Your friend claims that two isosceles triangles abc and def are congrent 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 is congruent to ef then it must follow that ca is congren to fd. explain the error in his reasonin

First of all, an isosceles triangle is that which has a pair of sides with the same lengths. Therefore, there could only be two measurements for the sides comprising the perimeter of the triangle. Those are the sides of the triangle and the base. 

The error in the reasoning is that there are 3 pairs of congruent sides where in fact there should only be 2.