In a circle with centre $P$,$AB$ and $CD$ are equal chords. If $\angle APB = 80^{\circ},$ then $\angle CPD =$ .......... (in $^{\circ}$)

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

Two circles intersect at two points $A$ and $B$. $AD$ and $AC$ are diameters to the two circles (see Fig.). Prove that $B$ lies on the line segment $DC$.

$AB$ 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 $\angle APB$.

In the figure,if $AOB$ is a diameter of the circle and $AC = BC$,then $\angle CAB$ is equal to (in $^{\circ}$)

Prove that equal chords of congruent circles subtend equal angles at their centres.