In the figure,$AB$ and $CD$ are two equal chords of a circle with center $O$. $OP$ and $OQ$ are perpendiculars on chords $AB$ and $CD$ respectively. If $\angle POQ = 150^{\circ}$,then $\angle APQ$ is equal to: (in $^{\circ}$)

State whether each of the following statements is true or false:
$(1)$ The region between a chord and either of its arcs is called a sector.
$(2)$ The region between an arc and the two radii,joining the centre to the end points of the arc,is called a segment.

In a circle with centre $P$,the radius is $13 \, cm$. The distance between two parallel chords $AB$ and $CD$ is $17 \, cm$. If $AB = 24 \, cm$,find the length of $CD$. (in $, cm$)

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 $^{\circ}$)

$ABCD$ is a quadrilateral such that $A$ is the centre of the circle passing through $B, C$ and $D$. Prove that $\angle CBD + \angle CDB = \frac{1}{2} \angle BAD$.

If $P, Q$ and $R$ are the mid-points of the sides $BC, CA$ and $AB$ of a triangle $ABC$ and $AD$ is the perpendicular from $A$ on $BC$,prove that $P, Q, R$ and $D$ are concyclic.