1
, CHAPTER 1 ap
Section 1.1 Solutions --------------------------------------------------------------------------------
ap ap ap
1
x 1 x
a p a p a p ap a p a p a p
1. Solve for x:
a p ap ap a p ap 2. Solve for x:
a p ap ap a p ap
2 360∘ 4 360∘
360∘ 2x, so that x 180∘ .
ap ap a p ap a p ap ap ap 360∘ 4x, so that x 90∘ .
ap ap a p ap a p ap ap ap
1 x 2
x
3. Solve for x: 4. Solve for x:
ap a p a p a p a p a p
a p ap ap a p ap ap a p ap ap a p a p ap
3 360∘ 3 360∘
360∘ 3x, so that x 120∘ . (Not
ap ap ap ap a p ap ap ap ap 720∘ 2(360∘ ) 3x, so that x 240∘ .
ap ap ap ap ap ap ap a p ap ap ap ap
e: The angle has a negative measure s
ap ap ap ap ap ap ap (Note: The angle has a negative measu
a p ap ap ap ap a p
ince it is a clockwise rotation.)
ap ap ap ap ap re since it is a clockwise rotation.)
ap ap ap ap ap ap
5
x 7 x
a p a p a p ap apa p a p a p
5. Solve for x:
a p ap ap a p ap 6. Solve for x:
a p ap ap a p ap
6 360∘ 12 360∘
1800∘ 5(360∘ ) 6x, so that x 300∘ .
ap ap ap ap ap ap ap a p ap ap ap 2520∘ 7(360∘ ) 12x, so that x 210∘ .
ap ap ap ap ap ap ap a p ap ap ap
4 x x 5
7. Solve for x: 8. Solve for x:
ap a p a p a p a p a p a p
a p ap ap a p ap ap a p ap ap a p ap ap
5 360∘ 9 360∘
1440∘ 4(360∘ ) 5x, so thatap ap ap ap ap ap ap 1800∘ 5(360∘ ) 9x, so that
ap ap ap ap ap ap ap
x 288∘ .
ap ap ap x 200∘ .
ap ap ap
(Note: The angle has a negative meas
a p ap ap ap ap ap (Note: The angle has a negative measur
a p ap ap ap ap ap
ure since it is a clockwise rotation.)
ap ap ap ap ap ap e since it is a clockwise rotation.)
ap ap ap ap ap ap
9. 10.
a) complement: 90∘ 18∘ 72∘ a p ap a p a p a) complement: 90∘ 39∘ 51∘ a p ap ap a p a p
b) supplement: 180∘ 18∘ 162∘ a p ap a p a p b) supplement: 180∘ 39∘ 141∘ a p ap ap a p a p
11. 12.
a) complement: 90∘ 42∘ 48∘ a p ap ap a p a p a) complement: 90∘ 57∘ 33∘ a p ap ap a p a p
b) supplement: 180∘ 42∘ 138∘ a p ap ap a p a p b) supplement: 180∘ 57∘ 123∘ a p ap ap a p a p
2
, Section 1.1 ap
13. 14.
a) complement: 90∘ 89∘ 1∘ a p ap ap a p a p a) complement: 90∘ 75∘ 15∘ a p ap ap a p a p
b) supplement: 180∘ 89∘ 91∘ a p ap ap a p a p b) supplement: 180∘ 75∘ 105∘ a p ap ap a p a p
15. Since the angles with measures 4x∘ and 6x∘ are assumed to be complemen
a p ap ap ap ap ap a p a p ap ap ap ap ap
tary, we know that 4x∘ 6x∘ 90∘. Simplifying this yields
ap ap ap ap ap ap ap ap a p ap ap
10x∘ 90∘, ap ap ap a p so that x 9. So, the two angles have measures 36∘and 54∘ .
ap a p ap ap a p ap ap ap ap ap a p ap ap
16. Since the angles with measures 3x∘ and 15x∘ are assumed to be supplemen
a p ap ap ap ap ap a p a p ap ap ap ap ap
tary, we know that 3x∘ 15x∘ 180∘. Simplifying this yields
ap ap ap ap ap ap ap ap a p ap ap
18x∘ 180∘, so that ap ap ap ap ap x 10. So, the two angles have measures 30∘ and 150∘ .
