RD Sharma Solutions Class 9 Lines And Angles Exercise 8.1

RD Sharma Class 9 Solutions Chapter 8 Ex 8.1 Free Download

RD Sharma Solutions Class 9 Chapter 8 Ex 8.1

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

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

Solution:

(i) Given, the angle is 20°

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

Therefore, the complement of 20° = 90° – 20° = 70°

(ii) Given, the angle is 35°

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

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

(iii) Given, the angle is 90°

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

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

(iv) Given, the angle is 77°

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

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

(v) Given, the angle is 30°

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

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

 

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

          (i)54° (ii)132° (iii)138°

Solution:

(i) Given here, the angle is 54°.

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

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

(ii) The given angle is 132°,

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

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

(iii) The given angle is 138°,

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

The supplement of angle of 138° = 180° – 138° = 42°

 

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

Solution: Say, the angle is measured by ‘ a ‘ in degrees

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

So, the Angle = Complement – 28

a = ( 90 – a ) – 28

2a = 62

a = 31

Hence, the angle measured is 31°.

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

Solution:  Say, the angle is measured by ‘ a ‘ in degrees

Thus, its complement will be (90-a)°

As per the given question;

Angle =30° + complement/2

a = 30° + ( 90 – a )° / 2

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

3a = 150°

a = 50°

Therefore, the measure of the angle is 50°.

 

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 = 180°

9a = 180°

a = 20°

So, 4a = 4 (20) = 80°

5(a) = 5 (20) = 100°

Therefore, angles are 80° and 100°.

 

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

Solution:  As per the given question, two supplementary angles differ by 48°.

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

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

According to the question;

(180 – a ) – x = 48

(180 – 48 ) = 2a

2 a = 132

a = 132/2

a = 66°

Therefore, 180 – a = 114°

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

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

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

Say, ‘ a ‘ is the angle to be measured, hence,

a = 8 times complement

a = 8 ( 90 – a )

a = 720 – 8a

a + 8a = 720

9a = 720

a = 80

Therefore, the required angle is 80°.

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

Solution: As per given question, (2x-10)° and (x-5)° are complementary to each other.

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

( 2x – 10 ) + ( x – 5 ) = 90

3x -15 = 90

3x = 90 + 15

3x = 105

x = 105/3

x = 35°

Therefore, the value of x = 35°

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

Solution: Let us assume, the measure of the angle be ‘ a ‘.

Therefore,

The complementary angle for angle a = ( 90 – a ) and

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

As per the given question;

( 90 – a ) = ( 180 – 3a )

3a – a = 180 – 90

2a = 90

a = 90/2

a = 45

Therefore, the measured angle, a = 45°

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

Solution: Let us assume the measured angle be ‘ a ‘

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

a – ( 90 – a ) = 10

a – 90 + a = 10

2a = 90 + 10

2a = 100

a = 100/2

a = 50

Hence, the angle is 50°.

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

Solution: 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 – a = 3 ( 90 – a )

3x – a = 270 – 180

2a = 90

a = 90/2

a = 45

Hence, the required angle 45°.

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 – a = 2/3a

( 180 – a ) 3 = 2a

540 -3a = 2a

5a = 540

a = 540/5

a = 108

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

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

Solution: Let the angle to be measured is a.

The complementary angle of ‘ a ’ is ( 90 – a )

According to the given question;

a – ( 90 – a ) = 14

a – 90 +x = 14

2a = 90 + 14

2a = 104

a =104/2

a = 52°

Therefore, the required angle is 52°.

 

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

Solution: Let the angle to be measured is a.

Therefore, supplementary of angle a is (180 – a)

As per given data;

a = 2 (180 – a)

a = 360 -2a

3a = 360

a = 360/3

a = 120

Therefore, the required angle is 120°.

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