# amscogeometry answer

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Answers for Textbook ExercisesFor each proof in this Answer Key, one method is provided. Valid alternative proofs may exist and should be considered acceptable.

Chapter 1. Essentials of Geometry1-1 Undefined Terms (page 3)Writing About Mathematics 1. It may be called a plane because it is flat. It is not a mathematical plane because it does not extend endlessly in all directions. Also, when taking into account the curvature of the earth or local geographic features such as hills, it is not really a flat surface. 2. A stretched string does not go on forever. It also has a measurable thickness. Developing Skills 3. True 4. False 5. False 6. True 7. False 8. True Applying Skills 9. Answers will vary. 10. Answers will vary. 11. Answers will vary. 11. a. 1x x b. Identity property of multiplication 12. a. 6(4 b) 6(4) 6(b) b. Distributive property 1 13. a. 17 A 17 B 1 b. Inverse property of multiplication 14. a. 1 (24) 24 A 1 B 4 4 b. Commutative property of multiplication 15. a. 12(4 10) 12(10 4) b. Commutative property of addition 1 16. a. p A p B 1 b. Inverse property of multiplication 17. x 1 3 18. If a 0, then b 0 by the multiplication property of zero. Therefore, a b a 0 a.

1-3 Definitions, Lines, and Line Segments (pages 1011)Writing About Mathematics 1. A hammer is a tool is not a good definition because it does not distinguish hammers from other tools. 2. A hammer is used to drive nails is not a good definition because it is not reversible. There are many other objects that are not hammers that can be used to drive nails. Developing Skills 3. a. A noncollinear set of points is a set of three or more points that do not all lie on the same straight line. b. The set of all points c. Noncollinear points do not all lie on the same straight line. 4. a. The distance between any two points on the number line is the absolute value of the difference of the coordinates of the points. b. The set of real numbers c. The distance between any two points must be a positive real number or 0. 5. a. A line segment is a set of points consisting of two points on a line, called endpoints, and all of the points on the line between the endpoints.

1-2 The Real Numbers and Their Properties (page 6)Writing About Mathematics 1. a. The set of positive real numbers is not closed under subtraction because, if the value of the minuend is less than or equal to the value of the subtrahend, then the difference is a negative real number or zero. b. Yes. Subtracting any two real numbers results in another real number. 2. The set of negative real numbers is not closed under multiplication because multiplying any two negative real numbers results in a positive real number. Developing Skills 3. 0 4. 1 5. a 11 6. 0 7. a. 7 14 14 7 b. Commutative property of addition 8. a. 7 ( 7) 0 b. Inverse property of addition 9. a. 3(4 6) (3 4)6 b. Associative property of multiplication 10. a. 11(9) 9(11) b. Commutative property of multiplication

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6.

7.

8. 11. 14. 15.

b. The set of all points c. The points in a line segment all lie on the same line and are included between two endpoints. a. The measure of a line segment is the distance between its endpoints. b. The set of all real numbers c. The measure of a line segment must be a positive real number or 0. a. Congruent segments are segments that have the same measure. b. The set of all pairs of line segments c. Congruent pairs of segments have the same measure. AB 2 9. BD 3 10. CD 2 FH 4 12. GJ 5 13. EJ 8 Answers will vary. Example: AB CD DE P, Q, and R are noncollinear. If the points were collinear, then PR PQ QR, but PR 18 while PQ QR 10 12 22. P R

8. 2.5 9. 10 10. a. Yes; PQ = 16, QR 2, PR 16 2 18. b. No; 1 Applying Skills 11. 305 12. a. 16 in. b. At 16 in. Hands-On Activity 12. C A D E B

18, and

3. D bisects AB; AD > DB

1-5 Rays and Angles (pages 1819)Writing About Mathematics 1. A half-line consists of the set of points that lie to one side of a point on a line. A ray consists of a point on a line and all the points to one side of this endpoint. Therefore, a ray is a half-line plus an endpoint.h h

Q Applying Skills 16. a. 12 in. b. He could either mark the board at 13 inches or at 5 inches. 17. No. If Troy, Albany, and Schenectady were in a straight line, then the distance traveled on the return trip would equal the distance traveled on the first trip.

2. PR and PS are the same ray: they both have endpoint P, and R and S are on the same side of P on the line. Developing Skillsh h

3. a. EF and ED b. Eh h

1-4 Midpoints and Bisectors (pages 1314)Writing About Mathematicsg g

c. EF and ED d. y, E, FED, 4. a. Ah h

DEF

1. Yes. The notation is valid when AB and CD name the same line. In this case, A, B, C, and D are collinear. 2. No. It is possible that A, M, and B are noncollinear. Developing Skills 3. 4. 5. 6. 7. AT RN BC SP a. TC; AT > TC NS; RN > NS CD; BC > CD PT; SP > PT

b. AD and AB c. CAD, BAD, d. CABh

CAB

A

B

C

D

e. f. g. h. i. 5. a. b. c. d.

