NCERT Solutions Class 8th Maths Chapter – 3 Understanding Quadrilaterals Exercise – 3.2

1. Find × in the following figures.

Solution:

(a)

125° + m = 180° ⇒ m = 180° – 125° = 55° (Linear pair)
125° + n = 180° ⇒ n = 180° – 125° = 55° (Linear pair)
x = m + n (exterior angle of a triangle is equal to the sum of 2 opposite interior 2 angles)
⇒ x = 55° + 55° = 110°

(b)

Two interior angles are right angles = 90°
70° + m = 180° ⇒ m = 180° – 70° = 110° (Linear pair)
60° + n = 180° ⇒ n = 180° – 60° = 120° (Linear pair) The figure is having five sides and is a pentagon.

Thus, sum of the angles of pentagon = 540° 90° + 90° + 110° + 120° + y = 540°
⇒ 410° + y = 540° ⇒ y = 540° – 410° = 130°
x + y = 180° (Linear pair)
⇒ x + 130° = 180°
⇒ x = 180° – 130° = 50°

2. Find the measure of each exterior angle of a regular polygon of

(i) 9 sides
(ii) 15 sides

Solution: 

Sum of angles a regular polygon having side n = (n – 2)×180°

(i) Sum of angles a regular polygon having side 9 = (9 – 2)×180°= 7×180° = 1260°
Each interior angle=1260/9 = 140°
Each exterior angle = 180° – 140° = 40°
Or, Each exterior angle = sum of exterior angles/Number of angles = 360/9 = 40°

(ii) Sum of angles a regular polygon having side 15 = (15-2)×180°
= 13×180° = 2340°
Each interior angle = 2340/15 = 156°
Each exterior angle = 180° – 156° = 24°
Or, Each exterior angle = sum of exterior angles/Number of angles = 360/15 = 24°

3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Solution:

Each exterior angle = sum of exterior angles/Number of angles
24°= 360/ Number of sides
⇒ Number of sides = 360/24 = i15
Thus, the regular polygon has 15 sides.

4. How many sides does a regular polygon have if each of its interior angles is 165°?

Solution: 

Interior angle = 165°
Exterior angle = 180° – 165° = 15°

Number of sides = sum of exterior angles/ exterior angles
⇒ Number of sides = 360/15 = 24
Thus, the regular polygon has 24 sides.

5. (a) Is it possible to have a regular polygon with the measure of each exterior angle as 22°?
(b) Can it be an interior angle of a regular polygon? Why?

Solution:

(a) Exterior angle = 22°
Number of sides = sum of exterior angles/ exterior angle
⇒ Number of sides = 360/22 = 16.36
No, we can’t have a regular polygon with each exterior angle as 22° as it is not a divisor of 360°.

(b) Interior angle = 22°
Exterior angle = 180° – 22°= 158°
No, we can’t have a regular polygon with each exterior angle as 158° as it is not a divisor of 360°.

6. (a) What is the minimum interior angle possible for a regular polygon? Why?

(b) What is the maximum exterior angle possible for a regular polygon?

Solution:

(a) Equilateral triangle is a regular polygon with 3 sides and has the least possible minimum interior angle because the regular with minimum sides can be constructed with 3 sides at least. Since the sum of interior angles of a triangle = 180°
Each interior angle = 180/3 = 60°

(b) Equilateral triangle is a regular polygon with 3 sides and has the maximum exterior angle because the regular polygon with the least number of sides has the maximum exterior angle possible. Maximum exterior possible = 180 – 60° = 120°