CHAPTER 4

CHAPTER 4 Chapter Opener

4.1 Practice and Applications (pp. 176–178)

Chapter Readiness Quiz (p. 172)

11. scalene

12. isosceles

1. D

14. equilateral

2. H; PQ **** is horizontal, so subtract the x-coordinates.

17. obtuse

PQ ⏐7 2⏐ ⏐5⏐ 5

0 2 6

24 2

3. B; M , (3, 3)

15. scalene 18. acute

20. acute

13. equilateral

21. right

16. isosceles

19. right 22. equiangular

23. An acute triangle has three acute angles, so the triangle is

not an acute triangle. An obtuse triangle has one obtuse angle and two acute angles.

Lesson 4.1

24. acute isosceles triangle

25. right isosceles triangle

4.1 Checkpoint (pp. 173–174)

26. obtuse isosceles triangle

27. right scalene triangle

28. obtuse isosceles triangle

29. acute scalene triangle

1. Because this triangle has 2 congruent sides, it is

isosceles. 2. Because this triangle has 3 congruent sides, it is

equilateral. 3. Because this triangle has no congruent sides, it is scalene. 4. Because this triangle has 3 angles with measures less

than 90 and 2 congruent sides, it is an acute isosceles triangle. 5. Because this triangle has 3 angles with measures less

than 90 and no congruent sides, it is an acute scalene triangle. 6. Because this triangle has one angle greater than 90

and 2 congruent sides, it is an obtuse isosceles triangle. 4.1 Guided Practice (p. 175) 1. An obtuse triangle has one angle that is greater than

90 and an acute triangle has no angles that are greater than 90. 2. QR **** is the side that is opposite a P.

30. B

31. E

32. A

33. D

34. G

35. C

36. F

37. acute triangle

38. right triangle

39. acute triangle

40. A, B, and E; A, C, and D; A, D, and E; or B, C, and E 41. B, C, and E; A, D, and E; A, B, and E; or B, D, and E 42. CB **** is opposite a A;

43. EF **** is opposite a D;

AC **** is opposite a B;

DE **** is opposite a F;

AB **** is opposite a C.

DF **** is opposite a E.

44. HJ **** is opposite a G;

45. LM **** is opposite a K;

GH **** is opposite a J;

KM **** is opposite a L;

GJ **** is opposite a H.

KL **** is opposite a M.

46. NQ **** is opposite a P;

47. ST **** is opposite a R;

NP **** is opposite a Q;

RT **** is opposite a S;

PQ **** is opposite a N.

RS **** is opposite a T.

48. Sample answer:

49.

50. Sample answer:

51. Sample answer:

3. PR **** is the side that is opposite a Q. 4. Because this triangle has 2 congruent sides, it is

isosceles. 5. Because this triangle has 2 congruent sides, it is

isosceles.

15

13

equilateral.

14

7. Because this triangle has no congruent sides, it is scalene. 8. Because this triangle has 3 angles with measures less

5

4

6. Because this triangle has 3 congruent sides, it is

3

52. Sample answer:

53. Sample answer:

that 90, it is an acute triangle. 120

9. Because this triangle has a right angle, it is a right

triangle.

24

10. Because this triangle has 3 congruent angles, it is an

equiangular triangle. 78 78

Copyright © McDougal Littell Inc. All rights reserved.

Geometry, Concepts and Skills Chapter 4 Worked-Out Solution Key

51

Chapter 4 continued 71. 2(x 1) 3x 7 180

4.1 Standardized Test Practice (p. 178) 54. C

2x 2 3x 7 180

55. F

x 5 180

4.1 Mixed Review (p. 178)

x 175

56. 4x (6x 10) 90

x 175

10x 10 90

72. 4(3x 1) 9x 10 180

10x 80

12x 4 9x 10 180

x8

3x 6 180

57. (11x 7) (5x 3) 180

3x 174

16x 4 180

x 58

16x 176 Lesson 4.2

x 11 58. 50 (8x) 90


8x 40

1. ma A ma B ma C 180

x 5;

ma A 65 50 180

50 8(5) (2y) 180

ma A 115 180

90 2y 180

ma A 65

2y 90

2. ma A ma B ma C 180

y 45

45 ma B 60 180

59. (2, 5) → (2 2, 5 4) → (0, 9)

ma B 105 180

60. (1, 3) → (1 2, 3 4) → (1, 1)

ma B 75

61. (1, 2) → (1 2, 2 4) → (3, 6)

3. ma A ma D 90

62. (0, 5) → (0 2, 5 4) → (2, 1)

ma A 50 90

63. (4, 2) → ( 4 2, 2 4) → (6, 2)

ma A 40

64. (0, 0) → (0 2, 0 4) → ( 2, 4)

ma A ma C 90

65. (6, 4) → (6 2, 4 4) → (8, 8)

40 ma C 90

66. (3, 1) → (3 2, 1 4) → (5, 3)

ma C 50 4. ma 2 60 60 120

4.1 Algebra Skills (p. 178)

5. ma 3 125 30 155

67. 5x 15 180

6. ma 4 55 58 113

5x 195 x 39

4.2 Guided Practice (p. 182)

68. x 2x 36 180

69. 3x 5x 20 180

3x 36 180

8x 20 180

3x 144

8x 160

x 48

x 20

70. 3x (x 8) 180

3x x 8 180 4x 8 180 4x 188 x 47

1.

