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 $\angle AOB = \angle AO^{\prime}B$.

In a circle with centre $P$,$AB$ is a diameter and $ABCD$ is a cyclic quadrilateral. If $\angle ADC = 150^{\circ}$,then find $\angle BAC$. (in $^{\circ}$)

In cyclic quadrilateral $PQRS$,if $5 \angle Q = 7 \angle S$,then find the value of $\angle Q$. (in $^{\circ}$)

$XY$ is a diameter of the circumcircle of $\Delta XYZ$. What is the value of $\angle XZY$ (in $^{\circ}$)?

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

In a circle with centre $P$,the length of a radius is $\sqrt{2} \ cm$. It is divided into two segments by chord $AB$ of length $2 \ cm$. If $C$ is a point on the major arc $AB$,then find the value of $\angle ACB$. (in $^{\circ}$)