In Fig. A, B and C are three points on a circle with centre O such that ∠BOC = 30° and ∠AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.

Given :

⇒ ∠BOC = 30° and ∠AOB = 60°

From figure,

⇒ ∠AOC = ∠AOB + ∠BOC

⇒ ∠AOC = 60° + 30° = 90°.

Arc AC subtends ∠AOC at center of circle and ∠ADC on point D.

We know that,

The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

⇒ ∠AOC = 2∠ADC

⇒ 2∠ADC = 90°

⇒ ∠ADC = $\dfrac{90°}{2}$ = 45°.

Hence, ∠ADC = 45°.