Yes the student is correct
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.
