In the given figure, AB is a side of a regular hexagon and AC is a side of regular eight sided polygon. Find :

(i) ∠AOB

(ii) ∠AOC

(iii) ∠BOC

(iv) ∠OBC

We know that,

The angle subtended by each side of an n-sided regular polygon at the center of circle = $\dfrac{360°}{n}$.

(i) Given,

AB is the side of the hexagon.

Angle subtended by each arm of the hexagon at the center of the circle is $\dfrac{360°}{6}$ = 60°.

Hence, ∠AOB = 60°.

(ii) Given,

AC is the side of a regular eight sided polygon.

Angle subtended by each arm of the regular eight sided polygon at the center of the circle is $\dfrac{360°}{8}$ = 45°.

Hence, ∠AOC = 45°.

(iii) From figure,

⇒ ∠BOC = ∠AOB + ∠AOC = 60° + 45° = 105°.

Hence, ∠BOC = 105°.

(iv) In △ OBC,

⇒ OB = OC (Radius of same circle)

⇒ ∠OBC = ∠OCB = x (let) [Angle opposite to equal sides are equal]

By angle sum property of triangle,

⇒ ∠OBC + ∠OCB + ∠BOC = 180°

⇒ x + x + 105° = 180°

⇒ 2x + 105° = 180°

⇒ 2x = 180° - 105°

⇒ 2x = 75°

⇒ x = $\dfrac{75°}{2}$

⇒ x = 37.5°

⇒ ∠OBC = 37.5°.

Hence, ∠OBC = 37.5°=37°30'.