −−− −− −− 8 (2) XY ⬜ MA 3 (4) AE BE 4 (2) (x, y) 9 (1) (4, 5) 5 (4) ~r → ~p 10 (2) dilation Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 106 Score Explanation 2 25, and an appropriate explanation is given. 1 25, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 12 13 14 Score Explanation 2 100, and an appropriate explanation is given. 1 100, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 a (3, 4) b (3, 4) 1 Student answers only two parts correctly. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score c (4, 3) d (3, 2) Explanation 2 39 and 51, and an appropriate explanation is given. 1 39 and 51, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 16 Score Explanation 4 45-45-90 isosceles triangle and an appropriate explanation is given. 3 An appropriate method is used, but student does not identify the type of triangle. 2 An appropriate method is used, but multiple computational errors are made. 1 45-45-90 isosceles triangle, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 36, and an appropriate explanation is given. 3 An appropriate method is used, but one computational error is made. 2 An appropriate method is used, but multiple computational errors are made. 1 36, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–6 107 17 Score 4 Explanation The following proof or another appropriate proof is given. Statements −− −− 1. AB EB −− −− 2. BC BD Reasons 1. Given. 3. ⬔B ⬔B 3. Reflexive property of congruence. 4. 䉭ABC 䉭EBD 4. SAS SAS. 5. ⬔1 ⬔2 5. CPCTC. 2. Given. 3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or multiple steps are missing or have errors. 1 ⬔1 ⬔2 and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 Score Explanation 1 a K(1, 1), A(5, 2), T(3, 5) 1 b K (1, 1) A(5, 2), T (3, 5) 2 c y T' T 5 4 3 A' 2 K' 1 A K 5 4 3 2 1 1 1 2 K" 2 3 4 5 x A" 3 4 5 108 T" 2 d r x-axis 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 19 Score 6 Explanation The following proof or another appropriate proof is given. Statements −− −− 1. Isosceles 䉭ABC, with AB BC −− 2. BD bisects ⬔ABC. 1. Given. 2. Given. 3. ⬔ABD ⬔CBD −− −− 4. BD BD 3. Definition of angle bisector. 5. 䉭ABD 䉭CBD −−− −−− 6. AD CD −− −− 7. BD bisects AC. 5. SAS SAS. 4. Reflexive property of congruence. 6. CPCTC. 7. Definition of bisector. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but more than two reasons are faulty or more than two steps are missing or have errors. −− −− BD bisects AC and a reason is given, but no appropriate method is shown. 1 0 20 Reasons A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 1 a C 1 b D 2 c B 2 d C 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–7 (pages 454–456) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. −− −−− 1 (3) supplementary 6 (4) AB RM 2 (4) (0, 6) 7 (2) 3(b a) 3 (1) 6 8 (1) ⬔B is the largest angle. 4 (3) I and III only 9 (1) 15 x 5 (2) 2 10 (1) _ 2 Chapters 1–7 109 Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 12 13 14 Score Explanation 2 6, and an appropriate explanation is given. 1 6, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 (7, 5), and an appropriate explanation is given. 1 (7, 5), but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 144, and an appropriate explanation is given. 1 144, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 (3, 3), and an appropriate explanation is given. 1 (3, 3), but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 Score 1 Explanation a (4, 0) b (4, 2) 1 c (6, 3) d (3, 6) 2 e (8, 3) f (6, 3) 0 110 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 16 17 Score Explanation 4 The following proof or another appropriate proof is given. −− −− Right angles B and A are congruent because AB ⬜ DB and −− −−− −−− −−− AC ⬜ DC. BW CW. ⬔BWA ⬔CWD because vertical angles are congruent. −−− −−− Therefore, by ASA 䉭ABW 䉭DCW. AW DW because of CPCTC. Hence, 䉭AWD is isosceles by definition of an isosceles triangle. 3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or multiple steps are missing or have errors. 1 䉭AWD is isosceles and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 4 Explanation The following proof or another appropriate proof is given. Statements Reasons 1. BC AB 1. Given. 2. m⬔BAC m⬔BCA 2. The measures of the angles opposite unequal sides are unequal. 1 m⬔BAC _ 1 m⬔BCA 3. _ 2 2 −−− 4. DA bisects ⬔BAC. 3. Division postulate. 1 m⬔BAC 5. m⬔DAC _ 2 −−− 6. DC bisects ⬔BCA. 5. Definition of a bisector. 1 m⬔BCA 7. m⬔DCA _ 2 8. m⬔DAC m⬔DCA 7. Definition of a bisector. 9. DC DA 9. The lengths of the sides opposite two unequal angles of a triangle are unequal. 4. Given. 6. Given. 8. Substitution postulate. 3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or multiple steps are missing or have errors. 1 DC DA and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–7 111 Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 Score 1 a (4, 1) 1 b √10 2 c y 3x 13 1 , slope of the median is 3. The slopes are negative _ −−− d Slope MD 3 −−− reciprocals; therefore, the median is perpendicular to MD. 2 0 19 20 Score A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Explanation 4 −− −−− 1 x 3. a Altitude from D to AC is x 0. Altitude from C to AD is y _ 2 −−− Altitude from A to CD is y x 3. 