Question

How can two isosceles triangles be congruent?

Answer 1

Hint: We first describe the congruence and the types of possible congruences. We then pick the most possible two congruences as $SAS$ and $AAS$ to find the two isosceles triangles to be congruent.

Complete answer:
Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure. In many cases it is sufficient to establish the equality of three corresponding parts and use one of the results to deduce the congruence of two triangles. There are in total four types of congruence which are $SAS,SSS,AAS,RHS$.

The most common ones for two isosceles triangles to be congruent is $SAS$ and $AAS$. Also, if one congruence holds then the other one holds automatically.
We can equate the two same sides and their common angle. We can also show two angles being equal and one more side.
We can also use the other two congruences of $SSS,RHS$ only if some specific information about the sides and the angles are given.

Note: We also need to remember that the Isosceles Triangle Theorem states: If two sides of a triangle are congruent, then angles opposite those sides are congruent. We know it's an isosceles triangle because it has two equal sides which gives two equal angles.