If bisectors of opposite angles of a cyclic quadrilateral $ABCD$ intersect the circle, circumscribing it at the points $P$ and $Q$, prove that $PQ$ is a diameter of the circle.

A circle has radius $\sqrt{2 }\mathrm{cm}.$ It is divided into two segments by a chord of length $2 \mathrm{cm}.$

Prove that the angle subtended by the chord at a point in major segment is ${45}^o$.

In Fig. $10.20$, $O$ is the centre of the circle, $\angle BCO={30}^o$. Find $x$ and $y$.

$O$is the centre of the circle, $BD=OD$ and $CD\perp AB$. If $\angle CAB=k°$, then find the value of $k.$