3 a An appropriate method is used, but one computational error is made. 2 a An appropriate method is used, but multiple computational errors are made. 1 a The altitudes are identified, but no explanation is given. 2 b (0, 3), and appropriate work is shown. 1 b (0, 3), but no work is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 6 112 Explanation Explanation The following proof or another appropriate proof is given. Statements −− −− 1. 䉭ABC, AB BC Reasons 1. Given. 2. 䉭ABC is isosceles. 2. Definition of isosceles triangle. 3. ⬔A ⬔C 3. Definition of isosceles triangle. 4. m⬔C m⬔T 4. Definition of an exterior angle. 5. m⬔A m⬔T 5. Substitution postulate. 6. PT AP 6. Greater side opposite greater angle. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or multiple steps are missing or have errors. Cumulative Reviews 1 PT AP and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–8 (pages 457–460) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (3) rotation of 180 6 (1) y 2 (1) 5 7 (3) (5, 3) 3 (1) x 2y 5 8 (2) negative and less than the x-intercept of m 4 (1) 23 9 (2) 2 5 (1) {1, 2, 3} 10 (4) (9, 7) Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 12 13 14 Score Explanation 2 m⬔1 36, m⬔2 72, m⬔3 36, m⬔AOE 144, and an appropriate explanation is given. 1 m⬔1 36, m⬔2 72, m⬔3 36, m⬔AOE 144, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 (0, 1), and an appropriate explanation is given. 1 (0, 1), but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 (5, 1), and an appropriate explanation is given. 1 (5, 1), but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 105, and an appropriate explanation is given. 1 105, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–8 113 Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 16 17 Score 4 2 : 3, and an appropriate explanation is given. 3 An appropriate method is used, but a single arithmetical error is made. 2 An appropriate method is used, but multiple arithmetical errors are made. 1 2 : 3, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 x 7, m⬔P 62, m⬔Q 28, and an appropriate explanation is given. 3 An appropriate method is used, but a single arithmetical error is made. 2 An appropriate method is used, but multiple arithmetical errors are made. 1 x 7, m⬔P 62, m⬔Q 28, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 4 114 Explanation Explanation The following proof or another appropriate proof is given. Statements −−− −− −− −− −− −− 1. AD BC, DE CE, EA EB Reasons 1. Given. 2. 䉭ADE 䉭BCE 2. SSS SSS. 3. ⬔1 ⬔2 3. CPCTC. 4. ⬔3 ⬔4 5. ⬔1 ⬔3 ⬔2 ⬔4 4. Base angles of an isosceles triangle are congruent. 5. Addition postulate. 6. ⬔DAB ⬔CBA 6. Partition postulate. 3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or multiple steps are missing or have errors. 1 ⬔DAB ⬔CBA and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 Score 6 Explanation 2. If two sides are congruent, the triangle is isosceles. 3. Base angles of isosceles triangles are congruent. 4. Exterior angle of a triangle is greater than either nonadjacent interior angle. 5. Substitution postulate. 6. Longest side of a triangle is opposite the angle with the largest measure. 5 One reason is faulty or missing. 4 Two reasons are faulty or missing. 3 Three reasons are faulty or missing. 2 More than three reasons are faulty or missing. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Check students’ graphs. The coordinates are listed below. 19 20 Score Explanation 1 a A(6, 6), B(12, 6), C(6, 15) 2 b A(6, 6), B(12, 6), C(6, 15) 2 c A (6, 4), B (6, 10), C (15, 4) 1 d D3 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 1 a Student graphs points A and B correctly. 3 b x 4 and an appropriate explanation is given. Student graphs point C correctly. 2 c 0, and an appropriate explanation is given. 1 Correct answers for b and c, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–8 115 Chapters 1–9 (pages 461–463) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (3) right 6 (1) 34 −− 2 (4) AE bisects ⬔BAC. 7 (3) SSS 3 (4) 4, 5, 6 8 (1) 290 4 (1) (5, 3) 9 (3) 3,600 5 (2) congruent 10 (1) 30 Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 12 13 14 116 Score Explanation 2 x is divisible by 2, and an appropriate explanation is given. 1 x is divisible by 2, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 155, and an appropriate explanation is given. 1 155, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 (9, 2), and an appropriate explanation is given. 1 (9, 2), but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 (4, 2), and an appropriate explanation is given. 1 (4, 2), but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 Score 4 16 17 Explanation a True b True c False d True e False f True 3 Student answers all but one part correctly. 2 Student answers two or three parts incorrectly. 1 Student answers four or five parts incorrectly. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 18, and an appropriate explanation is given. 3 An appropriate method is used, but a single arithmetical error is made. 2 An appropriate method is used, but multiple arithmetical errors are made. 1 18, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 4 Explanation a Yes b Yes c Yes d No e No f Yes 3 Student answers all but one part correctly. 