In a circle with centre $O$,$PQ$ and $RS$ are chords. If $\angle POQ = 90^{\circ}$,$\angle ROS = 90^{\circ}$ and $PQ = 6\, cm$,then $RS = \dots \, cm$.

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 chord and $PA = 4\, cm$. In a circle with centre $Q$,$XY$ is a chord and $QX = 4\, cm$. If $\angle APB = 80^{\circ}$,$\angle XQY = 50^{\circ}$ and $AB = 5\, cm$,then $XY = \dots\dots\dots\, cm$.

$A$ quadrilateral $ABCD$ is inscribed in a circle such that $AB$ is a diameter and $\angle ADC = 130^{\circ}$. Find $\angle BAC$.

$ABCD$ 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,$\angle ACB = 40^{\circ}$. Find $\angle OAB$.