Question

State whether the following statement is true or false.
If two angles of a triangle are congruent then the sides opposite to these angles are congruent.
A. TRUE
B. FALSE

Answer 1

Hint: We will prove the statement is true or false by dividing the triangle into halves by an angle bisector of the non-congruent angle and proving the two triangles congruent using AAS congruence rule.
* AAS : Angle Angle Side Congruence rule states that if two angles and the non-included side of one triangle are congruent to the two angles and the non-included side of the other triangle then the two triangles are congruent to each other.

Complete step-by-step answer:
We draw a triangle ABC having two congruent angles \[\angle B,\angle C\].
We will prove this statement is true or false by dividing the triangle into two parts by drawing an angle bisector of \[\angle A\] which divides the angle into two equal halves. Let the bisector be AD which meets the base BC at point D.

          

From the figure, we can say that \[\angle BAD = \angle CAD\] as AD bisects \[\angle A\].
Also we are given the two angles \[\angle B,\angle C\] congruent to each other.
Side AD is common in triangles \[\vartriangle ABD,\vartriangle ACD\]. Also it is the non-included side which means it does not lie between the two congruent angles .
Now we will prove the two triangles \[\vartriangle ABD,\vartriangle ACD\] congruent to each other.
In \[\vartriangle ABD,\vartriangle ACD\]
\[
  \angle B \cong \angle C \\
  \angle BAD \cong \angle CAD \\
  AD = AD \\
 \]
Therefore, using AAS congruence we can say \[\vartriangle ABD \cong \vartriangle ACD\]
Since, corresponding sides of congruent triangle are congruent
Therefore, \[AB \cong AC\]
So, the statement in the question is TRUE.

So, the correct answer is “Option A”.

Note: Students are likely to make mistake while writing which congruence rule they are using as they might see two angles and one side congruent and write ASA rule which is wrong as it means two angles and the included side of a triangle are congruent to two angles and the included side of another triangle.