**Chapter 3 Understanding Quadrilaterals Exercise 3.2**

Question 1.

Find x in the following figures.

Solution:

(a) We know that the sum of all the exterior angles of a polygon = 360°

125° + 125° + x = 360°

⇒ 250° + x = 360°

x = 360° – 250° = 110°

Hence x = 110°

(b) Here ∠y = 180° – 90° = 90°

and ∠z = 90° (given)

x + y + 60° + z + 70° = 360° [∵ Sum of all the exterior angles of a polygon = 360°]

⇒ x + 90° + 60° + 90° + 70° = 360°

⇒ x + 310° = 360°

⇒ x = 360° – 310° = 50°

Hence x = 50°

Question 3.

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

Solution:

Sum of all exterior angles of a regular polygon = 360°

Number of sides

Hence, the number of sides = 15

Question 4.

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

Solution:

Let re be the number of sides of a regular polygon.

Sum of all interior angles = (n – 2) × 180°

and, measure of its each angle

Hence, the number of sides = 24

Question 5.

(a) Is it possible to have a regular polygon with measure of each exterior angle a is 22°?

(b) Can it be an interior angle of a regular polygon? Why?

Solution:

(a) Since, the sum of all the exterior angles of a regular polygon = 360° which is not divisible by 22°.

It is not possible that a regular polygon must have its exterior angle 22°.

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

not a whole number.

Since number of sides cannot be in fractions.

It is not possible for a regular polygon to have its interior angle = 22°.

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

The measure of each interior angle

The minimum measure the angle of an equilateral triangle (n = 3) = 60°.

(b) From part (a) we can conclude that the maximum exterior angle of a regular polygon = 180° – 60° = 120°.