Question

$ABCD$ is a cyclic quadrilateral and $O$ is the centre of the circle through $A,B,C$ and $D$. If $\angle AOC={120}^o$, find $\angle ADC$.

Answer 1

By the inscribed angle theorem, we have,

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

$\Rightarrow \angle ABC={60}^o$.

Since quadrilateral $ABCD$ is a cyclic quadrilateral, we have,

$\angle ABC+\angle ADC={180}^o$ ...[Opposite angles of cyclic quadrilateral are supplementary]

$\Rightarrow \angle ADC={180}^o-{60}^o$

$\Rightarrow \angle ADC={120}^o$.