## RD Sharma Solutions Class 9 Chapter 8 Ex 8.1

*Q 1 :Â Â Â Â Â Â Write the complement of each of the following angles:*

*Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (i)20Â° (ii)35Â° (iii)90Â° (iv)Â°(v)30Â°*

*Solution: *

** (i) **Given, the angle is 20Â°

As we know, the sum of an angle and its complement is 90Â°

Therefore, the complement of 20Â° = 90Â° â€“ 20Â° = 70Â°

** (ii) **Given, the angle is 35Â°

As we know, the sum of an angle and its complement is 90Â°.

Therefore, the complement of 35Â° = 90Â° – 35Â° = 55

** (iii) **Given, the angle is 90Â°

As we know, the sum of an angle and its complement is 90Â°.

Therefore, the complement of 90Â° = 90Â° – 90Â° = 0Â° )

** (iv)** Given, the angle is 77Â°

As we know, the sum of an angle and its complement is 90Â°.

Therefore, the complement of 77Â° = 90Â° – 77Â° = 13Â°

** (v)** Given, the angle is 30Â°

As we know, the sum of an angle and its complement is 90Â°

Therefore, the complement of 30Â° = 90Â° – 30Â° = 60Â°

*Q 2 : Write the supplement of each of the following angles:*

*Â Â Â Â Â Â Â Â Â Â (i)54Â° (ii)132Â° (iii)138Â°*

*Solution:*

** (i) G**iven here, the angle is 54Â°.

As we know, the sum of an angle and its supplement is 180Â°, therefore;

The supplement of angle 54Â° = 180Â° – 54Â° = 126Â°

** (ii)** The given angle is 132Â°,

As we know, the sum of an angle and its supplement is 180Â°, therefore;

The supplement of angle 132Â° =Â 180Â° – 132Â° = 48Â°

** (iii)** The given angle is 138Â°,

As we know, the sum of an angle and its supplement is 180Â°, therefore;

The supplement of angle of 138Â° = 180Â° â€“ 138Â° = 42Â°

*Q 3 : If an angle is 28Â°Â less than its complement, find its measure?*

** Solution:Â **Say, the angle is measured by â€˜ a â€˜ in degrees

Thus, its complement will be (90 – a)Â°

So, the Angle = Complement â€“ 28

a = ( 90 â€“ a ) â€“ 28

2a = 62

a = 31

Hence, the angle measured is 31Â°.

*Q 4 : If an angle is 30Â°Â more than half of its complement, find the measure of the angle?*

**Solution:**Â Say, the angle is measured by â€˜ a â€˜ in degrees

Thus, its complement will be (90-a)Â°

As per the given question;

Angle =30Â° + complement/2

a = 30Â° + ( 90 â€“ a )Â° / 2

\(3\frac{a}{2}=30+45\)

3a = 150Â°

a = 50Â°

Therefore, the measure of the angle is 50Â°.

*Q 5 : Two supplementary angles are in the ratio 4:5.Find the angles?*

** Solution: **Given here, two supplementary angles are in the ratio 4:5.

Say, the angles are 4a and 5a

Since, it is already mentioned, they are supplementary angles;

Hence, 4a + 5a = 180Â°

9a = 180Â°

a = 20Â°

So, 4a = 4 (20) = 80Â°

5(a) = 5 (20) = 100Â°

Therefore, angles are 80Â° and 100Â°.

*Q 6 : Two supplementary angles differ by 48Â°.Find the angles ?*

** Solution:Â Â **As per the given question, two supplementary angles differ byÂ 48Â°.

Say, one of the angles is measured by aÂ°.

Hence, its supplementary angle will be equal to (180 – a)Â°

According to the question;

(180 â€“ a )Â â€“ x = 48

(180 â€“ 48 ) = 2a

2 a = 132

a = 132/2

a = 66Â°

Therefore, 180 â€“ a = 114Â°

Hence, the two angles are 66Â° and 114Â°.

*Q 7 : An angle is equal to 8 times its complement. Determine its measure?*

** Solution:Â **As per the question, required angle = 8 times of its complement

Say, â€˜ a â€˜ is the angle to be measured, hence,

a = 8 times complement

a = 8 ( 90 â€“ a )

a = 720 â€“ 8a

a + 8a = 720

9a = 720

a = 80

Therefore, the required angle is 80Â°.

*Q 8 : If the angles (2x-10)Â°and (x-5)Â°Â are complementary, find x?*

** Solution:Â **As per given question, (2x-10)Â°Â and (x-5)

**Â are complementary to each other.**

*Â°*As we know, if angles are complementary, their sum will be equal to 90*Â°.*

( 2x â€“ 10 ) + ( x â€“ 5 ) = 90

3x -15 = 90

3x = 90 + 15

3x = 105

x = 105/3

x = 35*Â°*

Therefore, the value of x = 35*Â°*

*Q.9: If the complement of an angle is equal to the supplement of Thrice of itself, find the measure of the angle?*

** Solution:Â **Let us assume, the measure of the angle be â€˜ a â€˜.

Therefore,

The complementary angle for angle a = ( 90 â€“ a ) and

The supplementary of thrice of angle a = ( 180 – 3a )

As per the given question;

( 90 â€“ a ) = ( 180 â€“ 3a )

3a â€“ a = 180 â€“ 90

2a = 90

a = 90/2

a = 45

Therefore, the measured angle, a = 45Â°

*Q 10 : If an angle differs from its complement by 10Â°, find the angle?*

** Solution:Â **Let us assume the measured angle be â€˜ a â€˜

As per the given question, the angle differs from its compliment by 10Â°. Therefore,

a â€“ ( 90 â€“ a ) = 10

aÂ â€“ 90 +Â a = 10

2a = 90 + 10

2a = 100

aÂ = 100/2

aÂ = 50

Hence, the angle is 50Â°.

*Q 11 : If the supplement of an angle is 3 times its complement, find its angle ?*

** Solution:Â **Let us say the angle is a.

*As per the given question;*

Supplement of an angle = 3 times its complementary angle

Therefore,

Supplementary angle = 180 – a

And Complementary angle = 90 – a

As per the given condition;

180 â€“ a = 3 ( 90 â€“ a )

3x â€“ a = 270 – 180

2a = 90

a = 90/2

a = 45

Hence, the required angle 45Â°.

*Q 12 : If the supplement of an angle is two-third of itself. Determine the angle and its supplement?*

*Solution: As per the given question,*

Supplementary of an angle = 2/3rd of the angle

Let us assume, the angle is a.

Now, Supplementary of angle a = ( 180 – x)

According to the given condition;

180 â€“ a = 2/3a

( 180 â€“ a ) 3 = 2a

540 -3a = 2a

5a = 540

a = 540/5

a = 108

Therefore, the angle is 108Â° and the supplementary angle will be = 180 – 108 = 72Â°

*Q 13 : An angle is 14Â°Â more than its complementary angle. What is its measure?*

** Solution:Â **Let the angle to be measured is a.

The complementary angle of â€˜ a â€™ is ( 90 â€“ a )

According to the given question;

a â€“ ( 90 â€“ a ) = 14

a â€“ 90 +x = 14

2a = 90 + 14

2a = 104

a =104/2

a = 52Â°

Therefore, the required angle is 52Â°.

*Q 14: The measure of an angle is twice the measure of its supplementary angle. Find the measure of the angles?*

** Solution:**Â Let the angle to be measured is a.

Therefore, supplementary of angle a is (180 â€“ a)

As per given data;

a = 2 (180 â€“ a)

a = 360 -2a

3a = 360

a = 360/3

a = 120

Therefore, the required angle is 120Â°.