In cyclic quadrilateral $PQRS$,if $\angle PRQ = 52^{\circ}$,then find $\angle PSQ$. (in $^{\circ}$)

In quadrilateral $ABCD$,if $\angle A : \angle B : \angle C : \angle D = 3 : 2 : 4 : 5$,then identify the type of quadrilateral $ABCD$.

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.

If $BM$ and $CN$ are the perpendiculars drawn on the sides $AC$ and $AB$ of the triangle $ABC$,prove that the points $B, C, M$ and $N$ are concyclic.

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 $AB = CD$.

If arcs $AXB$ and $CYD$ of a circle are congruent,find the ratio of $AB$ and $CD.$