Question

PQRS is a parallelogram. L and M are points on PQ and SR respectively such that PL = MR.
Show that LM and QS bisect each other.

Answer 1

Given: PL = MR           ...(i)

To prove SR = PQ   ...(ii) (parallelogram opposite sides)

SP - PQ and SR - MR

LQ = SM           ...(iii) 

In ΔLOQ & ΔMOS

∠LQO = ∠MSO     ....( alternate interior angles )

∠OLQ = ∠OMS     ....( alternate interior angles )

 LQ = SM            ...(from (iii))

ΔLOQ ≅ ∠MSO      ...(by ASA congruence)

Then, OL = OM

OQ = OS       ...(by c.p.c.t.c)

Hence, LM and QS bisect each other.