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

If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides,prove that the quadrilateral so formed is cyclic.

$ABCD$ is a parallelogram. The circle passing through $A, B$ and $C$ intersects $CD$ (produced if necessary) at $E$. Prove that $AE = AD$.

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

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

$ABCD$ is a parallelogram. $A$ circle through $A$ and $B$ is drawn such that it intersects $AD$ at $P$ and $BC$ at $Q$. Prove that $P, Q, C,$ and $D$ are concyclic.