$A$ piece of a circle between two points on a circle is called ........ of the circle.

In a circle with centre $P$,$AB$ is a chord and point $C$ is a point other than $A$ and $B$ on the major arc $AB$. If $\angle ACB + \angle APB = 150^{\circ}$,then find $\angle APB$. (in $^{\circ}$)

Write True or False and justify your answer.
The angles subtended by a chord at any two points of a circle are equal.

The line segment joining the centre and any point on the circle is called a .......... of the circle.

In cyclic quadrilateral $ABCD$,if $5 \angle A = 13 \angle C$,then find $\angle A$. (in $^{\circ}$)

If $ABC$ is an equilateral triangle inscribed in a circle and $P$ is any point on the minor arc $BC$ which does not coincide with $B$ or $C$,prove that $PA$ is the angle bisector of $\angle BPC$.