4 A

B 2 5

1

3 6 C

Answers may vary. 2. A 3. 70 49 x 180

119 x 180

4. 94 x 136

x 42

x 61 5. 55 x 90

x 35

52



Chapter 4 continued 4.2 Practice and Applications (pp. 182–184)

21. 45 50 (2x 5) 180

2x 100 180

6. ma 1 78 31 180

2x 80

ma 1 109 180

x 40;

ma 1 71 x y 90

7. ma 1 40 30 180

40 y 90

ma 1 70 180

y 50

ma 1 110 8. ma 1 38 90

22. For any position of point C,

ma PBC ma BAC ma BCA. This illustrates the Exterior Angle Theorem.

ma 1 52 9. ma 1 60 60 180

ma 1 120 180

23.

x x 26 180

26

2x 26 180

ma 1 60 10. ma 1 100 43 180

ma 1 143 180 ma 1 37 11. ma 1 45 90

2x 154 x 77

x x

24. P 36

ma 1 45 12. ma 2 98 50 148 13. ma 2 64 75 139

5x

x

R

5x x 36 180 6x 36 180

14. ma 2 102 145

6x 144

ma 2 43

x 24

15. ML **** is opposite a MNL.

ma R x 24;

16. 37 ma L 90

ma Q 5x 5(24) 120

ma L 53 17. 30 ma L 90

4.2 Standardized Test Practice (p. 184)

ma L 60

25. C; (2x 3) 128 23 180

45 ma L 90

2x 148 180

ma L 45

2x 32

The downstream angle should be between 45 and 60. 18. x 2x (2x 15) 180

5x 15 180 5x 165 x 33 19. 6x 38 82

6x 120

x 16 26. F; the exterior angle measures are as follows:

50 60 110, 60 70 130, and 50 70 120 4.2 Mixed Review (p. 184) 27. m n by the Corresponding Angles Converse Postulate. 28. m n by the Alternate Exterior Angles Converse Theorem.

x 20 20. x 42 90

x 48; y 100 48 180 y 148 180 y 32


29. m n by the Same-Side Interior Angles Converse Theorem

(118 62 180).

4.2 Algebra Skills (p. 184) 30. 1015 1051

31. 3.5 3.06

32. 8.09 8.1

33. 1.75 1.57

34. 0 0.5

35. 2.055 2.1


53

Chapter 4 continued 17. x 30 30 180

Quiz 1 (p. 184) 1. obtuse isosceles triangle

18. x 50 50 180

x 60 180

2. acute scalene triangle

x 100 180

x 120

3. right scalene triangle 4. ma 1 60 90

So, ma A 120.

5. ma 1 38 40 78

19. 2x x x 180

ma 1 30

x 80 So, ma A 80. 20. y 11

4x 180

6. ma 1 85 150

x 45

ma 1 65

So, ma A 2(45) 90.

Lesson 4.3

21. 2y 10

4.3 Geo-Activity (p. 185)

y5

Step 2. a H and a K are congruent.

22. 4y 3 2y 5

Step 3. For each isosceles triangle, a H and a K are

23. y 5

2y 8

congruent.

y4 3y 5y 14

24.

4.3 Checkpoint (p. 186) 1. y 50

2. y 9

3. y 4 16

25. 8y 10 4y 2

2y 14

4y 12

y7

y3

26. First, show the T XYZ is equiangular and therefore

y 12

ma X ma Y ma Z 60. Then use the Corresponding Angles Postulate to show that ma YJK ma XJL 60, ma YKJ ma LKZ 60, and ma XLJ ma ZLK 60. With these measures you can use the Triangle Sum Theorem three times to show that ma JKL ma LJK ma JLK 60. Then you can state that T JKL is equiangular and therefore is also equilateral.

4.3 Guided Practice (p. 188) 1. Equilateral means that all sides are congruent.

Equiangular means that all angles are congruent. 2. LM **** MN **** NL **** ; a L a M a N 3. **** ST RS **** ; a R a T

4. UW **** UV **** ; a W a V

5. x 50 by the Base Angles Theorem

27. No, because the triangle would not be isosceles.

6. x 8.8 by the Converse of the Base Angles Theorem

28. Yes, because when 2 sides of a triangle are congruent,

then the angles opposite them are congruent (Base Angles Theorem).

4.3 Practice and Applications (pp. 188–190) 7. x 55 by the Base Angles Theorem

29.

Z

8. x 68 by the Base Angles Theorem 9. x 45 by the Corollary to the Triangle Sum Theorem

and the Base Angles Theorem 10. x 4 11

V

Angles Theorem.

x2

31. T WXV, T VXY, T YXZ, T ZXW

12. x 13 13. 7x 5 19

W

30. Because VX **** WX **** , a XWV a XVW by the Base

11. 6x 12

x7

Y X

14. (5x 7) 52

7x 14

5x 45

x2

x9

15. 3x 3x 4x 180

10x 180 x 18 16. By definition, an isosceles triangle is a triangle with at

32. Yes, T ABC is isosceles. (Note that two sides of the trian-

gle are radii of the circle, and all radii of a circle have the same length.) 4.3 Standardized Test Practice (p. 190) 33. A; ma EFD ma EFG 180

ma EFD 125 180 ma EFD 55

least 2 congruent sides. Every equilateral triangle has 3 congruent sides (therefore it has at least 2 congruent sides.) So, every equilateral triangle is also isosceles.

54



Chapter 4 continued 34. G; ma DEF ma EDF ma EFD 180

4. AB (3 0 )2 (4 0 )2

ma DEF 55 55 180

3 2 42

ma DEF 110 180

9 6 1

ma DEF 70

2 5 5


The distance between A and B is 5 units.

35. ma DBC ma ABD 42;

ma ABC 2(ma ABD) 2(42) 84 1 1 36. ma DBC (ma ABD) (56) 28;ˆ 2 2 ma ABC ma DBC 28

5. DE (3 1 )2 ( 2 4 )2

2 2 ( 6 )2 4 6 3 4 0

37. ma DBC 2(ma ABD) 2(75) 150;

6.3

ma ABC ma ABD 75 38. (x 20) 55

39. (x 8) 42

x 35

x 50

The distance between D and E is about 6.3 units. 6. FG ( 3 ( 2)) 2 ( 3 2 )2

( 1 )2 ( 5 )2

40. (2x 1) 81

1 5 2

2x 80

2 6

x 40

5.1


The distance between F and G is about 5.1 units.

