$AB$ 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 $\angle ACB = 50^{\circ}$,then find $\angle APB$. (in $^{\circ}$)

$AB$ 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 $\angle APB = 130^{\circ}$,then find $\angle ACB$. (in $^{\circ}$)

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 $^{\circ}$)

The circumcentre of the triangle $ABC$ is $O$. Prove that $\angle OBC + \angle BAC = 90^{\circ}$.

If bisectors of opposite angles of a cyclic quadrilateral $ABCD$ intersect the circle circumscribing it at the points $P$ and $Q$,prove that $PQ$ is a diameter of the circle.

Two circles of radii $8 \ cm$ with centers $P$ and $Q$ are given. The chord $AB$ of the circle with center $P$ and the chord $CD$ of the circle with center $Q$ are equal. If $\angle PAB = 40^{\circ}$,then find $\angle CQD$. (in $^{\circ}$)