Question

Pqrs is a parallelogram and the bisector of Angle P bisects QR at M. Show that ps =2pq

Answer 1

PM is the bisector of = QR  (Given)

Thus,

QM = MR --- eq 1

Since, PM bisects ∠P

∠SPM = ∠MPQ --- eq 2

Now, PS || QR and PM is the transversal, thus -

∠SPM = ∠PMQ  (Alternate interior angles) --- eq 3

From equation 2 and 3

∠MPQ = ∠PMQ

QM = PQ ( In a traingle sides opposite to equal angles are equal) --- eq 4

From 1 and 4

PQ = QM = MR

QM + MR = QR

QR = PQ + PQ  ( As PQ = QM = MR)

or, QR = 2(PQ)

Also, QR = PS  (Opposite sides of the parallelogram are equal)

Thus,

PS = 2PQ