CHAPTER 1
Section 1.1 Solutions --------------------------------------------------------------------------------
1 x 1 x
1. Solve for x: = 2. Solve for x: =
2 360 4 360
360 = 2 x, so that x = 180 . 360 = 4x, so that x = 90 .
1 x 2 x
3. Solve for x: − = 4. Solve for x: − =
3 360 3 360
360 = −3x, so that x = −120 . 720 = 2(360 ) = −3x, so that x = −240 .
(Note: The angle has a negative (Note: The angle has a negative
measure since it is a clockwise measure since it is a clockwise rotation.)
rotation.)
5 x 7 x
5. Solve for x: = 6. Solve for x: =
6 360 12 360
1800 = 5(360 ) = 6x, so that x = 300 . 2520 = 7(360 ) = 12x, so that x = 210 .
4 x 5 x
7. Solve for x: − = 8. Solve for x: − =
5 360 9 360
1440 = 4(360 ) = −5x, so that 1800 = 5(360 ) = −9x, so that
x = −288 . x = −200 .
(Note: The angle has a negative (Note: The angle has a negative
measure since it is a clockwise measure since it is a clockwise rotation.)
rotation.)
9. 10.
a) complement: 90 − 18 = 72 a) complement: 90 − 39 = 51
b) supplement: 180 − 18 = 162 b) supplement: 180 − 39 = 141
11. 12.
a) complement: 90 − 42 = 48 a) complement: 90 − 57 = 33
b) supplement: 180 − 42 = 138 b) supplement: 180 − 57 = 123
1
,Chapter 1
13. 14.
a) complement: 90 − 89 = 1 a) complement: 90 − 75 = 15
b) supplement: 180 − 89 = 91 b) supplement: 180 − 75 = 105
15. Since the angles with measures ( 4x ) and ( 6x ) are assumed to be
complementary, we know that ( 4x ) + ( 6x ) = 90. Simplifying this yields
(10x ) = 90 , so that x = 9. So, the two angles have measures 36 and 54 .
16. Since the angles with measures ( 3x ) and (15x ) are assumed to be
supplementary, we know that ( 3x ) + (15x ) = 180. Simplifying this yields
(18x ) = 180 , so that x = 10. So, the two angles have measures 30 and 150 .
17. Since the angles with measures ( 8x ) and ( 4x ) are assumed to be
supplementary, we know that ( 8x ) + ( 4x ) = 180. Simplifying this yields
(12x ) = 180 , so that x = 15. So, the two angles have measures 60 and 120 .
18. Since the angles with measures ( 3x + 15 ) and (10x + 10 ) are assumed to be
complementary, we know that ( 3x + 15 ) + (10x + 10 ) = 90. Simplifying this yields
(13x + 25 ) = 90 , so that (13x ) = 65 and thus, x = 5. So, the two angles have
measures 30 and 60 .
19. Since α + β + γ = 180 , we know 20. Since α + β + γ = 180 , we know
that that
117
+ γ = 180 and so, γ = 30 .
+ 33 110
+ γ = 180 and so, γ = 25 .
+ 45
= 150 = 155
21. Since α + β + γ = 180 , we know 22. Since α + β + γ = 180 , we know
that that
(
4 β ) + β + ( β ) = 180 and so, β = 30.
(
3β ) + β + ( β ) = 180 and so, β = 36.
= 6β = 5β
Thus, α = 4 β = 120 and γ = β = 30 . Thus, α = 3β = 108 and γ = β = 36 .
2
, Section 1.1
23. α = 180 − ( 53.3 + 23.6 ) = 103.1 24. β = 180 − (105.6 + 13.2 ) = 61.2
25. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 4 2 + 32 = c 2 , which
simplifies to c 2 = 25, so we conclude that c = 5 .
26. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 32 + 32 = c 2 , which
simplifies to c 2 = 18, so we conclude that c = 18 = 3 2 .
27. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 6 2 + b 2 = 10 2 , which
simplifies to 36 + b 2 = 100 and then to, b2 = 64, so we conclude that b = 8 .
28. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes a 2 + 7 2 = 12 2 , which
simplifies to a 2 = 95, so we conclude that a = 95 .
29. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 82 + 52 = c 2 , which
simplifies to c 2 = 89, so we conclude that c = 89 .
30. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 6 2 + 52 = c 2 , which
simplifies to c 2 = 61, so we conclude that c = 61 .
31. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 7 2 + b 2 = 112 , which
simplifies to b 2 = 72, so we conclude that b = 72 = 6 2 .
32. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes a 2 + 52 = 92 , which
simplifies to a 2 = 56, so we conclude that a = 56 = 2 14 .
3
, Chapter 1
33. Since this is a right triangle, we know from the Pythagorean Theorem that
( 7)
2
a 2 + b2 = c 2 . Using the given information, this becomes a 2 + = 52 , which
simplifies to a 2 = 18, so we conclude that a = 18 = 3 2 .
34. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 52 + b 2 = 10 2 , which
simplifies to b 2 = 75, so we conclude that b = 75 = 5 3 .
35. If x = 10 in., then the hypotenuse 36. If x = 8 m, then the hypotenuse of
of this triangle has length this triangle has length 8 2 ≈ 11.31 m .
10 2 ≈ 14.14 in.
37. Let x be the length of a leg in the given 45 − 45 − 90 triangle. If the
hypotenuse of this triangle has length 2 2 cm, then 2 x = 2 2, so that x = 2.
Hence, the length of each of the two legs is 2 cm .
38. Let x be the length of a leg in the given 45 − 45 − 90 triangle. If the hypotenuse
10 10
of this triangle has length 10 ft., then 2 x = 10, so that x = = = 5.
2 2
Hence, the length of each of the two legs is 5 ft.
39. The hypotenuse has length 40. Since 2 x = 6m x = 6 2 2 = 3 2m,
( )
2 4 2 in. = 8 in. each leg has length 3 2 m.
41. Since the lengths of the two legs of the given 30 − 60 − 90 triangle are x and
3 x, the shorter leg must have length x. Hence, using the given information, we
know that x = 5 m. Thus, the two legs have lengths 5 m and 5 3 ≈ 8.66 m, and
the hypotenuse has length 10 m.
42. Since the lengths of the two legs of the given 30 − 60 − 90 triangle are x and
3 x, the shorter leg must have length x. Hence, using the given information, we
know that x = 9 ft. Thus, the two legs have lengths 9 ft. and 9 3 ≈ 15.59 ft., and
the hypotenuse has length 18 ft.
4
Section 1.1 Solutions --------------------------------------------------------------------------------
1 x 1 x
1. Solve for x: = 2. Solve for x: =
2 360 4 360
360 = 2 x, so that x = 180 . 360 = 4x, so that x = 90 .
1 x 2 x
3. Solve for x: − = 4. Solve for x: − =
3 360 3 360
360 = −3x, so that x = −120 . 720 = 2(360 ) = −3x, so that x = −240 .
(Note: The angle has a negative (Note: The angle has a negative
measure since it is a clockwise measure since it is a clockwise rotation.)
rotation.)
5 x 7 x
5. Solve for x: = 6. Solve for x: =
6 360 12 360
1800 = 5(360 ) = 6x, so that x = 300 . 2520 = 7(360 ) = 12x, so that x = 210 .
4 x 5 x
7. Solve for x: − = 8. Solve for x: − =
5 360 9 360
1440 = 4(360 ) = −5x, so that 1800 = 5(360 ) = −9x, so that
x = −288 . x = −200 .
(Note: The angle has a negative (Note: The angle has a negative
measure since it is a clockwise measure since it is a clockwise rotation.)
rotation.)
9. 10.
a) complement: 90 − 18 = 72 a) complement: 90 − 39 = 51
b) supplement: 180 − 18 = 162 b) supplement: 180 − 39 = 141
11. 12.
a) complement: 90 − 42 = 48 a) complement: 90 − 57 = 33
b) supplement: 180 − 42 = 138 b) supplement: 180 − 57 = 123
1
,Chapter 1
13. 14.
a) complement: 90 − 89 = 1 a) complement: 90 − 75 = 15
b) supplement: 180 − 89 = 91 b) supplement: 180 − 75 = 105
15. Since the angles with measures ( 4x ) and ( 6x ) are assumed to be
complementary, we know that ( 4x ) + ( 6x ) = 90. Simplifying this yields
(10x ) = 90 , so that x = 9. So, the two angles have measures 36 and 54 .
16. Since the angles with measures ( 3x ) and (15x ) are assumed to be
supplementary, we know that ( 3x ) + (15x ) = 180. Simplifying this yields
(18x ) = 180 , so that x = 10. So, the two angles have measures 30 and 150 .
