Question

In figure PQRS is a parallelogram QU and ST are perpendiculars on the diagonal PR prove that :
1. ∆STR is congruent to ∆QUP 2. ST = QU

Answer 1

In  ΔPXS and ΔRYQ

PS = QR  (opp sides of paralleogram)

angle PXS = angle RYQ = 90 degrees  

angle PSX = angle YQR (alternate interior angles)

By AAS two triangles are congruent.

By CPCT PX = RY

Answer 2

Given:

PQRS is a parallelogram

QU and ST are perpendiculars on the diagonal PR

To Find:

1. ∆STR ≅ ∆QUP

2. ST = QU

Solution:

1. In order to prove the congruency, we see that

∠STR =∠QUP = 90°         (Given)

SR = PQ                            (Opp.sides of parallelogram are equal )

∠SRT = ∠QPU                   (alternate interior angle)

∆STR ≅ ∆QUP                  (By AAS)

2. By corresponding parts of congruent triangles, we get

   ST = QU

Hence, ∆STR ≅ ∆QUP and ST = QU.