AB CAB and BAD BAD No; A, B, and D are not collinear. No; BAC is not the union of opposite rays. EAB, CAB, BAC, BAE DEC and CED CBA and ABC Eh h h h

b. Let x AB CD and y BC. Then AC AB BC x y and BD BC CD y + x.

e. ED and EB ; EA and EC f. DEB and AEC g. ABE and EBC

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1-6 More Angle Definitions (pages 2123)Writing About Mathematics 1. Disagree; a bisector is a ray. However, the angleg

1-7 Triangles (pages 2728)Writing About Mathematics 1. No. The statement does not specify that a triangle is a polygon. It is possible that three line segments form a figure that is not a triangle. 2. The legs of an isosceles triangle must be congruent while the legs of a right triangle may not be congruent. Developing Skills 3. Legs: CA, AB 4. Legs: JL, LK 5. Legs: NL, NM Vertex angle: N 6. Legs: RS, RT Vertex angle: R 710. Answers will vary. 11. BAC and BCA 12. EDF and DEF 13. SR and ST Hypotenuse: CB Hypotenuse: JK Base: LM Base angles: L, M Base: TS Base angles: S, T

bisector is on RST. 2. Acute; since an obtuse angle has a measure greater than 90 and less than 180, the two angles formed have measures greater than 45 and less than 90. Developing Skills 3. a. Acute 4. a. Obtuse b. 12 b. 49 5. a. Obtuse 6. a. Right b. 63 b. 45 7. a. Acute 8. a. Straight b. 41 b. 90 9. a. Acute 10. a. Acute b. 28.5 b. 1.5 11. 45 12. 30 13. 72 14. b. ACD DCB c. m ACD m DCB 15. b. DAC CAB c. m DAC m CAB 16. Two right angles 17. Two acute, 45 angles 18. m LMN m LMP m NMP 19. m LMP m LMN m NMP 20. m LMN m LMP m NMP 21. m ABE m EBC m ABC 22. m BEC m CED m DEB 23. m ADC m CDE m ADE 24. m AEC m AEB m CEB 25. a. ACD, ACF, DCB, BCFh h h h

g

14. a. Answers will vary. Examples: AB is on AB , ECB is adjacent to BCG, FCG b. Answers will vary. Examples: FCE is isosceles, AC CB, C is the midpoint of DE. Applying Skills 15. 57, 25 16. 22 17. 21, 10, 21 18. (1) With legs x 5 and 4x 11: 3, 7, 3 (2) With legs 3x 13 and 4x 11: 7, 19, 19

Review Exercises (pages 3132)1. Set, point, line, plane 2. It does not clearly define the class to which line belongs. 3. Collinear set of points 4. Distance between two points on the real number line 5. Triangle 6. Bisector 7. Opposite rays 8. Congruent angles 9. Isosceles triangle 10. Line segment 11. LN 12. a. RT b. S 13. a. DE = 4, EF 8, DF 12 b. 3 c. 5 14. The midpoint of a line is a single point while a bisector is any line that passes through the midpoint, and there are infinitely many lines passing through a point. 15. BED and AEC 16. ADC 17. BE and ED 18. AC 19. ABD and DBC 20. 45 21. BDA and BDC

b. 90 c. CD and CF d. CA and CB 26. x 15 27. m ABC 68 28. m QRS = 150 29. Yes. An angle bisector divides an angle into two congruent angles, which have equal measure. Let x m CBD m PMN. Then m ABC 2x and m LMN 2x. Therefore, m ABC m LMN and ABC LMN. Hands-On Activity 12. D A B E C

3. The angles have equal measure and are congruent: m ABD = m ABE m CBD m CBE ABD ABE CBD CBE 4. DE and AC are perpendicular.

90;

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22. No. In order for this equality to be true, A, B, and C would have to be collinear.

Exploration (pages 3233)1. The points are the same. Each represents a location on the surface. 2. A geodesic in Euclidean geometry extends infinitely in each direction. The geodesic on the sphere is a circle. 3. Two intersecting geodesics in Euclidean geometry form four angles. Two intersecting

geodesics on a sphere form eight angles. The sides of these angles do not extend infinitely. 4. The sum of the measures of the angles of a triangle in Euclidean geometry is 180. Drawing three geodesics on a sphere forms eight triangles. The sum of the measures of the angles of a triangle on sphere is greater than 180.

Chapter 2. Logic2-1 Sentences, Statements, and Truth Values (pages 3941)Writing About Mathematics 1. A sentence in grammar need only have a subject and a predicate. A mathematical sentence used in logic must state a fact or a complete idea that can be judged to be true or false. 2. Answers will vary. a. Example: Today is Wednesday. b. Example: I am six feet tall. c. Example: This country is in the southern hemisphere. Developing Skills 3. Mathematical sentence 4. Mathematical sentence 5. Not a mathematical sentence 6. Not a mathematical sentence 7. Not a mathematical sentence 8. Not a mathematical sentence 9. Mathematical sentence 10. Not a mathematical sentence 11. She 12. We 13. y 14. x 15. This 16. He 17. It 18. It 19. a. True 20. a. Open 21. a. False 22. a. True b. They 23. a. Open 24. a. False 25. a. True b. x 26. a. False 27. {New York} 28. {Nevada, Illinois} 29. {Massachusetts, New York} 30. {Nevada, Illinois, Massachusetts, Alaska, New York} 31. {Alaska} 32. {triangle} 33. 34. {triangle} 35. {square, rectangle} 36.