In a circle with centre $P$,the chord $AB = 8 \, cm$ and the radius $= 8 \, cm$. Then,$\angle APB = \dots$ (in $^{\circ}$)

If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection,prove that the chords are equal.

$AB$ is a chord of a circle with centre $P$. Point $C$ is a point other than $A$ and $B$ on the major arc $AB$. If $\angle ACB + \angle APB = 135^{\circ}$,then find $\angle ACB$ and $\angle APB$.

Prove that if the angles subtended by the chords of a circle (or of congruent circles) at the centre (or the corresponding centres) are equal,then the chords are equal.

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 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 ........... $cm$.