ap ap a p ap ap ap ap ap a p ap ap ap
17. Since the angles with measures 8x∘ and
a p ap ap ap ap a p ap a p 4x∘ are assumed to be supplementa ap ap ap ap ap
ry, we know that 8x∘ 4x∘ 180∘. Simplifying this yields
ap ap ap ap ap ap ap ap a p ap ap
12x∘ 180∘, ap ap a p so that x 15. So, the two angles have measures 60∘ and 120∘ .
ap a p ap ap a p ap ap ap ap ap a p ap ap ap
18. Since the angles with measures 3x 15∘and 10x 10∘are assumed to be co
a p ap ap ap ap a p ap ap a p ap ap ap ap ap ap
mplementary, we know that 3x 15∘ 10x 10∘ 90∘. Simplifying this yields ap ap ap ap ap ap ap ap ap ap a p ap ap
13x 25∘ 90∘, ap ap ap ap a p so that 13x∘ 65∘ and thus, x 5. So, the two angles have meas
ap ap ap ap a p ap a p ap ap a p ap ap ap ap ap
ures 30∘and 60∘ .
a p ap ap
19. Since 180∘, we know th
a p ap ap ap ap ap a p ap a p ap ap 20. Since 180∘, we know th
a p ap ap ap ap ap a p ap a p ap ap
at at
1 17∘ –33∘ 180∘ and so, 30∘ . 1 10∘ –45∘ 180∘ and so, 25∘ .
– –
ap ap ap ap ap ap ap a p ap ap ap ap ap ap ap ap ap ap ap ap ap ap
ap ap
ap150∘ ap155∘
21. Since 180∘, we know th
a p ap ap ap ap ap a p ap a p ap ap 22. Since 180∘, we know th
a p ap ap ap ap ap a p ap a p ap ap
at at
4 180∘ and so, 30∘.
ap ap ap ap ap ap ap ap ap ap ap ap ap 3 180∘ and so, 36∘.
ap ap ap ap ap ap ap ap ap ap ap ap ap
–– –– –– ––
ap6ap ap5
Thus, 4 120∘ and 30∘ .
a p ap ap a p ap a p ap a p ap a p ap ap Thus, 3 108∘ and 36∘ .
a p ap ap a p ap a p ap a p ap a p ap ap
3
,
, CHAPTER 1 ap
Section 1.1 Solutions --------------------------------------------------------------------------------
ap ap ap
1
x 1 x
a p a p a p ap a p a p a p
1. Solve for x:
a p ap ap a p ap 2. Solve for x:
a p ap ap a p ap
2 360∘ 4 360∘
360∘ 2x, so that x 180∘ .
ap ap a p ap a p ap ap ap 360∘ 4x, so that x 90∘ .
ap ap a p ap a p ap ap ap
1 x 2
x
3. Solve for x: 4. Solve for x:
ap a p a p a p a p a p
a p ap ap a p ap ap a p ap ap a p a p ap
3 360∘ 3 360∘
360∘ 3x, so that x 120∘ . (Not
ap ap ap ap a p ap ap ap ap 720∘ 2(360∘ ) 3x, so that x 240∘ .
ap ap ap ap ap ap ap a p ap ap ap ap
e: The angle has a negative measure s
ap ap ap ap ap ap ap (Note: The angle has a negative measu
a p ap ap ap ap a p
ince it is a clockwise rotation.)
ap ap ap ap ap re since it is a clockwise rotation.)
ap ap ap ap ap ap
5
x 7 x
a p a p a p ap apa p a p a p
5. Solve for x:
a p ap ap a p ap 6. Solve for x:
a p ap ap a p ap
6 360∘ 12 360∘
1800∘ 5(360∘ ) 6x, so that x 300∘ .
ap ap ap ap ap ap ap a p ap ap ap 2520∘ 7(360∘ ) 12x, so that x 210∘ .
ap ap ap ap ap ap ap a p ap ap ap
4 x x 5
7. Solve for x: 8. Solve for x:
ap a p a p a p a p a p a p
a p ap ap a p ap ap a p ap ap a p ap ap
5 360∘ 9 360∘
1440∘ 4(360∘ ) 5x, so thatap ap ap ap ap ap ap 1800∘ 5(360∘ ) 9x, so that
ap ap ap ap ap ap ap
x 288∘ .
ap ap ap x 200∘ .
ap ap ap
(Note: The angle has a negative meas
a p ap ap ap ap ap (Note: The angle has a negative measur
a p ap ap ap ap ap
ure since it is a clockwise rotation.)
