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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