41. 4 9 7 7 7

42. 1 21 1 1 1 1 11

43. 1 1 1 1

44. 4 00 2 0 0 2 20

Lesson 4.4

4.4 Guided Practice (p. 195) 1. Sample answer:

2. x2 82 102

x2 64 100

A

4.4 Activity (p. 191)

x2 36

c

b

x6

1. yes; 9 9 18 2. Yes, the sum of the areas from the two legs is equal to

the area of the square from the hypotenuse. 3. There are 9 full squares contained in the figure and

4 triangles. If you combine 2 triangles, they are 4 full squares. So, the area is (full squares) 2 (two triangles combined) 9 2(4) 17. 4. When squares are drawn from each side of a right

triangle, the sum of the area of the squares from the two legs is equal to the area of the square from the hypotenuse.

a 62 102 a2 36 100 a2 64 a 64 8

3. c2 72 82

c 49 64 2

a

B

If ma C 90, then a2 b2 c2. 3. x2 12 22

x2 1 4

x2 16 64

x 5 2

x 5 2.2

x2 48 x 48 6.9

5. AB (5 0 )2 (3 0 )2

5 2 32

2.

b2 152 172 b2 225 289 b2 64 b 6 4 8

3 4 5.8 units 6. CD (4 2 )2 (6 1 )2

2 2 52 4 5 2 2 9 5.4 units 7. FG (3 1 )2 (3 ( 3)) 2

2

2 2 62

c 113 10.6

4 6 3

c 113

4. x2 42 82

2 5 9

4.4 Checkpoint (pp. 193–194) 1.

C

4 0 6.3 Copyright © McDougal Littell Inc. All rights reserved.


55

Chapter 4 continued 4.4 Practice and Applications (pp. 195–198) 8. c2 92 122

a2 2304 2500

9. c2 92 402

c2 81 144

c2 81 1600

c 225

c 1681

2

a2 196 a 1 9 6 14;

2

c 225 15 10. c 65 72 2

2

2

c 1681 41 11. c 10 24 2

2

2

c2 4225 5184

c2 100 576

c2 9409

c2 676

c 9409 97 12. c2 122 352

c2 1369

c2 289

14. b2 242 252

b 576 625 2

23. Yes, because 202 212 400 441 841 292;

21

24. Yes, because 72 242 49 576 625 252;

c 289 17 a 1521 7921 2

b 49

a 6400

b7

a 80

2

24

25. No, because 52 122 25 144 169 142. 26. x2 23.262 47.572

x2 541.0276 2262.9049

16. b2 52 6 1 2

x2 2803.9325

b 25 61 2

x 2 8 0 3 .9 3 2 5 52.95

b2 36

Each support beam is approximately 52.95 m.

b6 17. b 3 5

18. c 4 6

2

2

b2 9 25

2

2

27. AB (5 2 )2 (2 ( 2)) 2

3 2 42

c2 16 36

b2 16

9 6 1

c2 52 c 52 7.2;

b 4; The side lengths form a Pythagorean Triple.

The side lengths do not form a Pythagorean Triple.

2 5 5 units 28. CD (6 0 )2 (8 2 )2 2 6 62

3 6 6 3

19. c2 72 112

c2 49 121 c2 170

7 2 8.5 units 29. EF (5 4 )2 (5 ( 1)) 2

1 2 62

c 170 13.04;

1 6 3

The side lengths do not form a Pythagorean Triple. 20. c 16 30 2

2

25

7

15. a2 392 892

2

2

29

20

13. c2 82 152

c2 64 225

c 1369 37

The side lengths form a Pythagorean Triple.

c 676 26

c2 144 1225

2

a2 482 502

22.

3 7 6.1 units

2

c 256 900 2

c2 1156 c 1156 34;

30.

y 1

x

1

R (1, 3) P (4, 4)

The side lengths form a Pythagorean Triple. 21. b2 92 242

b2 81 576 b2 495 b 495 22.2; The side lengths do not form a Pythagorean Triple.

(1, 6)

PQ (4 1 )2 ( 4 ( 6)) 2 32 22 9 4 1 3 3.6

1 1 )2 ( 3 ( 6)) 2 ( 2 )2 32 QR ( 4 9 1 3 3.6 So, PQ **** QR ****.

56



Chapter 4 continued 5 50 )2 (3 0 0 )2 34. A to B (6 31.

1 52 302 (2 2 5 0 9 0 )

y

(8, 5)

1 1 2 5 33.5 yd;

2 2

5 0 )2 (3 0 15 )2 B to C (6

x

6 52 152 4225 2 25

R (3, 2)

4 4 5 0 66.7 yd;

P (1, 6)

C to A (0 50 )2 (1 5 0 )2

PQ ( 8 ( 1)) 2 (5 ( 6)) 2 ( 7 )2 11 2

5 02 152 2500 2 25

49 2 11 170 13.04

2 7 2 5 52.2 yd

QR ( 8 3 )2 (5 ( 2)) 2 ( 11 )2 72

x2 82 162

35.

121 9 4 170 13.04

x2 64 256

So, PQ **** QR ****. 32.

x2 192 9 2 13.9 x 1

y

R (3, 6)

36.

P (5, 1)

2 2

y 62 102

37. y2 32 52

y 36 100

y2 9 25

2

2

x

y2 64

(5, 7)

y 6 4 8 12 8 4

PQ (5 ( 5) )2 (1 ( 7) 2 2

5 ( 3)) ( 7 6 ) QR ( 2

2

x 16 36

x2 49 9

x2 52

x2 58

2

x 5 2 7.2

( 2 ) ( 13 ) 2

11 4 7 x2 72 32

2

2

164 12.81

y 16 4

x 4 6 2

10 8 100 4 6 2

y2 16

x 58 7.6

2


41 69 173 13.2

1 3 )2 ( 4 5 )2 38. D; d (

So, PQ **** is not congruent to QR ****.

