Your friend claims that two isosceles triangles abc and def are congruent if two corresponding sides are congruent. he explains that there are only two different lengths of sides , so if ab is congruent to de and bc us congruent to ef, then it must follow that ca is congruent to fd. explain the error in his reasoning

Think about the fact that you can have two types of isosceles triangle: one of them that is a right triangle (isosceles right triangle) such as a 45-45-90 triangle and the other type can be just a regular triangle that has two sides that are congruent but it isn't a right triangle. 

Thus, you would need more info about the triangles in order to conclude that the two isosceles triangles are congruent to each other.