Question

Answer the following questions:
1) The measure of an angle is five times its compliment. What is the measure of the angle?

2) Two complementary angles are such that twice the measure of the one is equal to three times the measure of the other.What does the larger of the two measures?

3) An angle is one fifth of its supplement. What is the measure of the angle?

Answer 1

1 )

Let the angle be x .

x is 5 times the compliment .

Compliment of x = 90 - x

Hence :

x = 5 ( 90 - x )

⇒ x = 450 - 5 x

⇒ 6 x = 450

⇒ x = 450/6

⇒ x = 75

The angle is 75 degrees.

2 )

Let one angle be x.

Another angle is 90 - x

2 x = 3 ( 90 - x )

⇒ 2 x = 270 - 3 x

⇒ 5 x = 270

⇒ x = 270/5

⇒ x = 54

90 - x = 36

So the larger angle is 54 degrees

3 )

Let one angle be x .

Supplement = 180 - x

x = 5 ( 180 - x )

⇒ x = 900 - 5 x

⇒ 6 x = 900

⇒ x = 900/6

⇒ x = 150

The angle is 180 - 150 = 30

The answer is 30 degrees.

Answer 2

Some Definitions

a) Complementary Angles-

A pair of angle is said to be complementary, if their sum is 90°.

For two complementary angles, θ₁ and θ₂

⇒ θ₁ + θ₂= 90°

Also, complement of angle θ= 90- θ

b) Supplementary Angles-

A pair of angle is said to be supplementary, if their sum is 180°.

For two supplementary angles, θ₁ and θ₂

⇒ θ₁ + θ₂= 180°

Also, supplement of angle θ= 180- θ

$\rule{190}{2}$

1)

Given:

An angle is five times of its complement

To Find:

• Measure of given angle

Solution:

Let the required angle be θ

Then its complement= 90-θ

According to the question,

⇒ θ= 5(90-θ)

⇒ θ= 450-5θ

⇒ θ+5θ = 450

⇒ 6θ = 450

⇒ θ = 450/6

⇒ θ = 75°

$\rule{190}{2}$

2)

Given:

Two complementary angles are such that twice the measure of the one is equal to three times the measure of the other

To Find:

• Measure of larger of the two angles

Solution:

Let the first angle be θ

Then its complement or second angle= 90-θ

According to the question,

⇒ 2θ= 3(90-θ)

⇒ 2θ= 270-3θ

⇒ 2θ+3θ = 270

⇒ 5θ = 270

⇒ θ = 270/5

⇒ θ = 54°

So, First angle= θ = 54°

Second Angle= 90-θ = 90°-54°= 36°

Hence, the larger angle is 54°

$\rule{190}{2}$

3)

Given:

An angle is one-fifth times of its supplement

To Find:

• Measure of given angle

Solution:

Let the required angle be θ

Then its supplement= 180-θ

According to the question,

$\longrightarrow\sf{\theta=\dfrac{1}{5}(180-\theta)}$

$\longrightarrow\sf{5\theta=180-\theta}$

$\longrightarrow\sf{5\theta+\theta=180}$

$\longrightarrow\sf{6\theta=180}$

$\longrightarrow\sf{\theta=\dfrac{180}{6}}$

$\longrightarrow\sf\pink{\theta=30^{\circ}}$

Hence, the required angle is 30°.