The two similar triangles are equal in area. Prove that the triangles are congruent.

Let's consider two similar triangles as ∆ABC ~ ∆PQR

We know that,

The areas of two similar triangles are proportional to the squares of their corresponding sides.

So,

$\dfrac{\text{Area of ∆ABC}}{\text{Area of ∆PQR}} = \Big(\dfrac{AB}{PQ}\Big)^2 = \Big(\dfrac{BC}{QR}\Big)^2 = \Big(\dfrac{AC}{PR}\Big)^2$

Since,

Area of ∆ABC = Area of ∆PQR [Given]

Hence,

AB = PQ

BC = QR

AC = PR

So, as the respective sides of two similar triangles are all of same length.

We can conclude that,

∆ABC ≅ ∆PQR [By SSS rule]

Hence proved that both triangles are congruent.