Question

In the given figure,, BC is a diameter of circle and ∠BAO = 60∘ . Then, ∠ADC = __________.

Answer 1

Given:
BC is a diameter of circle
∠BAO = 60° ...(1)

In ∆OAB,
OA = OB
∴ ∠OAB = ∠OBA = 60° (angles opposite to equal sides are equal) ...(2)

Also,
∠AOC = ∠OAB + ∠OBA (exterior angle)
⇒ ∠AOC = 60° + 60° (From (2))
⇒ ∠AOC = 120° ...(3)

We know, the angle subtended by an arc at the centre is twice the angle subtended by it at the remaining part of the circle.
Thus, ∠AOC = 2∠ADC
⇒ 120° = 2∠ADC (From (3))
⇒ ∠ADC = 60°


Hence, ∠ADC = 60°.