Question

ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD( See the given figure). Show that

AP = CQ

Answer 1

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

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

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

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

By using this result
$\Delta APB\cong \Delta CQD$, we obtain

AP = CQ ( By CPCT)