In the figure,if $\angle ABC = 20^{\circ},$ then $\angle AOC$ is equal to (in $^{\circ}$)

$AC$ and $BD$ are chords of a circle which bisect each other. Prove that $AC$ and $BD$ are diameters.

In a circle with centre $P$,$AB$ and $CD$ are chords. If $\angle APB = 80^{\circ}$,$\angle CPD = 50^{\circ}$ and $AB = 7\, cm$,then find the length of chord $CD$. (in $, cm$)

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

In the figure,$A, B, C$ 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 $\angle BAC$. (in $^{\circ}$)

In a circle with centre $O$,$PQ$ and $XY$ are chords. If $\angle POQ = 120^{\circ}$,$\angle OXY = 30^{\circ}$ and $XY = 8 \, cm$,then $PQ$ is ........... $cm$.