( 4 )2 ( 9 )2

33. A to B AD DE EB

1 6 1 8

(1 00 50 )2 (0 0 )2

9 7

(1 00 100 )2 (0 30 )2

39. F; c 332 562 2

(1 00 65 )2 (30 30 )2

1089 3136

502 ( 30 )2 352

4225

50 30 35 115 yd;

2 2 5 65 c 4

B to C BF FC

5 0 )2 (30 30 )2 (6


(0 0 )2 (30 15 )2

40. 7

41. 1.05

44. x 5 8

65 15 2

2

x3

65 15

42. 0

43. 0.02 45. 10 x 12

x2

46. 4x 28

80 yd;

x7

C to A CG GA

0 ) (15 0 ) (0 50 ) (0 0 ) (0 2

152 ( 50 )2 15 50

2

2

2

47. 6x 11 11

6x 0 x 0

65 yd



57

Chapter 4 continued 5. 252 ⱨ 242 72


2 48. 5 1 52. 4

2 49. 25 173 53. 1000

27 50. 50 1 54. 3

3 51. 25 1 55. 9

Quiz 2 (p. 198) 1. 3x 5 13

6. 262 ⱨ 242 72

625 ⱨ 576 49 625 625

676 625

The triangle is right.

The triangle is obtuse.

4.5 Guided Practice (p. 203) 2. (2x 1) 55

3x 18

2x 54

x6

x 27

3. x 5 4x 16

1. Sample answer: If you square the lengths of the two

shortest sides of a triangle and add the results, and the sum is equal to the square of the length of the longest side of the triangle, then the triangle is a right triangle. 2. 52 ⱨ 42 42

3x 21 x7 4. AB ( 2 3 )2 (3 0 )2

3. 142 ⱨ 62 122

25 ⱨ 16 16

196 ⱨ 36 144

25 32

196 180

The triangle is acute.

( 5 )2 32

2

2

225 ⱨ 144 81

34 5.8 units 5. AB ( 3 1 )2 ( 1 ( 2)) 2

( 4 )2 12

225 225 The triangle is right. 5. C; 112 ⱨ 22 102

16 1 17 4.1 units 6. AB ( 1 1 )2 ( 1 2 )2

( 2 )2 ( 3 )2

6. B; 82 ⱨ 52 72

121 ⱨ 4 100

64 ⱨ 25 49

121 104

64 74

7. D; The side lengths are equal so the triangle is equiangular. 8. A; 102 ⱨ 62 82

4 9

100 ⱨ 36 64

13 3.6 units

100 100

7. d 2 52 6 2

d 2 25 36

4.5 Practice and Applications (pp. 203–205)

d 11 2

9. 252 ⱨ 202 152

d 11 3.3 ft Lesson 4.5

676 ⱨ 100 576

625 625

676 676

11. 1 7 ⱨ 1 4 2

1. ma ACB 90

2. ma ACB 90

3. ma ACB 90

4. no

2

2

17 17

529 562

2

2

1. 62 ⱨ 52 22

2. 172 ⱨ 82 152

36 ⱨ 25 4

289 ⱨ 64 225

36 29;

289 289;

49 52



4. 242 ⱨ 242 72

12. 232 ⱨ 212 112

529 ⱨ 441 121

13. 7 ⱨ 6 4 2


17 ⱨ 1 16 The triangle is right.

4.5 Checkpoint (p. 202)

10. 262 ⱨ 102 242

625 ⱨ 400 225 The triangle is right.

4.5 Technology Activity (p. 199)



4. 15 ⱨ 12 9 2

25 9

3. 72 ⱨ 72 72

676 ⱨ 576 49

49 ⱨ 36 16

The triangle is acute. 14. 2 9 2 ⱨ 52 32

29 ⱨ 25 9 29 34 The triangle is acute.

15. 72 ⱨ 22 62

16. 122 ⱨ 82 62

49 ⱨ 4 36

144 ⱨ 64 36

49 ⱨ 49 49

576 ⱨ 576 49

49 98;

576 625

49 40

144 100





58



Chapter 4 continued 17. 222 ⱨ 162 132

18. 232 ⱨ 122 202

32. 142 ⱨ 102 112

33. 52 ⱨ 42 52

529 ⱨ 144 400

196 ⱨ 100 121

25 ⱨ 16 25

484 425

529 544

196 221

25 41





484 ⱨ 256 169

19. 212 ⱨ 162 92

20.

441 ⱨ 256 81

392 ⱨ 362 152

34.

1521 ⱨ 1296 225

21,025 ⱨ 289 20,736

2500 ⱨ 100 2401

1521 1521

21,025 21,025

2500 2501




22.

522 ⱨ 202 482

36.

5.52 ⱨ 5 2 52

676 ⱨ 324 289

2704 ⱨ 400 2304

30.25 ⱨ 5 25

676 613

2704 2704

30.25 30



The triangle is obtuse. 37.

7142 ⱨ 4032 5992

256 ⱨ 196 100

509.796 ⱨ 162,409 358,801

256 296

509,796 521,210


The triangle formed between the 3 cities is not a right triangle.

120 119 ⱨ 169 2

2

2

38. No; For Tallahassee to be directly south of Cincinnati, the

14,400 14,161 ⱨ 28,561

triangle formed by the three cities would have to be a right triangle. As shown in Exercise 37, the triangle is not a right triangle.

28,561 28,561; 48002 46012 ⱨ 66492 23,040,000 21,169,201 ⱨ 44,209,201

39. No; Sample answer: A counterexample is the right

triangle with side lengths 3, 4, and 5. Double each side to get 6, 8, and 10.

