Question : In $\triangle ABC$ and $\triangle PQR, \angle B=\angle Q, \angle C=\angle R$ and $AB=2PQ$, then the two triangles are:
Option 1: congruent as well as similar
Option 2: neither similar nor congruent
Option 3: similar but not congruent
Option 4: congruent but not similar
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Correct Answer: similar but not congruent
Solution :
Given: In the triangles ABC and PQR, $\angle B = \angle Q$, $\angle C = \angle R$
Also, AB = 2PQ
We have to find if the triangles are similar and congruent or not.
Angle-Angle-Angle(AAA) criterion states that if two angles of a triangle are respectively equal to two angles of another triangle, then by the angle sum property of a triangle their third angle will also be equal.
By AAA criterion,
As $\angle B = \angle Q$ and $\angle C = \angle R$, the third angle will also be equal.
$\therefore \angle A = \angle P$
Therefore, the triangles ABC and PQR are similar.
It is given that AB = 2PQ
It is clear that the sides AB and PQ are not equal,
So, the triangles are not congruent.
Therefore, the triangles ABC and PQR are similar but not congruent.
Hence, the correct answer is 'similar but not congruent'.
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