In Fig. 9.25, $\angle ABC = 69^{\circ}$, $\angle ACB = 31^{\circ}$, find $\angle BDC$.

If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

$ABC$ and $ADC$ are two right triangles with common hypotenuse $AC$. Prove that $\angle CAD = \angle CBD$.

Prove that a cyclic parallelogram is a rectangle.

In Fig. 9.23, $A, B$ and $C$ are three points on a circle with centre $O$ such that $\angle BOC = 30^{\circ}$ and $\angle AOB = 60^{\circ}$. If $D$ is a point on the circle other than the arc $ABC$, find $\angle ADC$.

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

In Fig. 9.26, $A$, $B$, $C$ and $D$ are four points on a circle. $AC$ and $BD$ intersect at a point $E$ such that $\angle BEC = 130^{\circ}$ and $\angle ECD = 20^{\circ}$. Find $\angle BAC$.

$ABCD$ is a cyclic quadrilateral whose diagonals intersect at a point $E$. If $\angle DBC = 70^{\circ}$, $\angle BAC$ is $30^{\circ}$, find $\angle BCD$. Further, if $AB = BC$, find $\angle ECD$.

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

In Fig. 9.24, $\angle PQR = 100^{\circ}$, where $P$, $Q$ and $R$ are points on a circle with centre $O$. Find $\angle OPR$.

Two circles intersect at two points $B$ and $C$. Through $B$, two line segments $ABD$ and $PBQ$ are drawn to intersect the circles at $A$, $D$ and $P$, $Q$ respectively (see Fig. 9.27). Prove that $\angle ACP = \angle QCD$.