Question

“If two angles and a side of one triangle are equal to two angles and a side of another triangle, then the two triangles must be congruent”. Is this statement true? Why?

Answer 1

Hint:Here we are to use the congruence rules of triangles. We know, if two angles and one side of a triangle is equal to corresponding angles and side of the other triangle, then the two triangles are congruent. So, in this question we are to find whether the given conditions satisfy the conditions of congruency so that we can conclude that the two triangles are congruent.

Complete step by step solution:
Here, given, two angles and a side of one triangle are equal to two angles and a side of another triangle.
Here, there can be two cases.
Case I: The two angles and one side of one triangle are equal to the corresponding angles and side of the other triangle.
If so, then they satisfy the conditions of congruency and the two triangles are congruent.
Case II: The two angles and one side of one triangle are equal to any two angles and side of the other triangle.
Then this doesn’t satisfy the condition of congruency.
Hence, we can’t say that they are congruent.
As in the question, it is nowhere mentioned that the two angles and one side of one triangle are equal to the corresponding angles and sides of the other triangle.
So, it doesn’t always satisfy the conditions of congruency.
So, the statement is False that, “If two angles and a side of one triangle are equal to two angles and a side of another triangle, then the two triangles must be congruent”.

Note:
Congruency can also be established by the various other congruence methods like side, side, side, SSS congruence, by side, angle, side congruence, SAS congruence, right angle, hypotenuse, side congruence for right angled triangles, RHS congruence.