Question

In the given figure, O is the centre of the circle. Find $x$.

Answer 1

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

Given$\angle ABC={50}^{\circ },$
$\therefore \angle AOC=2\angle ABC=2\times {50}^{\circ }={100}^{\circ }$

Also, since OA and OC are radii of the circle, OA=OC.
$⟹\angle OAC=\angle OCA$

$\text{In}△AOC,$

$x+x+\angle AOC={180}^{\circ }.$

$⟹2x={180}^{\circ }-\angle AOC={180}^{\circ }-{100}^{\circ }={80}^{\circ }$

$⟹x={40}^{\circ }$