Question

Question 10
In $\Delta ABC$ and $\Delta PQR$, if $AB=AC,\angle C=\angle P$ and $\angle B=\angle Q$. The two triangles are
(A) isosceles but not congruent
(B) isosceles and congruent
(C) congruent but not isosceles
(D) neither congruent nor isosceles

Answer 1

In $\Delta ABC$ AB =AC [given]
$\Rightarrow \angle C=\angle B$ [angles opposite to equal sides are equal]
so $\Delta ABC$ is an isosceles triangle
But it is given that $\angle B=\angle Q$
$\angle C=\angle P$
$\therefore \angle P=\angle Q[\because \angle C=\angle B]$
$\Rightarrow QR=PR$ [sides opposite to equal angles are equal]
So, $\Delta PQR$ is also an isosceles traingle

Therefore, both triangles are isosceles but not congruent. As, we know that AAA is not a criterion for congruence of triangles.