Question

In two triangles, two angles and one side of the first triangle are equal to the two angles and one side of the second triangle. Will these two triangles always be congruent?[3 MARKS]

Answer 1

Proof: 1 Mark
Steps: 2 Marks

Consider two triangles, $\Delta ABC$ and $\Delta PQR$ in which,



$\angle ABC=\angle PQR$

$\angle ACB=\angle PRQ$

$AB=PQ$

We know that,

$\angle ABC+\angle ACB+\angle BAC={180}^0$

$\angle BAC={180}^0-(\angle ABC+\angle ACB)....\left(i\right)$

Similarly,

$\angle QPR={180}^0-(\angle PQR+\angle PRQ)......\left(ii\right)$

From (i) and (ii),

$\angle BAC=\angle QPR$

Now, In $\Delta ABC$ and $\Delta PQR$

$\angle ABC=\angle PQR$ [Given]

$\angle BAC=\angle QPR$ [ Proved above]

$AB=PQ$ [Given]

$\Rightarrow \Delta ABC\cong \Delta PQR$ [ ASA congruency rule]

$\Rightarrow$ These triangles are always congruent.