Question

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

Answer 1

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.