Question

If PM and RN are perpendiculars on the diagonal QS of a parallelogram PQRS. Prove that PM=RNQ.

Answer 1

Step-by-step explanation:

SK = KL = QL … given … (i)

Consider ΔPKQ and ΔRLS

PQ = SR … opposite sides of parallelogram PQRS

∠PQK = ∠RSL … alternate pair of interior angles for parallel lines PQ and SR with transversal as SQ

SL = SK + KL and KQ = KL + LQ so using (i) we can say that

SL = KQ

Therefore, ΔPQK ≅ ΔRSL

⇒ ∠PKQ = ∠RLS … corresponding angles of congruent triangles

Thus PM || NR because ∠PKQ and ∠RLS are pair of alternate interior angles with transversal as KL

⇒ PM || NR … (ii)