In Fig., PSDA is a parallelogram. Points Q and R are taken on PS such that PQ = QR = RS and . Prove that ar (PQE) = ar (CFD).
Solution.
Given: a parallelogram PSDA with points Q and R taken on PS such that PQ = RS = QR
and
To prove: ar(PQE) = ar(CFD)
Proof:
PQ = QR = RS
PS = PQ + QR + RS = 3 PQ
Also, and
So PABQ, QBCR, RCDS – all are parallelograms.
Hence,
We know that opposite sides of a parallelogram are equal, so in APSD:
PS = AD
…(1)
Consider and
and PD is transversal
\ [Alternate interior angles] …(2)
Now, and AD is a transversal
\ [Corresponding angles] …(3)
and PD is transversal
[Alternate interior angles] …(4)
From (3) and (4)
…(5)
From (1), (2), (5)
[ASA congruence rule]
Hence, [Congruent figures have equal area]
Hence proved