2 Student answers two or three parts incorrectly. 1 Student answers four or five parts incorrectly. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 Score Explanation 6 , AC √ AB √ 85 , BC √85 68 , and an appropriate explanation is given. 5 An appropriate method is shown, but one computational mistake is made. 4 An appropriate method is shown, but two computational mistakes are made. 3 An appropriate method is shown, but wrong sides of the triangle are shown congruent. 2 An appropriate method is shown, but more than two computational mistakes are made. 1 An appropriate method is shown, but no conclusion is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–9 117 19 Score 6 Explanation The following proof or another appropriate proof is given. Statements 1. 䉭ABC is equilateral. 1. Given. 2. m⬔BAC 60 2. Interior angles of an equilateral triangle are 60. 3. ⬔BCA and ⬔BCE are a linear pair. 3. Definition of linear pair. 4. ⬔BCE is supplementary to ⬔BCA. 4. If two angles form a linear pair, then they are supplementary. 5. m⬔BCA m⬔BCE 180 5. Definition of supplementary angles. 6. m⬔BCE 120 −−− 7. CD bisects ⬔BCE. 1 m⬔BCE 8. m⬔DCE _ 2 9. m⬔DCE 60 6. Subtraction postulate. 7. Given. 8. Definition of angle bisector. 9. Substitution. 10. m⬔DCE m⬔BAC 10. Substitution. 11. ⬔DCE ⬔BAC 11. Angles with equal measures are congruent. −− AB 12. CD 12. Lines with congruent corresponding angles are parallel. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but more than two reasons are faulty or more than two steps are missing or have errors. −− AB and a reason is given, but no appropriate method is shown. CD 1 0 118 Reasons A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 20 Score 6 Explanation The following proof or another appropriate proof is given. Statements −− −−− 1. BC AD −− −−− 2. BC AD Reasons 1. Given. 2. Given. 3. ⬔EAD ⬔FCB −− −− 4. BF ED 4. Given. 5. ⬔BFC ⬔DEA 5. Alternate interior angles are congruent. 6. 䉭BFC 䉭DEA −− −− 7. BF ED 6. AAS AAS. 3. Alternate interior angles are congruent. 7. CPCTC. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but more than two reasons are faulty or more than two steps are missing or have errors. −− −− BF ED and a reason is given, but no appropriate method is shown. 1 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–10 (pages 464–466) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 10 1x _ 1 (2) 360 7 (1) y _ 3 3 2 (4) 140 8 (4) The diagonals bisect the angles of 3 (1) 22 the parallelogram. 4 (3) y x 9 (1) adjacent 5 (3) 8 10 (3) m⬔4 m⬔B 6 (3) a rhombus Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 Score Explanation 2 120, and an appropriate explanation is given. 1 120, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–10 119 12 13 14 Score Explanation 2 31, and an appropriate explanation is given. 1 31, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 7 √ 2 , and an appropriate explanation is given. 1 7 √2, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 9, and an appropriate explanation is given. 1 9, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 16 120 Score Explanation 4 6, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but multiple computational errors are made. 1 6, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 A(3, 6), B(2, 1), C(9, 1), and the transformations are performed correctly. 3 Student arrives at the correct answer, but performs one incorrect transformation. 2 Student performs transformations in the wrong order. 1 A(3, 6), B(2, 1), C(9, 1), but no appropriate explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 17 Score Explanation 4 51, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but multiple computational errors are made. 1 51, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 Score 3 19 Explanation −− 1 , slope −− _ 2 , slope −− _ 1 , TD _ −−− is a vertical line. Since a Slope DA RT AR 3 3 3 −−− −− two sides have the same slope, DA RT. DART is a trapezoid. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 3 b AK TA, and an appropriate explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 3 Explanation a The following proof or another appropriate proof is given. Statements −−− −− 1. AD BE −− −− 2. AE BD 2. Given. 3. DE DE 3. Reflexive property of congruence. 4. 䉭ADE 䉭BED 4. SSS SSS. 5. ⬔BDE ⬔AED 5. CPCTC. ___ ___ Reasons 1. Given. 1 ⬔BDE ⬔AED and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 3 b The following proof or another appropriate proof is given. Statements −−− −− 1. AD BE −− −− 2. AE BD −− −− 3. AB AB Reasons 1. Given. 2. Given. 3. Reflexive property of congruence. 4. 䉭ADB 䉭BEA 4. SSS SSS. 5. ⬔FAB ⬔FBA 5. CPCTC. 1 ⬔FAB ⬔FBA and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–10 121 20 Score 6 Explanation The following proof or another appropriate proof is given. Statements Reasons 1. Quadrilateral ABCD and −−−− diagonal AFEC −− −− −− −− 2. BF ⬜ AC, DE ⬜ AC 1. Given. 3. ⬔BFA and ⬔DEC are right angles. 3. Definition of perpendicular lines. 4. ⬔BFA ⬔DEC −− −− −− −− 5. BF DE, AE CF 4. Right angles are congruent. 6. 䉭ADE 䉭CBF −−− −− 7. AD BC 8. ⬔DAE ⬔BCF −−− −− 9. AD BC 10. ABCD is a parallelogram. 