In this section, you will learn how to solve word problems on complementary and supplementary angles.
Problem 1 :
Two angles are complementary. If one of the angles is double the other angle, find the two angles.
Solution :
Let x be one of the angles.
Then the other angle is 2x.
Because x and 2x are complementary angles, we have
x + 2x = 90°
3x = 90
Divide each side by 3.
x = 30
And,
2x = 2(30) = 60
So, the two angles are 30° and 60°.
Problem 2 :
Two angles are complementary. If one angle is two times the sum of other angle and 3, find the two angles.
Solution :
Let x and y be the two angles which are complementary.
So, we have
x + y = 90° -----> (1)
Given : One angle is two times the sum of other angle and 3.
Then,
x = 2(y + 3)
x = 2y + 6 ----->(2)
Now, substitute (2y + 6) for x in (1).
(1)-----> 2y + 6 + y = 90
3y + 6 = 90
Subtract 6 from each side.
3y = 84
Divide each side by 3.
y = 28
Substitute 28 for y in (2).
(2)-----> x = 2(28) + 6
x = 56 + 6
x = 62
So, the two angles are 62° and 28°.
Problem 3 :
The measure of an angle is 3/4 of 60°. What is the measure of the complementary angle ?
Solution :
Let x be the measure of a complementary angle required.
Given : The measure of an angle is 3/4 of 60°.
Then,
3/4 ⋅ 60° = 45°
Because x and 45° are complementary angles, we have
x + 45 = 90
Subtract 45° from each side.
x = 45
So, the measure of the complementary angle is 45°.
Problem 4 :
Two angles are supplementary. If one angle is 36° less than twice of the other angle, find the two angles.
Solution :
Let x and y be the two angles which are supplementary.
Then,
x + y = 180° ----->(1)
From the information, "one angle is 36
Given : One angle is 36° less than twice of the other angle.
Then,
x = 2y - 36 ----->(2)
Now substitute (2y - 6) for x in (1),
(1)-----> 2y - 36 + y = 180
3y - 36 = 180
Add 36 to each side.
3y = 216
Divide each side by 3.
y = 72
Now, Substitute 72 for y in (2).
(2)-----> x = 2(72) - 36
x = 144 - 36
x = 108
So, the two angles are 108° and 72°.
Problem 5 :
An angle and its one-half are complementary. Find the angle.
Solution :
Let x be the required angle.
Then, one-half of the angle is x/2.
Given : An angle and its one-half are complementary.
Then,
x + x/2 = 90°
2x/2 + x/2 = 90
(2x + x) / 2 = 90
3x/2 = 90
Multiply each side by 2.
3x = 180
Divide each side by 3.
x = 60
So, the required angle is 60°.
Problem 6 :
Two angles are supplementary. If 5 times of one angle is 10 times of the other angle. Find the two angles.
Solution :
Let x and y be the two angles which are supplementary.
Then,
x + y = 180 -----(1)
Given : 5 times of one angle is 10 times of the other angle.
Then,
5x = 10y
Divide each side by 5.
x = 2y ----(2)
Substitute 2y for x in (1).
2y + y = 180
3y = 180
Divide each side by 3.
y = 60
Substitute 60 for y in (2).
x = 2(60)
x = 120
So, the two angles are 60° and 120°.
Problem 7 :
The measure of the supplement of angle A is 40 degrees larger than twice the measure of the complement of angle of A. What is the sum in degrees, of the measures of the supplement and complement of angle A ?
Solution :
Let x be the measure of angle A.
Then,
supplement of angle A = 180° - x
complement of angle A = 90° - x
Given : The measure of the supplement of angle A is 40 degrees larger than twice the measure of the complement of angle of A.
That is,
(180° - x) is 40° more than 2(90° - x)
180 - x = 2(90 - x) + 40
180 - x = 180 - 2x + 40
Subtract 180 from each side.
- x = - 2x + 40
Add 2x to each side.
x = 40
Then, supplement of angle A is
= 180 - x
= 180 - 40
= 140°
Complement of angle A is
= 90 - x
= 90 - 40
= 50°
The sum of supplement and complement of angle A is
= 140° + 50°
= 190°
So, the sum of supplement and complement of angle A is 190° degrees.
Problem 8 :
Twice the complement of an angle is 24 degrees less than its supplement. What is the measure of the angle ?
Solution :
Let x be the measure of angle.
Then,
supplement of angle = 180° - x
complement of angle = 90° - x
Given : Twice the complement of an angle is 24 degrees less than its supplement.
That is,
2(90° - x) is 24 degrees less than (180° - x)
Then,
2(90 - x) = (180 - x) - 24
180 - 2x = 180 - x - 24
Subtract 180 from each side.
- 2x = - x - 24
Add x to each side.
- x = - 24
Multiply each side by (-1).
x = 24
So, the measure of the angle is 24 degrees.
Problem 9 :
Two angles are supplementary. If the ratio of the measure of the smaller angle to that of the larger angle is 5:7. What is the measure of the smaller angle ?
Solution :
Given : The ratio of the measure of the smaller angle to that of the larger angle is 5 : 7.
Then,
measure of the smaller angle = 5x
measure of the larger angle = 7x
Given : Two angles are supplementary.
Then,
5x + 7x = 180
12x = 180
Divide each side by 12.
x = 15
Measure of the smaller angle is
= 5x
= 5(15)
= 75°
So, the measure of the smaller angle is 75°.
Problem 10 :
The measure of supplement of an angle is equal to the twice the measure of the angle. What is the measure in degrees, of the compliment of the angle ?
Solution :
Let x be the measure of the angle.
Then,
measure of supplement of the angle = 180° - x
Given : The measure of supplement of the angle is equal to the twice the measure of the angle.
That is,
(180° - x) is equal to 2x
Then,
180 - x = 2x
Add x to each side.
180 = 3x
Divide each side by 3.
60 = x
So, the measure of the angle is 60°.
Then, the measure of complement of the angle is
= 90 - 60
= 30°
So, the measure of compliment of the angle is 30 degrees.
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