Question

$ABCD$ is a parallelogram and $AP$ and $CQ$ are perpendiculars from vertices $A$ and $C$ on diagonal $BD$.Show that $AP=CQ$

Answer 1

In $△APB$ and $△CQD,$

$\angle APB=\angle CQD$ (Each ${90}^{\circ }$)

$AB=CD$ (Opposite sides of parallelogram $ABCD$)

$\angle ABP=\angle CDQ$ (Alternate interior angles for $AB\parallel CD$)

$\therefore △APB\cong △CQD$ (By AAS congruency)

$\therefore AP=CQ$ by CPCT

Hence proved.