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.

Parallelogram PQRS is shown in the figure below:

We know that,

Opposite angles of a parallelogram are equal.

∠Q = ∠S = x (let).

Diagonals of a parallelogram bisect the interior angles.

∴ QS bisects interior angles Q and S.

∴ ∠LQN = $\dfrac{∠Q}{2} = \dfrac{x}{2}$ and ∠NSM = $\dfrac{∠S}{2} = \dfrac{x}{2}$.

∴ ∠LQN = ∠NSM.

We know that,

Opposite sides of a parallelogram are equal.

∴ PQ = SR = a (let)

Given,

⇒ PL = MR = b (let)

From figure,

⇒ LQ = PQ - PL = a - b

⇒ MS = SR - MR = a - b

∴ LQ = MS.

In △ LNQ and △ MNS,

⇒ ∠LNQ = ∠MNS (Vertically opposite angles are equal)

⇒ LQ = MS (Proved above)

⇒ ∠LQN = ∠NSM (Proved above)

∴ △ LNQ ≅ △ MNS (By A.A.S. axiom).

We know that,

Corresponding parts of congruent triangles are equal.

∴ QN = NS and LN = NM.

Hence, proved that LM and QS bisect each other at point of intersection.