Quadrilaterals question bank (Maths)

$ABC$ is an isosceles triangle in which $AB = AC$. $AD$ bisects exterior angle $PAC$ and $CD \parallel AB$ (see Fig). Show that $ABCD$ is a parallelogram.

$ABCD$ is a parallelogram in which $P$ and $Q$ are mid-points of opposite sides $AB$ and $CD$ respectively. If $AQ$ intersects $DP$ at $S$ and $BQ$ intersects $CP$ at $R$,show that $DPBQ$ is a parallelogram.