In cyclic quadrilateral $ABCD$,$\angle B = \angle D - 30^{\circ}$,then find $\angle B$. (in $^{\circ}$)

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

$AB$ and $CD$ are two parallel chords of a circle with centre $P$. Centre $P$ is not between the two chords $AB$ and $CD$. If $AB = 40\,cm$,$CD = 30\,cm$ and the radius of the circle is $25\,cm$,find the distance between $AB$ and $CD$. (in $,cm$)

$A$ line joining the midpoints of two chords passes through the centre of a circle. Prove that the chords are parallel.

$ABCD$ is a cyclic quadrilateral. If $AB \parallel CD$,then prove that $BC = AD$.

In a circle with centre $P$,the chord $AB = 8 \, cm$ and the radius $= 8 \, cm$. Then,$\angle APB = \dots$ (in $^{\circ}$)