2. Given. 5. Given. 6. SAS SAS. 7. CPCTC. 8. CPCTC. 9. Alternate interior angles of two parallel lines are congruent. 10. Definition of a parallelogram. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but more than two reasons are faulty or more than two steps are missing or have errors. 1 ABCD is a parallelogram and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–11 (pages 467–469) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 2 b 6 (3) 160 cm 1 (1) _ a 7 (1) 2 √6 2 (3) equilateral and equiangular 3 3 (3) 120 8 (3) _ −− 5 4 (1) AC 9 (3) The sphere is inscribed in the cube. 5 (2) 9 10 (1) parallelogram 122 Cumulative Reviews Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 12 13 14 Score Explanation 2 x 116, y 32, and an appropriate explanation is given. 1 x 116, y 32, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 n 1, and an appropriate explanation is given. 1 n 1, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 116.5, and an appropriate explanation is given. 1 116.5, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 50, and an appropriate explanation is given. 1 50, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 Score Explanation 4 10 , and an appropriate explanation is given. 3 √ 3 An appropriate method is used, but a single arithmetical error is made. 2 An appropriate method is used, but multiple arithmetical errors are made. 1 3 √ 10 , but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–11 123 16 17 124 Score Explanation 4 DN √ 106 , NA √ 106 , making 䉭DNA isosceles. 9 5 _ _ −−− −−− and slope NA . Since the slopes are negative reciprocals of each Slope DN 5 9 −−− −−− other, DN ⬜ NA. Therefore, 䉭DNA is an isosceles right triangle. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but 䉭DNA is not proven to be an isosceles right triangle. 1 An appropriate method is used, but multiple computational errors are made. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 6 √ , and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but multiple computational errors are made. 1 6 √ , but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 Score 6 Explanation The following proof or another appropriate proof is given. Statements −−− −− −−− −− 1. DA ⬜ AB, DC ⬜ CB 5 19 Reasons 1. Given. 2. ⬔BAD and ⬔BCD are right angles. 2. Definition of perpendicular lines. 3. ⬔BAD ⬔BCD 3. Right angles are congruent. 4. ⬔1 ⬔2 4. Given. 5. ⬔BAE ⬔BCE −− −− 6. BD ⬜ AC at E 5. Subtraction postulate. 7. ⬔BEA and ⬔BEC are right angles. 7. Definition of perpendicular lines. 8. ⬔BEA ⬔BEC −− −− 9. BE BE 8. Right angles are congruent. 6. Given. 9. Reflexive property of congruence. 10. 䉭ABE 䉭CBE 10. AAS AAS. 11. CPCTC. 11. ⬔3 ⬔4 An appropriate proof with correct conclusion is shown, but one reason is faulty and/ or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but more than two reasons are faulty or more than two steps are missing or have errors. 1 ⬔3 ⬔4 and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 a Student proves that MIKE is a rhombus by showing that 40 or by some other appropriate method. MI IK KE EM √ 1 a The correct conclusion is reached and a reason is given, but no appropriate method is shown. 2 b Student shows that MIKE is not a square by showing that the 1 and the slope −− 3. _ −− diagonals are not perpendicular. Slope MI IK 3 1 b The correct conclusion is reached and a reason is given, but no appropriate method is shown. 2 c 32, and an appropriate explanation is given. 1 c 32, but no appropriate explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–11 125 20 Score Explanation 2 a 3, and an appropriate explanation is given. 1 3, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 2 b 36, and an appropriate explanation is given. 1 36, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 1 , and an appropriate explanation is given. c _ 4 1 , but no explanation is given. _ 4 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 2 1 0 Chapters 1–12 (pages 470–472) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (3) It has a slope of 0. 2 (2) are congruent 3 (3) 50 4 (3) cylinder 5 (4) 130 2 6 (3) 16x 2 7 (2) 10 √ 8 (1) 1 : 3 4 9 (1) _ 25 10 (3) 4x y 2 Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 126 Score Explanation 2 225, and an appropriate explanation is given. 1 225, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 12 13 Score 2 (0, 4) and (6, 0), and an appropriate explanation is given. 1 (0, 4) and (6, 0), but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 1 Explanation 9 a _, and an appropriate explanation is given. 16 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 1 27 , and an appropriate explanation is given. b _ 64 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 0 14 Explanation Score Explanation 2 102, and an appropriate explanation is given. 1 102, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 16 Score Explanation 2 a 22, and an appropriate explanation is given. 1 a 22, but no appropriate explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 2 b 78, and an appropriate explanation is given. 