Question

In the given figure, ABCD, DCFE and ABFE are parallelograms. Show that \[ar\left( {ADE} \right) = ar\left( {BCF} \right)\].

Answer 1

Hint: The area of two congruent triangles is always equal. So, we will prove that ADE and BCF are congruent and we will get our required result.

Complete Step-by-step Solution
Given: ABCD, ABFE and DCFE are parallelograms.
If we observe the diagram, then we can say that triangle ADE and BCF are congruent from SSS congruence criteria because three sides of both the triangles are equal.
Since, we know that opposite sides of parallelogram are equal, we get,
$\begin{array}{l}
AD = BC\\
DE = CF\\
AE = BF
\end{array}$
$\Delta ADE \cong \Delta BCF$ is congruent triangle from SSS congruence criteria.
Also, by the rule of CPCT the area of triangles ADE and BCF are equal, that is,
$ar(ADE) = ar(BCF)$
Hence, it is proved that the area of the triangle is equal by $ar(ADE) = ar(BCF)$.
Note: In such types of problems, keep in mind that the above thing holds only for congruent triangles not for similar triangles. Most students are confused between similar triangles and congruence triangles but they are not the same. The triangles are said to be similar when the shape of both the triangles is the same. The triangles are said to be congruent when both the triangles have the same size as well as same shape.