Show that the diagonals of a rhombus divide it into four congruent triangles.

In the adjacent figure $ABCD$ is a parallelogram and $E$ is the midpoint of the side $BC$. If $DE$ and $AB$ are produced to meet at $F$, show that $AF=2AB$

In the adjacent figure, $ABCD$ is a parallelogram $P, Q$ are the midpoints of sides $AB$ and $DC$ respectively. Show that $PBCQ$ is also a parallelogram.

$ABC$ is an isosceles triangle in which $AB=AC$. $AD$ bisects exterior angle $QAC$ and $CD\parallel BA$ as shown in the figure. Show that $\angle DAC=\angle BCA$.

$ABC$ is an isosceles triangle in which $AB=AC$. $AD$ bisects exterior angle $QAC$ and $CD\parallel BA$ as shown in the figure. Show that $ABCD$ is a parallelogram.

$ABCD$ is a parallelogram $AP$ and $CQ$ are perpendiculars drawn from vertices $A$ and $C$ on diagonal $BD$, show that $∆APB\cong ∆CQD$

$ABCD$ is a parallelogram $AP$ and $CQ$ are perpendiculars drawn from vertices $A$ and $C$ on diagonal $BD$, show that $AP=CQ$