1 b 78, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 2 √ 3 , and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but multiple computational errors are made. 1 2 √ 3 , but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–12 127 17 Score Explanation 4 6 √ 5 , and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but multiple computational errors are made. 1 6 √ 5 , but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 128 Score Explanation 6 Student uses an appropriate method, such as showing that both pairs of opposite sides b , slope −− _ b. _ −−− 0, slope −−− 0; slope −−− are parallel, by showing that slope AD ME ED MA a a 5 An appropriate method is used, but one computational error is made. 4 An appropriate method is used, but two computational errors are made. 3 An appropriate method is used, but three computational errors are made. 2 More than three computational errors are made. 1 An appropriate method is used, but does not prove that MADE is a parallelogram. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 19 Score 6 Explanation The following proof or another appropriate proof is given. Statements 1. 䉭ABC is isosceles. −− −− 2. AB AC Reasons 1. Given. 2. Definition of isosceles triangle. 3. 䉭AMC is isosceles. −−− −−− 4. AM CM −−− −−− 5. PM QM −−− −−− −−− −−− 6. AM QM CM PM −−− −− 7. AQ CP 5. Given. 8. ⬔BAC ⬔BCA 8. Isosceles triangle theorem. 9. ⬔MAC ⬔MCA 9. Isosceles triangle theorem. 3. Given. 4. Definition of isosceles triangle. 6. Addition postulate. 7. Partition postulate. 10. ⬔BAC ⬔MAC ⬔BCA ⬔MCA 10. Subtraction postulate. 11. ⬔BAM ⬔BCP 11. Partition postulate. 12. 䉭ABQ 䉭CBP 12. SAS SAS. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but more than two reasons are faulty or more than two steps are missing or have errors. 1 䉭ABQ 䉭CBP and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–12 129 20 Score 6 Explanation The following proof or another appropriate proof is given. Statements Reasons 1. Parallelogram ABCD 1. Given. 2. ⬔A ⬔C 2. Opposite angles of a parallelogram are congruent. 3. m⬔1 m⬔2 3. Given. 4. ⬔1 ⬔2 4. Angles with equal measures are congruent. 5. ⬔1 and ⬔4 are a linear pair. ⬔2 and ⬔3 are a linear pair. 5. Definition of linear pair. 6. ⬔1 and ⬔4 are supplementary. ⬔2 and ⬔3 are supplementary. 6. Two angles that form a linear pair are supplementary. 7. ⬔3 ⬔4 7. Supplements of congruent angles are congruent. 8. 䉭AFE 䉭CHG 8. AA. GH FE _ 9. _ FA CH 9. Corresponding parts of similar triangles are in proportion. 10. FE CH GH FA 10. The product of means equals the product of extremes. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but more than two reasons are faulty or more than two steps are missing or have errors. 1 FE CH GH FA and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–13 (pages 473–475) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (4) 䉭ABD 䉭CDB 6 (3) R 270 2 (2) a median 7 (1) 30 3 (1) 12 8 (1) 12 4 (1) 1 : 2 9 (3) III and IV −− −− 5 (1) 2 10 (4) TA TO 130 Cumulative Reviews Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 12 13 14 Score Explanation 2 (16, 14), and an appropriate explanation is given. 1 (16, 14), but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 55, and an appropriate explanation is given. 1 55, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 38, and an appropriate explanation is given. 1 38, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 30.5, and an appropriate explanation is given. 1 30.5, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 Score Explanation 4 y x 3, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but the equation of a parallel line is given. 1 y x 3, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–13 131 16 17 Score Explanation 4 m⬔ADE 93, m⬔BGA 87, m⬔ABC 87, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but one of the angle measures found is incorrect. 1 m⬔ADE 93, m⬔BGA 87, m⬔ABC 87, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 4 3 2 1 0 Explanation r h_ , and an appropriate explanation is given. An appropriate method is used, but a single computational error is made. An appropriate method is used, but h is found in terms of r and s. r h_ , but no explanation is given. A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part IV For each question, use the specific criteria to award a maximum of 6 credits. 18 132 Score Explanation 6 h 12, and an appropriate explanation is given. 5 An appropriate method is used, but one computational error is made. 4 An appropriate method is used, but two computational errors are made. 3 An appropriate method is used, but three computational errors are made. 2 More than three computational errors are made. 1 h 12, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 19 Score 6 Explanation The following proof or another appropriate proof is given. Statements Reasons 1. Parallelogram ABCD; −−− −−− EAB; DCF −− −−− 2. AB DC −− −− 3. EB DF 1. Given. 4. ⬔AEH ⬔CFG 4. Alternate interior angles are congruent. 5. ⬔BAH ⬔BCD 5. Opposite angles of a parallelogram are congruent. 6. ⬔GCF and ⬔BCD are a linear pair. ⬔HAE and ⬔BAH are a linear pair. 6. Definition of linear pair. 7. ⬔GCF and ⬔BCD are supplementary. ⬔HAE and ⬔BAH are supplementary. 7. Two angles that form a linear pair are supplementary. 