In the given figure, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that `square` PQRS is a parallelogram.
ABCD is a parallelogram
So, opposite pair of sides will be congruent and parallel.
⇒ AD ≅ BC and AB ≅ CD
Given, AP = BQ = CR = DS
AD - SD = BC - BQ
⇒ AS = CQ ....(1)
In Δ APS and Δ RCQ
AS = CQ (From (1))
AP = CR (Given)
∠PAS = ∠RCQ
(Opposite angles of a parallelogram are equal)
Thus , Δ APS ≅ Δ RCQ (SAS congruency)
⇒ SP = RQ (CPCT)
Similarly, PQ = SR
Thus, opposite pair of sides are congruent.
Hence, PQRS is a parallelogram.