In 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

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Solution

ABCD is a parallelogram
So,opposite pair of sides will be congruent and parallel.
$\Rightarrow A D ≅ B C and A B ≅ C D$
Given, AP = BQ = CR = DS
AD−SD=BC−BQ
$\Rightarrow A S = C Q . . . . . (1)$
$In △ A P S and △ R C Q$
AS = CQ (From (1))
AP = CR (Given)
$∠ P A S = ∠ R C Q$ (Opposite angles of a parallelogram are equal)
$Thus, △ A P S ≅ △ R C Q$ (SAS congruency)
$\Rightarrow S P = R Q (C P C T)$
Similarly, PQ = SR
Thus, opposite pair of sides are congruent.
Hence, PQRS is a parallelogram.

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Q. 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

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PQRS is a parallelogram.

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