8. ⬔GCF ⬔HAE 8. Supplements of congruent angles are congruent. 9. a 䉭EAH 䉭FCG FC EA _ 10. _ AH CG 11. b EA CG AH CG 2. Definition of a parallelogram. 3. Collinearity. 9. AA. 10. Corresponding parts of similar triangles are in proportion. 11. The product of means equals the product of extremes. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 An appropriate proof with correct conclusion is shown for part a only. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is used, but more than two reasons are faulty or more than two steps are missing or have errors. 1 䉭EAH 䉭FCG and EA CG AH CG, and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–13 133 20 Score Explanation 1 72, and an appropriate explanation is given. a mBC 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 1 b m⬔EDC 108, and an appropriate explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 1 c m⬔BCR 36, and an appropriate explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 1 d m⬔AKB 72, and an appropriate explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 2 e m⬔RF 72, and an appropriate explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–14 (pages 476–479) Part I Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 2 3 4 5 (1) x 3 (3) 3 : 5 (3) 125 (2) 45 2 (3) 2 √ 6 7 8 9 10 (2) 8.8 (2) 2x 3 (1) {C, D, G, H} (4) y 3x 5 (4) similar triangles Part II For each question, use the specific criteria to award a maximum of 2 credits. 11 12 134 Score Explanation 2 720, and an appropriate explanation is given. 1 720, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 144, and an appropriate explanation is given. 1 144, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 13 14 Score Explanation 2 13, and an appropriate explanation is given. 1 13, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 80, and an appropriate explanation is given. 1 80, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part III For each question, use the specific criteria to award a maximum of 4 credits. 15 16 Score 4 Each side is 2b, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but h is found in terms of r and s. 1 Each side is 2b, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 4 3 2 1 0 17 Explanation Score Explanation 3 _ , and an appropriate explanation is given. 10 An appropriate method is used, but a single computational error is made. An appropriate method is used, but h is found in terms of r and s. 3 _ , but no explanation is given. 10 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Explanation 4 30,776.3 in. 3, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but h is found in terms of r and s. 1 30,776.3 in. 3, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–14 135 Part IV For each question, use the specific criteria to award a maximum of 6 credits. a y 6 x 10 y 9 2x 3 Explanation 8 7 Score 7 3y 18 6 5 4 3 2 1 7 6 5 4 3 2 1 1 y 2 1 2 3 4 5 6 7 8 9 10 x 2 3 4 5 6 7 19 136 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 3 b 45, and an appropriate explanation is given. 2 An appropriate method is used, but a single computational error is made. 1 45, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 −−− 1 and slope −−− 1. Since the slopes are negative reciprocals of each a Slope AX ME −−− other, the lines are perpendicular. The midpoint of AX is −−− (4, 3), which is point E. Therefore, ME is the perpendicular bisector. 3 An appropriate method is used, but a single arithmetical error is made. 2 An appropriate method is used to show the lines are perpendicular, but no midpoint is found. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 2 b Shows that 䉭MAX is isosceles by MX MA √ 68 , and an appropriate explanation is given. 1 An appropriate method is used, but a single computational error is made. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Cumulative Reviews 20 Score 6 Explanation The following proof or another appropriate proof is given. Statements Reasons CE 1. AE 1. Given. 2. AE CE 2. Congruent arcs have congruent chords. 3. ⬔ABE ⬔EBD 3. Congruent central angles have congruent chords. ___ ___ −− 4. FE is tangent to O at E. 4. Given. 5. ⬔FEB is a right angle. 5. Definition of a tangent. 6. ⬔FEB and ⬔BED are a linear pair. 6. Definition of linear pair. 7. ⬔FEB and ⬔BED are supplementary. 7. Two angles that form a linear pair are supplementary. 8. ⬔BED is a right angle. 8. A supplement to a right angle is a right angle. −−− 9. Diameter BOE 9. Given. is a semicircle. 10. BCE 10. Definition of a semicircle. 11. ⬔BAE is a right angle. 11. An angle inscribed in a semicircle is always a right angle. 12. ⬔BAE ⬔BED 12. Right angles are congruent. 13. 䉭ABE 䉭EBD 13. AA. BD BE _ 14. _ BE BA 14. Corresponding parts of similar triangles are in proportion. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or two steps are missing. 2 An appropriate proof with correct conclusion is shown, but multiple reasons are faulty or multiple steps are missing or have errors. BD , and a reason is given, but no appropriate method is shown. BE _ _ BE BA A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 1 0 Chapters 1–14 137 2%40!