In a polygon with all angles equal, one outer angle is twice an inner angle.

i) How much is each of its angle?

ii) How many sides does it have?

Let x be the measure of inner angle and 2x be the measure of outer angle.

Assume that the regular polygon has n sides (or angles)

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

n × x = (n – 2) × 180°

⇒ nx = (n – 2) × 180° … (1)

Sum of the exterior angle = 360°

⇒ n × 2x = 360°

⇒ 2nx = 360°

⇒

Substitute this value for x in equation (1)

⇒

⇒ 180° = 180°n – 360°

⇒ 180° = 180°n – 360°

⇒ 180°n = 180° + 360°

⇒ 180°n = 540°

⇒

⇒ n = 3

Sum of the angles of n-sided polygon = (n – 2) × 180°

⇒ S = (3 – 2) × 180°

⇒ S = 1 × 180°

⇒ S = 180°

Measure of each interior angle

⇒ x°

i. Measure of each interior angle is 60°

ii. Number of sides of this polygon is 3.