AB, BC and CD are the three consecutive sides of a regular polygon. If ∠BAC = 15°; find,

(i) each interior angle of the polygon.

(ii) each exterior angle of the polygon.

(iii) number of sides of the polygon.

(i) In △ ABC,

⇒ AB = BC (As, ABCD is a regular polygon)

⇒ ∠BCA = ∠BAC = 15° (In a triangle angles opposite to equal sides are equal)

By angle sum property of triangle,

⇒ ∠BCA + ∠BAC + ∠ABC = 180°

⇒ 15° + 15° + ∠ABC = 180°

⇒ 30° + ∠ABC = 180°

⇒ ∠ABC = 180° - 30° = 150°.

Since, each interior angle of a regular polygon are equal.

Hence, each interior angle of a regular polygon = 150°.

(ii) We know that,

At each vertex of every polygon,

⇒ Exterior angle + Interior angle = 180°

⇒ Exterior angle + 150° = 180°

⇒ Exterior angle = 180° - 150° = 30°.

Hence, each exterior angle of a regular polygon = 30°.

(iii) Let n be the number of sides in the polygon.

By formula,

Sum of interior angles of an 'n' sided polygon = (2n - 4) × 90°.

∴ 150°.n = (2n - 4) × 90°

⇒ 150°.n = 180°.n - 360°

⇒ 180°.n - 150°.n = 360°

⇒ 30°.n = 360°

⇒ n = $\dfrac{360°}{30°}$ = 12.

Hence, no. of sides in polygon = 12.