(# Practice Geometry Regents Examinations Practice Regents Examination One (pages 483–489) Part I Allow a total of 56 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 1 (3) _ 23 (4) 4 11 (3) (9, 3) 2 24 (3) right 12 (1) 6 2 (2) 8 √ 2 25 (2) 9 13 (4) 16 : 81 1 b2 3 3 (1) _ 26 (3) x 14 (3) 18 2 27 (4) I and II only 15 (4) 1 4 (1) 48 1V 16 (2) 26 28 (3) _ 3 5 (2) 8 √ 8 17 (4) 16 6 (2) 12 18 (2) y 7 (4) r : 2 19 (2) 6 √2 8 (4) 106 20 (1) 86 9 (1) 8 21 (2) 18 10 (3) ⬔AEC is a 22 (2) (0, 1) right angle. Part II For each question, use the specific criteria to award a maximum of 2 credits. 29 138 Score Explanation 2 2 2 (x 2) (y 4) 2.25, and an appropriate explanation is given. 1 (x 2) 2 (y 4) 2 2.25, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. 30 31 32 33 34 Score Explanation 2 100, and an appropriate explanation is given. 1 100, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 30, and an appropriate explanation is given. 1 30, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 3 √ 2 , and an appropriate explanation is given. 1 3 √ 2 , but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 16, and an appropriate explanation is given. 1 16, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 (1, 0), and an appropriate explanation is given. 1 (1, 0), but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part III For each question, use the specific criteria to award a maximum of 4 credits. 35 Score Explanation 4 y x 1, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but multiple computational errors are made. 1 y x 1, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Practice Regents Examination One 139 36 37 Score Explanation 4 36, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but multiple computational errors are made. 1 36, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 27, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but multiple computational errors are made. 1 27, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part IV For each question, use the specific criteria to award a maximum of 6 credits. 38 Score 6 140 Explanation The following proof or another appropriate proof is given. Statements −− −− 1. LA 储 EG Reasons 1. Given. 2. ⬔LAR ⬵ ⬔GER 2. Alternate interior angles are congruent. 3. ⬔ALR ⬵ ⬔EGR 3. Alternate interior angles are congruent. 4. 䉭ALR ⬃ 䉭EGR RE AR _ 5. _ RL RG 6. Rhombus PARL 4. AA⬃. 7. AR PA; RL PL 7. Definition of a rhombus. 8. AR RG RL RE 8. The product of means equals the product of extremes. 9. PA RG RE PL 9. Substitution postulate. 5. Corresponding parts of similar triangles are in proportion. 6. Given. 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from the statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or three steps are missing. 2 An appropriate proof with correct conclusion is used, but more than three reasons are faulty or more than three steps are missing or have errors. 1 PA RG RE PL and a reason is given, but no appropriate method is shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Practice Geometry Regents Examinations Practice Regents Examination Two (pages 490–496) Part I Allow a total of 56 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. −− −− 1 (1) EK ⬜ EY 10 (2) 120 21 (2) 5 3 2 (4) 150 11 (2) 20 √3 22 (1) 64 cm 3 12 (3) 24 3 (1) 4 √ 23 (4) 4 1 13 (1) 28 24 (3) perpendicular 4 (3) _ 2 14 (3) 160 bisectors of the sides 5 (3) 36 15 (3) 2 of the triangle 6 (4) R 90 3 4 _ 25 (2) 5 √2 units 16 (1) 7 (4) Use the slope formula 3 26 (4) 36 17 (4) 6 s 2x to prove diagonals are 27 (1) BC AB 18 (4) 11 perpendicular. 28 (3) equal in area 19 (3) 8 8 (2) 105 29 } 20 (3) 15 9 (4) {2, 5, √ Part II For each question, use the specific criteria to award a maximum of 2 credits. 29 30 31 Score Explanation 2 69, and an appropriate explanation is given. 1 69, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 (x 4) 2 (y 5) 2 9, and an appropriate explanation is given. 1 (x 4) 2 (y 5) 2 9, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 (2x, 3y), and an appropriate explanation is given. 1 (2x, 3y), but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Practice Regents Examination Two 141 32 33 34 Score Explanation 2 108, and an appropriate explanation is given. 1 108, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 6.8, and an appropriate explanation is given. 1 6.8, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 117, and an appropriate explanation is given. 1 117, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part III For each question, use the specific criteria to award a maximum of 4 credits. 35 36 142 Score Explanation 4 11.55, and an appropriate explanation is given. 3 An appropriate method is used, but answer is not rounded to the nearest hundredth. 2 An appropriate method is used, but multiple computational errors are made. 1 11.55, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 OC 21, m⬔COD 69.5, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but only one of the two answers is found. 1 OC 21, m⬔COD 69.5, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Practice Geometry Regents Examinations 37 Score Explanation 4 Secant points of intersection are (0, 5) and (2, 4), and an appropriate explanation is given. 3 Secant is given, but graph is incorrect. 