Question

In the given figure, O is the centre of the circle. If ∠AOC = 160°, find: 

1. ∠ABC 
2. ∠ADC

Answer 1

Given:

O is the center of the circle.

∠AOC = 160°.

(i) For ∠ABC:

The angle at the center (O) is twice the angle at the circumference subtended by the same arc.

∠AOC = 2 × ∠ABC

\[\angle ABC = \frac{1}{2} \times \angle AOC = \frac{160^\circ}{2} = 80^\circ\]

(ii) For ∠ADC:

Angles subtended by the same chord/arc at the circumference on the opposite side of the circle add up to 180° because ABCD is a cyclic quadrilateral.

∠ABC + ∠ADC = 180°

ADC = 180°

Using ∠ABC = 80°,

∠ADC = 180° − 180°

= 100°