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In Fig., PSDA is a parallelogram. Points Q and R are taken on PS such that PQ = QR = RS and PA \parallel QB \parallel RC. Prove that ar (PQE) = ar (CFD).

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Solution.

Given: a parallelogram PSDA with points Q and R taken on PS such that PQ = RS = QR
and PA\parallel QB\parallel RC
To prove: ar(PQE) = ar(CFD)
Proof:

PQ = QR = RS
PS = PQ + QR + RS = 3 PQ
Also, PA \parallel QB \parallel RC and PS \parallel AD

So PABQ, QBCR, RCDS – all are parallelograms.
Hence,  AB=BC=CD=\frac{1}{3}AD
We know that opposite sides of a parallelogram are equal, so in \parallel gm APSD:
PS = AD
\Rightarrow \frac{1}{3}PS=\frac{1}{3}AD
\Rightarrow PQ=CD                     …(1)

Consider \triangle PEQ and \triangle CFD
PS\parallel AD and PD is transversal
\ \angle QPE = \angle CDF            [Alternate interior angles] …(2)
Now, QB \parallel RC and AD is a transversal
\\angle QBD = \angle RCD             [Corresponding angles] …(3)
PS \parallel AD and PD is transversal
\therefore \angle PQB = \angle QBC             [Alternate interior angles] …(4)
From (3) and (4)
\angle PQE=\angle FCD\angle PQE = \angle FCD???????\angle PQE = \angle FCD…(5)
From (1), (2), (5)
\triangle PEQ\cong \triangle CFD                [ASA congruence rule]

Hence, ar (\triangle PQE) = ar(\triangle CED)[Congruent figures have equal area]
Hence proved

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