Question

In Fig. $ABCD$, $DCFE$ and $ABFE$ are parallelograms. Show that $ar\left(ADE\right)$$=$$ar\left(BCF\right)$ .

Answer 1

Prove the area of given triangles equal

Given, $ABCD$, $DCFE$ and $ABFE$ are parallelogram

We need to prove, $ar\left(ADE\right)$$=$$ar\left(BCF\right)$ .

So,

In $∆ADE$ and $∆BCF$

$$\begin{gathered} AD=BC........\left[\mathrm{opposite}\mathrm{side}\mathrm{of}\mathrm{parallelogram}ABCD\right] \\ DE=CF........\left[\mathrm{opposite}\mathrm{side}\mathrm{of}\mathrm{parallelogram}DCFE\right] \\ AE=BF........\left[\mathrm{opposite}\mathrm{side}\mathrm{of}\mathrm{parallelogram}ABFE\right] \\ \therefore ∆ADE\cong ∆BCF\left(S.S.S\mathrm{Congruence}\right) \end{gathered}$$

Since congruent triangle are equal in area.

Therefore, $ar\left(ADE\right)$$=$$ar\left(BCF\right)$ .

Hence proved.