AB, BC and CD are three consecutive sides of a regular polygon. If angle BAC = 20°, find:

(i) its each interior angle

(ii) its each exterior angle

(iii) the number of sides in the polygon.

(i) As it is given that the polygon is regular,

AB = BC

∠ BCA = ∠ BAC

∠ BCA = 20°

Thus, ∠ BAC = 20°

In triangle ABC, the sum of all angles is 180°.

∠ BAC + ∠ BCA + ∠ CBA = 180°

⇒ 20° + 20° + ∠ CBA = 180°

⇒ 40° + ∠ CBA = 180°

⇒ ∠ CBA = 180° - 40°

⇒ ∠ CBA = 140°

Hence, each interior angle is 140°.

(ii) As we know that sum of the interior angle and the exterior angle of a polygon is 180°.

Thus, Exterior angle = 180° - Interior angle

= 180° - 140°

= 40°

Hence, each exterior angle is 40°.

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

⇒ $\dfrac{(2n - 4) \times 90°}{n}$ = 140°

By cross-multiplying, we get

⇒ (2n - 4) x 90° = 140°n

⇒ 180°n - 360° = 140°n

⇒ 180°n - 140°n = 360°

⇒ 40°n = 360°

⇒ n = $\dfrac{360°}{40°}$

⇒ n = 9

Hence, the number of sides is 9.