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

In a circle with centre $P$,$AB$ is a diameter and $ABCD$ is a cyclic quadrilateral. If $\angle ADC = 150^{\circ}$,then find $\angle BAC$. (in $^{\circ}$)

$ABCD$ is a parallelogram. $A$ circle through $A$ and $B$ is drawn such that it intersects $AD$ at $P$ and $BC$ at $Q$. Prove that $P, Q, C,$ and $D$ are concyclic.

In a cyclic quadrilateral $ABCD$,$\angle A = 4 \angle C$ and $\angle B = 3 \angle D$. Find all the angles of $ABCD$.

State whether each of the following statements is true or false:
$(1)$ $A$ line segment joining any two points of a circle is a diameter of the circle.
$(2)$ For any circle, $\text{diameter} = 2 \times \text{radius}$.
$(3)$ In a circle with radius $14 \text{ cm}$, the length of a chord can be $32 \text{ cm}$.

If a pair of opposite sides of a cyclic quadrilateral are equal,prove that its diagonals are also equal.