Question

$AOCB$ is a quadrilateral in the circle with centre $O$. If $\angle AOC=130°$, then find $\angle ABC$.

Answer 1

Find $\angle ABC$:

Given, $\angle AOC=130°$

We know that the angle subtended by the chord on the major area is half of the subtended on the centre.

$\angle ADC=\frac{1}{2}\angle AOC$

$\angle ADC=\frac{1}{2}\times 130°=65°$

It is given that $ADCB$ is a cyclic quadrilateral.

Therefore, the sum of opposite angles of a cyclic quadrilateral is $180°$

$\angle ABC+\angle ADC=180°$

$\Rightarrow \angle ABC+65°=180°$

$\Rightarrow \angle ABC=180°-65°$

$\Rightarrow \angle ABC=115°$

Therefore, $\angle ABC=115°$