2 Graph is shown, but secant is not given. 1 Secant is given, but graph is not shown. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Part IV For each question, use the specific criteria to award a maximum of 6 credits. 38 Score Explanation 6 Any appropriate method is used to prove that ABCD is a parallelogram, but not a rectangle. For example: showing that the diagonals bisect each other but are not congruent, or showing that no consecutive sides have negative reciprocal slopes. 5 An appropriate method is used, but a single computational error is made. 4 An appropriate method is used, but two computational errors are made. 3 An appropriate method is used to prove that ABCD is a parallelogram but does not show that it is not a rectangle. 2 An appropriate method is used to prove that ABCD is a parallelogram, but more than two computational errors are made and student does not show that it is not a rectangle. 1 An appropriate method is used, but ABCD is not shown to be a parallelogram, and multiple computational errors are made. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Practice Regents Examination Three (pages 497–504) Part 1 Allow a total of 56 credits, 2 credits for each of the following correct answers. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (2) 50 22 (1) perpendicular bisectors 11 (2) (x 3) 2 (y 7) 2 2 1 2 (4) 16 : 25 of the sides 12 (3) y _x 4 2 3 (4) (7, 7 √3) 23 (3) 120 13 (3) d 5 2 4 (2) 18 cm 24 (4) obtuse 14 (4) 11 5 (3) {2, 7, 10} 25 (3) 60 15 (1) 60 6 (3) (3, 6.5) 26 (4) 210 3 16 (2) 32 √ 3 7 (3) 80 √ 27 (1) The perpendicular 17 (2) 16 16 √2 8 (3) ABCD is a parallelobisector of the chord 18 (4) x 1 gram whose diagonals of a circle bisects the 19 (1) 5 bisect each other. intercepted arc. 20 (2) y 5x 2 2 _ 28 (1) (3, 5) 9 (2) 3 21 (3) 4 √ 3 10 (2) 156 Practice Regents Examination Three 143 Part II For each question, use the specific criteria to award a maximum of 2 credits. 29 30 31 32 Score 2 216 cubic inches, and an appropriate explanation is given. 1 216 cubic inches, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score a PA 9, and an appropriate explanation is given. 1 53 and an appropriate explanation is given. b mCB 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 16 inches, and an appropriate explanation is given. 1 16 inches, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 0 144 Explanation 2 1 34 Explanation 1 2 33 Explanation Score Explanation 60 _ , and an appropriate explanation is given. 13 60 _ , but no explanation is given. 13 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Explanation 2 32, and an appropriate explanation is given. 1 32, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 2 4, and an appropriate explanation is given. 1 4, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Practice Geometry Regents Examinations Part III For each question, use the specific criteria to award a maximum of 4 credits. 35 36 37 Score Explanation 4 16, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but incorrect radius is found. 1 16, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score Explanation 4 40, and an appropriate explanation is given. 3 An appropriate method is used, but a single computational error is made. 2 An appropriate method is used, but multiple computational errors are made. 1 40, but no explanation is given. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Score 4 Explanation 2. Reflexive property of congruence 3. Perpendicular lines meet to form right angles. 4. All right angles are congruent. 5. Two right triangles are similar if an acute angle of one triangle is congruent to an acute angle of the other. 6. Corresponding sides of similar triangles are in proportion. 7. The product of means equals the product of extremes. 3 One reason is incorrect. 2 Two reasons are incorrect. 1 More than two reasons are incorrect. 0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Practice Regents Examination Three 145 Part IV For each question, use the specific criteria to award a maximum of 6 credits. 38 Score 6 Explanation The following proof or another appropriate proof is given. Statements Reasons 1. Parallelogram ABCD, −−− with ABG −− −−− −−− −−− 2. AB 储 CD, AG 储 CD 1. Given. 3. ⬔MGB ⬵ ⬔MDC 3. Alternate interior angles are congruent. 4. ⬔BMG ⬵ ⬔CMD 4. Vertical angles are congruent. 2. Definition of a parallelogram. −− 5. M is the midpoint of BC. −−− −−− 6. BM ⬵ MC 7. a 䉭BGM ⬵ 䉭CDM −− −−− 8. BG ⬵ CD −− −−− 9. AB ⬵ CD −− −− 10. b AB ⬵ BG 6. Definition of midpoint. 7. AAS ⬵ AAS. 8. CPCTC. 9. Opposite sides of a parallelogram are congruent. 10. Transitive postulate of congruence 5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or one statement is missing. 4 A faulty conclusion or no conclusion is drawn from correct statements and reasoning. 3 An appropriate proof with correct conclusion is shown, but no conclusion is made −− −− for AB ⬵ BG. 2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or multiple steps are missing or have errors. −− −− 䉭BGM ⬵ 䉭CDM and AB ⬵ BG, and a reason is given, but no appropriate proof is shown. 1 0 146 5. Given. A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Practice Geometry Regents Examinations