#### Chapter 8 Lines and Angle R.D. Sharma Solutions for Class 9th Math Exercise 8.1

Exercise 8.1

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)

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

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°

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

Solution

Let the measure of the required angle be x°.
Thus its complement becomes (90-x)°
According to the statement, the required angle is 30 more than half of its complementary angle that is the required angle x becomes, 5. Two supplementary angles are in the ratio 4:5. Find the angles.

Solution

Supplementary angles are in the ratio: 4:5
Let the angle be 4x and 5x.
It is given that they are supplementary angles.
Hence, 4x + 5x = 180
⇒ 9x = 180
⇒ x  = 20
Therefore, 4x  = 4×20 = 80
5x  = 5×20 = 100
Hence, angles are 80° and 100°.

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

Solution

Give that two supplementary angle will be (180-x)°
Let the angle measured be x°
Therefore, its supplementary angle will be (180-x)°
Now,
(180-x) - x = 48
⇒ (180-48) = 2x
⇒ 2x = 132
⇒ x = 132/2
⇒ x = 66
Hence, the angles are 66° and 114°.

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

Solution

Required angle be x.
A/q,
Required is 8 times its complement.
Now,
⇒ x = 8 time s complement
⇒ x = 8(90-x)
⇒ x = 720-8x
⇒ x+8x = 720
⇒ 9x = 720
⇒ x = 80
Therefore, measured angle is 80°

8. If the angles (2x − 10)° and (x − 5)° are complementary angles, 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 will be 35°.

9. If the complement of an angle is equal to the supplement of the thrice of it. Find the measure of the angle.

Solution

Let the angle measured be 'x'.
Its complementary angle is (90-x)° and,
Its supplementary angle will be (180-3x)°
Given that,
Supplementary of 4 times the angle = (180-3x)
A/q,
⇒ (90-x) = (180-3x)
⇒ 3x-x = 180-90
⇒ 2x = 90
⇒ x = 90/2
⇒ x = 45
Therefore, the measured angle x  = 45°

10. If an angle differs from its complement by 10°, find the angle.

Solution

Let the measured angle be 'x'.
Given that,
An angle is differ by 10°.
A/q,
x-(90-x) = 10
⇒ 2x = 90+10
⇒ 2x = 100
⇒ x = 100/2
⇒ x = 50
Therefore, the measure of the angle will be 50°.

11. If the supplement of an angle is three times its complement, find the angle.

Solution

Let the required angle be x.
Given,
Supplement of angle = 3 times the complement angle.
Supplementary angle = 180-x
Complementary angle = 90-x
A/q,
180-x = 3(90-x)
⇒ 180 -x = 270 - 3x
⇒ -x + 3x = 270 - 180
⇒ 2x = 90
⇒ x = 90/2
⇒ x = 45
Therefore, required angle will be 45°.

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

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

Solution

Let the required angle be x.
Complementary angle of 'x' is (90-x)°.
A/q,
x-(90-x) = 14
⇒ x-90+x = 14
⇒ 2x - 90 = 14
⇒2x = 90+14
⇒ x = 104/2
⇒ x = 52
Hence, the angle will be 52°.

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

Solution

Let the required angle be x.
Supplement of the angle x = (180-x)
A/q,
x = 2(180-x)°
⇒ x = 360-2x
⇒ x+2x = 360
⇒ 3x = 360
⇒ x = 360/3
⇒ x = 120
Hence, the value of the angle will be 120°.