$XY$ is a diameter of the circumcircle of $\Delta XYZ$. What is the value of $\angle XZY$ (in $^{\circ}$)?

Write True or False and justify your answer in each of the following: Two chords $AB$ and $CD$ of a circle are each at distances $4 \ cm$ from the centre. Then $AB = CD$.

State whether each of the following statements is true or false:
$(1)$ For any two points on a circle,the line segment joining them is called a chord of the circle.
$(2)$ The length of a diameter of a circle is half the length of its radius.

Prove that two circles cannot intersect at more than two points.

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 $\angle BPC$.

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