Answer 1

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.