44,209,201 44,209,201; 13,5002 12,7092 ⱨ 18,5412

102 ⱨ 62 82

182,250,000 161,518,681 ⱨ 343,768,681

100 ⱨ 36 64

343,768,681 343,768,681 1012 ⱨ 202 992

26.

1225 ⱨ 441 784

10,201 10,201

1225 1225


The triangle is right. 28. 112 ⱨ 72 102

676 ⱨ 100 289

121 ⱨ 49 100

676 389

121 149



29. 92 ⱨ 6 7 2 42

30. 72 ⱨ 1 3 2 62

81 ⱨ 67 16

49 ⱨ 13 36

81 83

49 49



757 ⱨ 468 595 2

2

2

573,049 ⱨ 219,024 354,025 573,049 573,049 The triangle is right.

100 100

352 ⱨ 212 282

10,201 ⱨ 400 9801

27. 262 ⱨ 102 172

31.

502 ⱨ 102 492


23. 162 ⱨ 142 102

25.

35.

441 337 21. 262 ⱨ 182 172

24.

1452 ⱨ 172 1442

The side lengths 6, 8, and 10 form a right triangle, not an obtuse triangle. 40.

y

P (3, 4) 2

(5, 0) 4

x

R (6, 2)

PQ ( 3 5 )2 (4 0 )2 ( 8 )2 42 6 4 6 1 8 0 QR (5 ( 6) )2 (0 ( 2) )2 112 22 1 2 1 4 1 2 5 RP ( 3 ( 6)) 2 (4 ( 2)) 2 32 62 9 6 3 4 5 1 2 5 2 ⱨ 8 0 2 4 5 2 125 ⱨ 80 45 125 125 So, T PQR is right.



59

Chapter 4 continued 41.

Lesson 4.6

y

P (1, 2)

4.6 Activity (p. 206)

(4, 1) 1 1

1. Answers will vary, but in each column the second and

x

R (0, 1)

third entries should be approximately equal. 2. Yes; the distance from P to a vertex is equal to two

thirds the distance from that vertex to the midpoint of the opposite side.

PQ ( 1 4 ) (2 1 ) 2

2

( 5 )2 12

3. The results are the same for any triangle.

25 1 26


QR (4 0 )2 (1 ( 1) )2

1. Sample answer:

42 22 16 4 20 RP (0 ( 1)) ( 1 2 ) 2

3

12 ( 3 )2

5 5 3

So, T PQR is acute.

422 902 ⱨ 1002

b.

422 902 ⱨ 992

1764 8100 ⱨ 10,000

1764 8100 ⱨ 9801

9864 10,000

9864 9801

T ABC is obtuse.

T ABC is acute.

c. Sample answer:

16 ED 24 ED 8 2 3 KG JK JG 2 JG 4 JG 3 1 JG 4 3 JG 12; 2 KG (12) 8 3 2 6. QN PN 3 2 20 PN 3 3 (20) PN 2 30 PN; 5. KG JG

Choose AB to be 42 feet. AC 2 422 902 1764 8100 9864 99.3 feet AC 9864 If AB 42 feet and AC 99.3 feet, then T ABC would be a right triangle. 4.5 Mixed Review (p. 205) 43. x 67 45. 5x 5x 2x 180

3x 12

12x 180

x4

x 15

PQ QN PN


60

2 3 2 (24) 16; 3 BE ED BD

4. BE BD


51. 14

5

2

26 30

9 47. 32

2.5

2

26 ⱨ 20 10

2 46. 5 1 50. 4 2

3. Sample answer: 2.5

26 2 ⱨ 20 2 10 2

44. 3x 4 8

3

2. Sample answer:

1 9 10

42. a.

5

3

2

1 48. 4 16 52. 27

1 49. 3 1 53. 3 2


PQ 20 30 PQ 10


Chapter 4 continued 4.6 Guided Practice (p. 209) 1. The segment from a vertex of a triangle to the midpoint

of the opposite side is a median. 2. The centroid is the point where the three medians intersect. 3.

2 3 CD DE CE 2 CE 5 CE 3 1 5 CE 3 3(5) CE

14. CD CE

B 5 9

15 CE;

5

A

C

8

4. AD 4

5. AD 11

6. AD 3

33 CE;

CD DE CE

CD DE CE

CD 5 15

CD 11 33

CD 10 2 16. CD CE 3 CD DE CE 2 CE 9 CE 3 1 9 CE 3 3(9) CE

2 7. PT PS 3 2 (33) 22; 3 PT ST PS 22 ST 33 ST 11 2 BE BD 3 2 12 BD 3

8.

2 3 CD DE CE 2 CE 11 CE 3 1 11 CE 3 3(11) CE

15. CD CE

CD 22

27 CE; CD DE CE CD 9 27

3 (12) BD 2 18 BD;

CD 18 17. No. The median starts at a A and goes to BC ****. The

segment that starts at a A and goes to the centroid 2 D is the one that is the median. 3 2 2 So, AD AE (18) 12 and 3 3 AD DE AE

BE ED BD 12 ED 18 ED 6 4.6 Practice and Applications (pp. 210–211) 9. F

10.

12 DE 18

K

DE 6. 18. Sample answer:

E

G

2 3 2 (9) 6; 3 PN QP QN

11. PN QN

6 QP 9 QP 3 2 13. PN QN 3 2 (30) 20; 3 PN QP QN 20 QP 30 QP 10


J

L

2 3 2 (21) 14; 3 PN QP QN

12. PN QN

14 QP 21 QP 7

P

1 2 11 22 2 1 5 2 6 R , (2, 2) 2 2 5 11 6 2 S , (8, 4) 2 2

19. Q , (5, 0)

20. MS ( 1 8 )2 ( 2 4 )2 ( 9 )2 ( 6 )2

8 1 6 3 1 1 7 10.8 units 2 NR (11 2 )2 (2 2 )2 9 02 8 1 9 units 2 (5 5 )2 (6 0 )2 0 62 3 6 6 units PQ


61

Chapter 4 continued 2 2 3 3 Because PQ is vertical, T is 4 units directly below P.