17. Since the angles with measures ( 8x ) and ( 4x ) are assumed to be
supplementary, we know that ( 8x ) + ( 4x ) = 180. Simplifying this yields
(12x ) = 180 , so that x = 15. So, the two angles have measures 60 and 120 .
18. Since the angles with measures ( 3x + 15 ) and (10x + 10 ) are assumed to be
complementary, we know that ( 3x + 15 ) + (10x + 10 ) = 90. Simplifying this yields
(13x + 25 ) = 90 , so that (13x ) = 65 and thus, x = 5. So, the two angles have
measures 30 and 60 .
19. Since α + β + γ = 180 , we know 20. Since α + β + γ = 180 , we know
that that
117
+ γ = 180 and so, γ = 30 .
+ 33 110
+ γ = 180 and so, γ = 25 .
+ 45
= 150 = 155
21. Since α + β + γ = 180 , we know 22. Since α + β + γ = 180 , we know
that that
(
4 β ) + β + ( β ) = 180 and so, β = 30.
(
3β ) + β + ( β ) = 180 and so, β = 36.
= 6β = 5β
Thus, α = 4 β = 120 and γ = β = 30 . Thus, α = 3β = 108 and γ = β = 36 .
2
, Section 1.1
23. α = 180 − ( 53.3 + 23.6 ) = 103.1 24. β = 180 − (105.6 + 13.2 ) = 61.2
25. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 4 2 + 32 = c 2 , which
simplifies to c 2 = 25, so we conclude that c = 5 .
26. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 32 + 32 = c 2 , which
simplifies to c 2 = 18, so we conclude that c = 18 = 3 2 .
27. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 6 2 + b 2 = 10 2 , which
simplifies to 36 + b 2 = 100 and then to, b2 = 64, so we conclude that b = 8 .
28. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes a 2 + 7 2 = 12 2 , which
simplifies to a 2 = 95, so we conclude that a = 95 .
29. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 82 + 52 = c 2 , which
simplifies to c 2 = 89, so we conclude that c = 89 .
30. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 6 2 + 52 = c 2 , which
simplifies to c 2 = 61, so we conclude that c = 61 .
31. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 7 2 + b 2 = 112 , which
simplifies to b 2 = 72, so we conclude that b = 72 = 6 2 .
32. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes a 2 + 52 = 92 , which
simplifies to a 2 = 56, so we conclude that a = 56 = 2 14 .
3
, Chapter 1
33. Since this is a right triangle, we know from the Pythagorean Theorem that
( 7)
2
a 2 + b2 = c 2 . Using the given information, this becomes a 2 + = 52 , which
simplifies to a 2 = 18, so we conclude that a = 18 = 3 2 .
34. Since this is a right triangle, we know from the Pythagorean Theorem that
a 2 + b2 = c 2 . Using the given information, this becomes 52 + b 2 = 10 2 , which
simplifies to b 2 = 75, so we conclude that b = 75 = 5 3 .
35. If x = 10 in., then the hypotenuse 36. If x = 8 m, then the hypotenuse of
of this triangle has length this triangle has length 8 2 ≈ 11.31 m .
10 2 ≈ 14.14 in.
37. Let x be the length of a leg in the given 45 − 45 − 90 triangle. If the
hypotenuse of this triangle has length 2 2 cm, then 2 x = 2 2, so that x = 2.
Hence, the length of each of the two legs is 2 cm .
38. Let x be the length of a leg in the given 45 − 45 − 90 triangle. If the hypotenuse
10 10
of this triangle has length 10 ft., then 2 x = 10, so that x = = = 5.
2 2
Hence, the length of each of the two legs is 5 ft.
39. The hypotenuse has length 40. Since 2 x = 6m x = 6 2 2 = 3 2m,
( )
2 4 2 in. = 8 in. each leg has length 3 2 m.
41. Since the lengths of the two legs of the given 30 − 60 − 90 triangle are x and
3 x, the shorter leg must have length x. Hence, using the given information, we
know that x = 5 m. Thus, the two legs have lengths 5 m and 5 3 ≈ 8.66 m, and
the hypotenuse has length 10 m.
42. Since the lengths of the two legs of the given 30 − 60 − 90 triangle are x and
3 x, the shorter leg must have length x. Hence, using the given information, we
know that x = 9 ft. Thus, the two legs have lengths 9 ft. and 9 3 ≈ 15.59 ft., and
the hypotenuse has length 18 ft.
4