ap ap ap ap ap ap e since it is a clockwise rotation.)
ap ap ap ap ap ap
9. 10.
a) complement: 90∘ 18∘ 72∘ a p ap a p a p a) complement: 90∘ 39∘ 51∘ a p ap ap a p a p
b) supplement: 180∘ 18∘ 162∘ a p ap a p a p b) supplement: 180∘ 39∘ 141∘ a p ap ap a p a p
11. 12.
a) complement: 90∘ 42∘ 48∘ a p ap ap a p a p a) complement: 90∘ 57∘ 33∘ a p ap ap a p a p
b) supplement: 180∘ 42∘ 138∘ a p ap ap a p a p b) supplement: 180∘ 57∘ 123∘ a p ap ap a p a p
2
, Section 1.1 ap
13. 14.
a) complement: 90∘ 89∘ 1∘ a p ap ap a p a p a) complement: 90∘ 75∘ 15∘ a p ap ap a p a p
b) supplement: 180∘ 89∘ 91∘ a p ap ap a p a p b) supplement: 180∘ 75∘ 105∘ a p ap ap a p a p
15. Since the angles with measures 4x∘ and 6x∘ are assumed to be complemen
a p ap ap ap ap ap a p a p ap ap ap ap ap
tary, we know that 4x∘ 6x∘ 90∘. Simplifying this yields
ap ap ap ap ap ap ap ap a p ap ap
10x∘ 90∘, ap ap ap a p so that x 9. So, the two angles have measures 36∘and 54∘ .
ap a p ap ap a p ap ap ap ap ap a p ap ap
16. Since the angles with measures 3x∘ and 15x∘ are assumed to be supplemen
a p ap ap ap ap ap a p a p ap ap ap ap ap
tary, we know that 3x∘ 15x∘ 180∘. Simplifying this yields
ap ap ap ap ap ap ap ap a p ap ap
18x∘ 180∘, so that ap ap ap ap ap x 10. So, the two angles have measures 30∘ and 150∘ .
ap ap a p ap ap ap ap ap a p ap ap ap
17. Since the angles with measures 8x∘ and
a p ap ap ap ap a p ap a p 4x∘ are assumed to be supplementa ap ap ap ap ap
ry, we know that 8x∘ 4x∘ 180∘. Simplifying this yields
ap ap ap ap ap ap ap ap a p ap ap
12x∘ 180∘, ap ap a p so that x 15. So, the two angles have measures 60∘ and 120∘ .
ap a p ap ap a p ap ap ap ap ap a p ap ap ap
18. Since the angles with measures 3x 15∘and 10x 10∘are assumed to be co
a p ap ap ap ap a p ap ap a p ap ap ap ap ap ap
mplementary, we know that 3x 15∘ 10x 10∘ 90∘. Simplifying this yields ap ap ap ap ap ap ap ap ap ap a p ap ap
13x 25∘ 90∘, ap ap ap ap a p so that 13x∘ 65∘ and thus, x 5. So, the two angles have meas
ap ap ap ap a p ap a p ap ap a p ap ap ap ap ap
ures 30∘and 60∘ .
a p ap ap
19. Since 180∘, we know th
a p ap ap ap ap ap a p ap a p ap ap 20. Since 180∘, we know th
a p ap ap ap ap ap a p ap a p ap ap
at at
1 17∘ –33∘ 180∘ and so, 30∘ . 1 10∘ –45∘ 180∘ and so, 25∘ .
– –
ap ap ap ap ap ap ap a p ap ap ap ap ap ap ap ap ap ap ap ap ap ap
ap ap
ap150∘ ap155∘
21. Since 180∘, we know th
a p ap ap ap ap ap a p ap a p ap ap 22. Since 180∘, we know th
a p ap ap ap ap ap a p ap a p ap ap
at at
4 180∘ and so, 30∘.
ap ap ap ap ap ap ap ap ap ap ap ap ap 3 180∘ and so, 36∘.
ap ap ap ap ap ap ap ap ap ap ap ap ap
–– –– –– ––
ap6ap ap5
Thus, 4 120∘ and 30∘ .
a p ap ap a p ap a p ap a p ap a p ap ap Thus, 3 108∘ and 36∘ .
a p ap ap a p ap a p ap a p ap a p ap ap
3
,