21. PT PQ (6) 4

P(5, 6) so T(5, 6 4) (5, 2) 4.6 Standardized Test Practice (p. 211)

2 2 22. B; KN KM (36) 24; 3 3 KN NM KM 24 NM 36 NM 12 2 3 2 VT 12 3 3 VT (12) 18 2

23. G; VT PT


2. PR PQ, so ma Q ma R.

PQ QR, so ma R ma P. The order of the angles from largest to smallest is a Q, a R, a P. 3. ST TU, so ma U ma S.

TU US, so ma S ma T. The order of the angles from largest to smallest is a U, a S, a T. 4. ma J ma H, so HG GJ.

ma H ma G, so GJ JH. The order of the sides from longest to shortest is HG **** , GJ **** , JH **** . 5. ma F ma D, so DE EF.

ma D ma E, so EF FD. The order of the sides from longest to shortest is DE **** , EF **** , FD **** .

24. ma 1 45 25 180

6. ma B ma C, so AC BA.

ma 1 70 180

ma C ma A, so BA CB.

ma 1 110 25. ma 1 ma 1 60 180

2(ma 1) 60 180 2(ma 1) 120 ma 1 60 26. ma 1 35 90

ma 1 55 27. ma 1 60 65 125 28. ma 1 40 43 83 29. ma 1 32 115 147 4.6 Algebra Skills (p. 211)

In Exercises 30–41, sample answers are given. 3 10 2 5 12 21 30. , 31. , 32. , 15 50 4 10 16 28 1 6 4 48 24 120 33. , 34. , 35. , 10 60 7 84 52 260 2 4 2 30 2 56 36. , 37. , 38. , 9 135 5 140 9 18 8 48 3 90 2 200 39. , 40. , 41. , 11 66 4 120 3 300 Lesson 4.7 4.7 Checkpoint (pp. 213–214) 1. LM MN, so ma N ma L.

MN NL, so ma L ma M. The order of the angles from largest to smallest is a N, a L, a M.

62


The order of the sides from longest to shortest is AC **** , BA **** , CB **** . 7. The side lengths 5, 7, 13 can not form a triangle because

5 7 13. 8. The side lengths 6, 9, 12 can form a triangle because

6 9 12, 6 12 9, and 9 12 6. 9. The side lengths 10, 15, 25 can not form a triangle

because 10 15 25. 4.7 Guided Practice (p. 214) 1. The symbol “” means greater than, and the symbol

“” means less than. 2. a A is opposite the shortest side so it is the smallest angle. 3. AC **** is opposite the largest angle so it is the longest side. 4. ma D ma E ma F 180

32 ma E 103 180 ma E 135 180 ma E 45 a D is the smallest angle of T DEF. a F is the largest angle of T DEF. 5. EF **** is the shortest side of T DEF.

DE **** is the longest side of T DEF. 6. 1, 2, 3 can not form a triangle because 1 2 3. 7. 6, 10, 15 can form a triangle because 6 10 15,

6 15 10, and 10 15 6.

8. 12, 16, 30 can not form a triangle because 12 16 30. 9. 7, 8, 13 can form a triangle because 7 8 13,

7 13 8, and 8 13 7.


Chapter 4 continued 10. 4, 9, 16 can not form a triangle because 4 9 16. 11. 5, 5, 10 can not form a triangle because 5 5 10. 4.7 Practice and Applications (pp. 215–218) 12. a C is the smallest angle of T ABC;

a B is the largest angle of T ABC. 13. a R is the smallest angle of T PQR;

a Q is the largest angle of T PQR. 14. a H is the smallest angle of T GHF;

a F is the largest angle of T GHF. 15. RT **** is the shortest side of T RST;

TS is the longest side of T RST. **** 16. ma A ma B ma C 180

ma A 42 71 180 ma A 113 180 ma A 67 AC **** is the shortest side of T ABC. BA **** is the longest side of T ABC. 17. ma H ma J 90

35 ma J 90 ma J 55 KJ **** is the shortest side of T JKH. JH **** is the longest side of T JKH. 18. LK LM, so ma M ma K.

LM MK, so ma K ma L. a M, a K, a L 19. NQ PN, so ma P ma Q.

PN PQ, so ma Q ma N. a P, a Q, a N 20. TS TR, so ma R ma S.

TR RS, so ma S ma T. a R, a S, a T 21. AB BC, so ma C ma A.

BC AC, so ma A ma B. a C, a A, a B 22. XW YW, so ma Y ma X.

YW XY, so ma X ma W. a Y, a X, a W 23. EF DF, so ma D ma E.

DF DE, so ma E ma F. a D, a E, a F

24. Since 82 is the largest angle; the side opposite it (from

the sink to the refrigerator) should be the longest. But the labels show that the line from the refrigerator to the stove is longer. 25. No. A kitchen triangle could not have the side lengths of

9 ft, 3 ft, and 5 ft because 3 5 9. 26. ma B ma A, so AC BC.

ma A ma C, so BC AB. AC **** , CB **** , BA **** 27. ma E ma F 90

30 ma F 90 ma F 60 ma D ma F, so EF DE. ma F ma E, so DE DF. EF **** , DE **** , FD **** 28. ma H ma G ma J 180

35 ma G 120 180 ma G 155 180 ma G 25 ma J ma H, so HG GJ. ma H ma G, so GJ JH. HG **** , GJ **** , JH **** 29. ma A ma B ma C 180

44 95 ma C 180 139 ma C 180 ma C 41 ma B ma A, so AC BC. ma A ma C, so BC AB. AC **** , CB **** , BA **** 30. ma R ma Q 90

ma R 50 90 ma R 40 ma P ma Q, so QR PR. ma Q ma R, so PR PQ. RQ **** , PR **** , QP **** 31. ma F ma G ma H 180

ma F 58 62 180 ma F 120 180 ma F 60 ma H ma F, so FG GH. ma F ma G, so GH FH. GF **** , HG **** , FH **** 32. The side lengths do not satisfy the triangle inequality

because 2 3 5.



