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Constructions
Question
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In figure, ABCD, DCFE and ABFE are parallelograms. Show that ar(ΔADE) = ar(ΔBCF).

Answer

Given: ABCD, DCFE and ABFE are parallelograms.
To Prove: ar(ΔADE) = ar(ΔBCF).
Proof: ∵ ABCD is a || gm
∴ AB || DC    ...(1)
| Opposite sides of a || gm are ||
∵    DCFE is a || gm
∴ DC || EF    ...(2)
| Opposite sides of a || gm are ||
From (1) and (2),
AB || EF    ...(3)
∵    ABCD is a || gm
∴ AD = BC    ...(4)
| Opposite sides of a || gm are equal
∵    ΔADE and ΔBCF are on equal bases (∵ AD = BC) and between the same parallels AB and EF.
□∴ ar(ΔADE) = ar(ΔBCF).
∵ Two triangles on the same base (or equal bases) and between the same parallels are equal in areas

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