Chapter 8: Lines And Angles Exercise – 8.1

Question: 1

Write the complement of each of the following angles:

(i) 20°

(ii) 35°

(iii) 90°

(iv) 77°

(v) 30°

Solution:

(i) Given angle is 20

Since, the sum of an angle and its compliment is 90

Hence, its compliment will be (90 - 20 = 70)

(ii) Given angle is 35

Since, the sum of an angle and its compliment is 90

Hence, its compliment will be (90 - 35 = 55)

(iii) Given angle is 90

Since, the sum of an angle and its compliment is 90

Hence, its compliment will be (90 - 90 = 0)

(iv) Given angle is 77

Since, the sum of an angle and its compliment is 90

Hence, its compliment will be (90 - 77 = 13)

(v) Given angle is 30

Since, the sum of an angle and its compliment is 90

Hence, its compliment will be (90 - 30 = 60)

Question: 2

Write the supplement of each of the following angles:

(i) 54°

(ii) 132°

(iii) 138°

Solution:

(i) The given angle is 54,

Since the sum of an angle and its supplement is 180,

Hence, Its supplement will be (180 - 54 = 126)

(ii) The given angle is 132,

Since the sum of an angle and its supplement is 180,

Hence, its supplement will be 180 - 132 = 48

(iii) The given angle is 138,

Since the sum of an angle and its supplement is 180,

Hence, Its supplement will be 180 - 138 = 42

Question: 3

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

Solution:

Let the angle measured be 'x' in degrees

Hence, Its complement will be 90 − x°

Angle = Complement - 28

x = (90 - x) - 28

2x = 62

x = 31

Therefore, angle measured is 31°

Question: 4

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

Solution:

Let the measured angle be 'x'

Hence its complement will be (90 - x)

It is given that,

Angle = 30 + complement/2

x = 30 + (90 - x) /2

3x = 150

x = 50

Therefore the angle is 50°

Question: 5

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

Solution:

Supplementary angles are in the ratio 4: 5

Let the angles be 4x and 5x

It is given that they are supplementary angles

Hence 4x + 5x = 180

9x = 180

x = 20

Hence, 4x = 4 (20) = 80

5(x) = 5 (20) = 100

Hence, angles are 80 and 100

Question: 6

Two supplementary angles differ by 48°.Find the angles?

Solution:

Given that two supplementary angles differ by 48°

Let the angle measured be x°

Therefore, Its supplementary angle will be (180 - x)°   

It is given that:

(180 - x) - x = 48

(180 - 48)  = 2x

2x = 132

x = 132/2

x = 66

Hence, 180 - x = 114°

Therefore, the angles are 66 and 114.

Question: 7

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

Solution:

It is given that required angle = 8 times its complement

Let 'x' be the measured angle

angle = 8 times complement

angle = 8 (90 - x)

x = 8(90 - x)

x = 720 - 8x

x + 8x = 720

9x = 720

x = 80

Therefore measured angle is 80.

Question: 8

If the angles (2x − 10)° and (x − 5)° are complementary, find x?

Solution:

Given that (2x − 10)° and (x − 5)° are complementary

Since angles are complementary, their sum will be 90

(2x - 10) + (x - 5) = 90

3x -15 = 90

3x = 90 + 15

3x = 105

x = 105/3

x = 35

Hence, the value of x = (35)°

Question: 9

If the compliment of an angle is equal to the supplement of Thrice of itself, find the measure of the angle?

Solution:

Let the angle measured be 'x' say.

Its complementary angle is (90 - x) and

Its supplementary angle is (180 - 3x)

Given that, Supplementary of 4 times the angle = (180 - 3x)

According to the given information;

(90 - x) = (180 - 3x)

3x - x = 180 - 90

2x = 90

x = 90/2

x = 45

Therefore, the measured angle x = (45)°

Question: 10

If an angle differs from its complement by (10)°, find the angle ?

Solution:

Let the measured angle be 'x' say

Given that,

The angles measured will differ by (20)°

x - (90 - x) = 10

x - 90 + x = 10

2x = 90 + 10

2x = 100

x = 100/2

x = 50

Therefore the measure of the angle is (50)°

Question: 11

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

Solution:

Let the angle in case be 'x'

Given that,

Supplement of an angle = 3 times its complementary angle

Supplementary angle = 180 - x

Complementary angle = 90 - x

Applying given data,

180 - x = 3 (90 - x)

3x - x = 270 - 180

2x = 90

x = 90/2

x = 45

Therefore, the angle in case is 45°

Question: 12

If the supplement of an angle is two third of itself. Determine the angle and its supplement?

Solution:

Supplementary of an angle = 2/3 angle

Let the angle in case be 'x',

Supplementary of angle x will be (180 - x)

It is given that

180 - x = 2/3 x

(180 - x) 3 = 2x

540 - 3x = 2x

5x = 540

x = 540/5

x = 108

Hence, supplementary angle = 180 - 108 = 72

Therefore, angles in case are 108° and supplementary angle is 72°

Question: 13

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

Solution:

Let the angle in case be 'x'

Complementary angle of 'x' is (90 - x)

From given data,

x - (90 - x) = 14

x - 90 + x = 14

2x = 90 + 14

2x = 104

x = 104/2

x = 52

Hence the angle in case is found to be 52°

Question: 14

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

Solution:

Let the angle in case be 'x'

The supplementary of a angle x is (180 - x)°

Applying given data:

x = 2 (180 - x)

x = 360 - 2x

3x = 360

x = 360/3

x = 120

Therefore the value of the angle in case is 120°