In the given figure, ABCD, DCFE and ABFE are parallelograms. Show that ar (∆ADE) = ar (∆BCF).

Given ABCD, DCFE and ABFE are parallelograms.

To prove ar (∆ADE) = ar (∆BCF)

Proof Since, ABCD is a parallelogram.

∴AD = BC …(i)

[∵ opposite sides of a parallelogram are equal]

Similarly, as DCFE and ABFE are parallelograms.

∴ DE = CF and AE = BF …(ii)

[∵ opposite sides of a parallelogram are equal]

Thus, in ∆ADE and ∆BCF, we have

AD = BC [from Eq. (i)]

DE=CF [from Eq. (ii)]

and AE = BF [from Eq. (ii)]

∴∆ADE = ∆BCF

[by SSS congruence criterion] => ar(∆ADE) = ar(∆BCP)

[∵ congruent figures have equal areas]