Question

If two sides of a triangle are congruent, then are the angles opposite those sides always, sometimes or never congruent?

Answer 1

Hint: First we will draw a triangle and then we will draw a median from the interior angle of two adjacent sides which are equal. Then we will use the property of congruence and find out the relation between the angles opposite to the congruent sides.

Complete step by step answer:
In the above question, it is given that two of the three sides of a triangle are congruent.Therefore, it is an isosceles triangle. Now, we will draw a line from the interior angle of two equal sides as shown below:

In the above figure, there are two triangles, one is $\vartriangle ABD$ and the second one is $\vartriangle ACD$, where AD is the median. Therefore,
$BD = CD$
Now, we will prove that $\vartriangle ABD$ and $\vartriangle ACD$ are congruent.
In $\vartriangle ABD$ and $\vartriangle ACD$,
$AB = AC$(given)
$\Rightarrow AD = AD$.......(common)
$\Rightarrow BD = CD$......(D is the midpoint)
Therefore, $\vartriangle ABD$and $\vartriangle ACD$ are congruent by SSS (side side side) property. Hence, $\angle ABD = \angle ACD$. Therefore, the angles are congruent. So, we can say that if two sides of the triangle are congruent, then the triangle is isosceles, and the angles at the third side (base) are congruent. If the triangle is not only isosceles but equilateral the third angle is also congruent.

Note: An Isosceles Triangle has the following properties:
-Two sides are congruent to each other.
-The third side of an isosceles triangle which is unequal to the other two sides is called the base of the isosceles triangle.
-The two angles opposite to the equal sides are congruent to each other. That means it has two congruent base angles and this is called an isosceles triangle base angle theorem.
-The angle which is not congruent to the two congruent base angles is called an apex angle.