Exercise 3.2 Page: 44

1. Find x 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 = 15

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 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 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 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 regular polygon with 3 sides has the least possible minimum interior angle because the regular with minimum sides can be constructed with 3 sides at least.. Since, sum of interior angles of a triangle = 180°

Each interior angle = 180/3 = 60°

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