63

Chapter 4 continued 33. The side lengths do not satisfy the Triangle Inequality

d. Sample answer: 1 in., 7 in., 10 in.; 3 in., 5 in., 10 in.;

2 in., 7 in., 9 in.

because 3 10 14. 34.

C

4.7 Mixed Review (p. 218) 45. RT **** is the hypotenuse of T RST.

10 cm 7 cm

A

46. In T RST, RT **** is the side opposite a RST. 47. The legs of T RST are RS ST . **** and ****

B

4 cm

35. 10, 12, 22 can not form a triangle because 10 12 22. 36. 17, 9, 30 can not form a triangle because 17 9 30.

48. RT **** is the base of T RST. 49. ma 1 79 43 180

ma 1 122 180

37. 75

9.7 cm

ma 1 58

11.5 cm

59

50. ma 2 28 25 180

46

ma 2 53 180

13 cm

38.

5 cm

135

15

ma 2 127

2.6 cm 30

51. ma 3 56 90

7.1 cm

ma 3 34

39. Cutting across the empty lot is shorter than taking the

sidewalks because the path through the lot is the hypotenuse of a right triangle and the sum of the two legs must be greater than the hypotenuse.

4.7 Algebra Skills (p. 218) 52.

40. No, your friend can not be right because 1.8 1.5 4.6.

By the Triangle Inequality, you must be more than 4.6 1.8 2.8 miles from camp. 41. When the boom lines are shortened, the boom is raised. 42. 150 43. Yes, a ACB can be larger than a BAC; the maximum

length for AB **** is 150 ft. Since BC is only 100 ft, when AB BC, the angle opposite AB **** (a ACB) would be larger than the angle opposite BC **** (a BAC).

18 6 3 x 18x 18

x 6 5 15 15x 30

53.

x2 x 6 54. 7 7 7x 42

x1 27 9 55. 21 x 27x 189

x6 x 5 56. 72 8 8x 360

x7 7 49 57. 10 x 7x 490

x 45


x 70

Quiz 3 (p. 218)

44. a. 6 in. 5 in.

1. 142 ⱨ 62 112

6 in.

5 in. 8 in.

6 in.

2. 162 ⱨ 152 72

196 ⱨ 36 121

256 ⱨ 225 49

196 157

256 274

The triangle is obtuse. 7 in.

7 in.

8 in.

3.

8 in.


82 ⱨ 18 80 2

2

2

6724 ⱨ 324 6400 6724 6724

4 in.

2 in.

b.

c. 6 in.

5 in. 7 in.

8 in. 7 in.

3 in.

64


The triangle is right. 2 4. KN KM 3 2 (6) 4; 3 KN MN KM

2 3 2 (39) 26; 3 KN MN KM

5. KN KM

4 MN 6

26 MN 39

MN 2

MN 13


Chapter 4 continued 19. x 8 13

2 3 2 (60) 40; 3 KN MN KM

6. KN KM

x5 20. (2x 5) 49

2x 44

40 MN 60

x 22

MN 20

21. 4x 16

x4

7. ma L ma Q, so MQ LM.

22. x x 120

ma Q ma M, so LM QL.

2x 120

MQ **** , LM **** , QL ****

x 52

24. 4x 5x 4

ma Q ma P, so PM MQ.

4x

QP **** , PM **** , MQ ****

25. x2 402 302

9. ma P ma M, so MN NP.

26. x2 102 222

x2 1600 900

ma M ma N, so NP PM.

x2 100 484

x2 2500

MN **** , NP **** , PM ****

x2 384

5 0 0 50 x 2

Chapter 4 Summary and Review (pp. 219–223) 1. A triangle is a figure formed by three segments joining

three noncollinear points. 2. The side opposite the right angle is the hypotenuse of a

x 384 19.6

27. x 6 16 2

2

2

x2 36 256 x2 220 x 2 2 0 14.8 28. AB ( 3 0 )2 (4 0 )2

right triangle. 3. A corollary to a theorem is a statement that can be

proved easily using the theorem. 4. The congruent sides of an isosceles triangle are called

legs, and the third side is called the base. 5. A point that joins two sides of a triangle is called

a vertex. 6. A segment from a vertex of a triangle to the midpoint of

its opposite side is called a median. called the centroid of a triangle. 9. equilateral 12. acute

10. scalene 13. isosceles

14. ma 1 52 63 115 15. ma 1 102 40 142 16. ma 1 83 54 137 17. 16 x x 180

2x 16 180 2x 164 x 82 The other interior angles each measure 82. 18. 31 x 90

x 59 One of the interior angles is 90 and the other is 59.


( 3 )2 42 9 6 1 2 5 5 29. AB (6 2 )2 ( 4 5 )2

4 2 ( 9 )2 1 6 1 8 9 7 9.8

7. The point at which the medians of a triangle intersect is

11. right

x 128 180

x 60

8. ma M ma Q, so QP PM.

8. isosceles

23. 64 64 x 180

30. AB (3 ( 8)) 2 (7 7 )2

1 12 02 1 2 1 11 31. AB (0 ( 4)) 2 (6 ( 1)) 2

4 2 72 1 6 9 4 6 5 8.1 32. AB ( 6 ( 2)) 2 ( 7 ( 1)) 2

( 4 )2 ( 6 )2 1 6 6 3 5 2 7.2 33. AB ( 2 8 )2 (4 ( 3)) 2

( 10 )2 72 1 0 0 9 4 1 4 9 12.2


65

Chapter 4 continued 34. AB ( 3 9 )2 ( 6 1 )2

47.

