The measure of each interior angle of a regular polygon is five times the measure of its exterior angle. Find:

(i) measure of each interior angle,

(ii) measure of each exterior angle and

(iii) number of sides in the polygon

(i) It is given that each interior angle of a regular polygon is five times the measure of its exterior angle.

Let the exterior angles of the polygon be a.

Then, the interior angle = 5 x a = 5a

We know that the sum of the interior angle and the exterior angle is 180°.

⇒ a + 5a = 180°

⇒ 6a = 180°

⇒ a = $\dfrac{180°}{6}$

⇒ a = 30°

So, the exterior angle is 30°.

The interior angle = 5a = 5 x 30° = 150°

Hence, the interior angle of the polygon is 150°.

(ii) Exterior angle = a = 30°

Hence, the exterior angle of the polygon is 30°.

(iii) According to the properties of polygons, each interior angle of a regular polygon with n sides is $\dfrac{(2n - 4) \times 90°}{n}$.

⇒ $\dfrac{(2n - 4) \times 90°}{n}$ = 150°

⇒ (2n - 4) x 90° = 150°n

⇒ 180°n - 360° = 150°n

⇒ 180°n - 150°n = 360°

⇒ 30°n = 360°

⇒ n = $\dfrac{360°}{30°}$

⇒ n = 12

Hence, the number of sides of the polygon is 12.