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Answer 1

$\mathbb{ELLO \ THERE!}$

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Question: ABCD, DCFE and ABFE are parallelograms. Prove that ar(ADE) = ar(BCF)

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Given

ABCD is a parallelogram.

DCFE is a parallelogram.

ABFE is a parallelogram.

To Prove

ar(ADE) = ar(BCF)

Proof

ABCD is a parallelogram

∴ AD = BC

(in a parallelogram, opposite sides are equal)

DCFE is a parallelogram.

∴ DE = CF

(in a parallelogram, opposite sides are equal)

ABFE is a parallelogram.

∴ AE = BF

(in a parallelogram, opposite sides are equal)

Now in ΔADE and ΔBCF,

AD = BC (proved)

DE = CF (proved)

AE = BF (proved)

∴ΔADE ≅ ΔBCF

We know that Congruent figures have equal areas, therefore,

ar(ADE) = ar(BCF)

Hence proved!

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