In a cyclic quadrilateral $ABCD$,$\angle A = 2x - 10^{\circ}$ and $\angle C = 3x - 35^{\circ}$,then $\angle A =$ .......... (in $^{\circ}$)

$ABCD$ is a cyclic quadrilateral in which $\angle BAC = 45^{\circ}$ and $\angle ADB = 55^{\circ},$ then $\angle ABC = \dots$ (in $^{\circ}$)

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

Prove that a cyclic parallelogram is a rectangle.

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.

In a cyclic quadrilateral $ABCD$,$\angle A : \angle C = 4 : 5$ and $\angle B : \angle D = 5 : 7$. Find all the angles of $ABCD$.