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 □ Pqrs is a Parallelogram. - Geometry

Sum

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.

Solution

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.

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Balbharati Mathematics 2 Geometry 9th Standard Maharashtra State Board