Write True or False and justify your answer in each of the following:
Two congruent circles with centres $O$ and $O^{\prime}$ intersect at two points $A$ and $B$. Then $\angle AOB = \angle AO^{\prime}B$.

Prove that two circles cannot intersect at more than two points.

In the figure,two congruent circles have centers $O$ and $O'$. Arc $AXB$ subtends an angle of $75^{\circ}$ at the center $O$ and arc $A'YB'$ subtends an angle of $25^{\circ}$ at the center $O'$. Then the ratio of the lengths of arcs $AXB$ and $A'YB'$ is:

$ABCD$ is a quadrilateral such that $A$ is the centre of the circle passing through $B, C$ and $D$. Prove that $\angle CBD + \angle CDB = \frac{1}{2} \angle BAD$.

In the figure,$\angle OAB = 30^{\circ}$ and $\angle OCB = 57^{\circ}$. Find $\angle BOC$ and $\angle AOC$.

$AB$ and $AC$ are two chords of a circle with centre $P$. If the bisector of $\angle BAC$ passes through the centre $P$,prove that $AB = AC$.