( 12 )2 ( 7 )2 144 9 4

3 (28) CE 2 42 CE;

193 13.9 35. AB (0 5 )2 (6 4 )2

( 5 )2 22

CD DE CE

25 4

28 DE 42

29 5.4 36. 152 ⱨ 122 92

37. 162 ⱨ 72 112

225 ⱨ 144 81

256 ⱨ 49 121

225 225

256 170

right

obtuse

38. 222 ⱨ 182 192

39.

442 ⱨ 182 422

484 ⱨ 324 361

1936 ⱨ 324 1764

484 685

1936 2088

acute

acute

40. 122 ⱨ 102 32

41. 312 ⱨ 152 212

144 ⱨ 100 9

961 ⱨ 225 441

144 109

961 666

obtuse 2 42. KP KM 3 2 (18) 12; 3 KP PM KM 12 PM 18 PM 6 2 44. KP KM 3 2 (120) 80; 3 KP PM KM

obtuse 2 3 2 (42) 28; 3 KP PM KM

43. KP KM

48. QR PR, so ma P ma Q.

PR PQ, so ma Q ma R. a P, a Q, a R 49. TS US, so ma U ma T.

US UT, so ma T ma S. a U, a T, a S 50. XZ YZ, so ma Y ma X.

YZ XY, so ma X ma Z. a Y, a X, a Z 51. ma B ma A, so AC CB.

ma A ma C, so CB BA. AC **** , CB **** , BA **** 52. ma E ma D, so DF FE.

ma D ma F, so FE ED. DF **** , FE **** , ED **** 53. ma J ma H, so GH JG.

PM 14

ma H ma G, so JG HJ. GH **** , JG **** , HJ **** 54. Yes, the side lengths can form a triangle because

10 11 20, 10 20 11, and 11 20 10. 55. Yes, the side lengths can form a triangle because

21 23 25, 21 25 23, and 23 25 21. 56. No, the side lengths can not form a triangle because

3 10 15. 2 46. CD CE 3 2 16 CE 3 3 (16) CE 2 24 CE;

CD DE CE

CD DE CE

8 DE 12

16 DE 24

DE 4

DE 14

28 PM 42

80 PM 120 PM 40 2 45. CD CE 3 2 8 CE 3 3 (8) CE 2 12 CE;

2 CD CE 3 2 28 CE 3

DE 8

57. No, the side lengths can not form a triangle because

6 6 12. 58. Yes, the side lengths can form a triangle because

13 14 15, 13 15 14, and 14 15 13. 59. Yes, the side lengths can form a triangle because

2 3 4, 2 4 3, and 3 4 2. 60. No, the side lengths can not form a triangle because


11 11 20 and 11 20 11. 62. No, the side lengths can not form a triangle because

14 20 38.

66



Chapter 4 continued Chapter 4 Test (p. 224)

14.

1. T JLM

2. T JKL

3. T JKL

4. T JLM

5. x (2x 3) (3x 15) 180

6x 18 180

2 DC EC 3 2 (33) 22; 3 DC DE EC 22 DE 33 DE 11

6x 198 x 33 6. 12x 54 78

7. 3x 15

12x 132

x5

x 11

15. ma B ma A, so AC BC.

ma A ma C, so BC AB. AC **** , CB **** , BA **** 16. ma 3 44 90

8. x 102 142 2

ma 3 46;

x 100 196

ma 2 31;

x 296

31 ma 2 ma 1 180

2 2

x 296 17.2 9.

31 31 ma 1 180 62 ma 1 180

y

P (0, 0)

2

ma 1 118

2 x


5 8 18. (6, 8)

18. Yes, the side lengths can form a triangle because

20 24 40, 20 40 24, and 24 40 20. PQ ( 6 0 )2 ( 8 0 )2 ( 6 )2 ( 8 )2

31 45 50, 31 50 45, and 45 50 31.

100 10

Chapter 4 Standardized Test (p. 225)

y

(2, 6)


36 4 6 10.


1. D 2. H; ma BCD 90 35 125

P (2, 4)

3. A; 14 8 22

1 1

4. F

x

PQ (2 ( 2)) 2 (4 6 )2

5. C; JK (8 3 )2 ( 2 5 )2

5 2 ( 7 )2 2 5 9 4

42 ( 2 )2

7 4

16 4 20 4.5 11. 242 ⱨ 172 182

12. 132 ⱨ 62 92

576 ⱨ 289 324

169 ⱨ 36 81

576 613

169 117

acute

obtuse

13. 202 ⱨ 122 142

400 ⱨ 144 196 400 340

6. G; 8 ⱨ 72 42

7. B; x 3 4

64 ⱨ 49 16

x7

2

64 65 acute 8. x y 90

9. 10 8, so x y.

10. G

Chapter 4 Algebra Review (p. 227) 9 hours 24 hours

9 3 24 3

3 8

1.

obtuse



67

Chapter 4 continued 10 inches 2 12 inches

10 inches 2 feet

2.

10 inches 10 2 5 24 2 12 24 inches 40 minutes 4 hours

40 minutes 4 60 minutes

3.

40 minutes 40 40 1 240 minutes 240 40 6 6 16 ounces 20 ounces

6 pounds 20 ounces

4.

96 ounces 96 4 24 20 ounces 20 4 5 56 hours 8 days

56 hours 8 8 days 8

7 hours 1 day

60 miles 2 2 hours 2

30 miles 1 hour

5.

7 hours/day 60 miles 2 hours

6.

30 miles/hour $38 4 4 hours 4

$38 4 hours

$9.50 1 hour

7.

$9.50/hour $5.88 12 bagels

$5.88 12 12 bagels 12

$.49 1 bagel

8.

$.49/bagel

68



CHAPTER 4

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