Question

In two right triangles one pair of acute angles and the sides opposite to them are equal. Prove that the triangles are congruent.

Answer 1

Given that in both the right triangles one side and one acute angle are equal to the corresponding side and angles of the other.

Let us consider the two right angled triangles $ABC$ & $EDF$ such that

$\angle B=\angle C={90}^0⟶\left(1\right)$

$AB=DE⟶\left(2\right)$

$\angle C=\angle F⟶\left(3\right)$

In $\Delta ABC$ and $\Delta EDF$

$\angle C=\angle F$ [from (3)]

$\angle B=\angle E$ [from (1)]

$AB=DE$ [from (2)]

So, by $AAS$ eongruence criterion, we have shown that

$\Delta ABC\cong \Delta EDE$

$\therefore$ The two triangles are congruent.

Hence proved.