In cyclic quadrilateral $PQRS$,if $5 \angle Q = 7 \angle S$,then find the value of $\angle Q$. (in $^{\circ}$)

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

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.

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.

$A$ quadrilateral $ABCD$ is inscribed in a circle such that $AB$ is a diameter and $\angle ADC = 130^{\circ}$. Find $\angle BAC$.

State whether the following statement is True or False and justify your answer: $ABCD$ is a cyclic quadrilateral such that $\angle A = 90^{\circ}, \angle B = 70^{\circ}, \angle C = 95^{\circ}$ and $\angle D = 105^{\circ}$.