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 is congruent to Ef then it must follow that CA is congren to FD. Explain the error in his reasoning

A) He incorrectly assumes that all sides of an isosceles triangle are congruent.
B) He misinterprets the relationship between corresponding sides of isosceles triangles.
C) He overlooks the importance of angles in determining congruence.
D) His reasoning is correct; there is no error.

Answer 1

The error is that the importance of angles in determining congruence was overlooked; it's necessary to have two congruent sides and the included angle to prove isosceles triangles are congruent.

Correct option is B

Step-by-step explanation:

The error in your friend's reasoning is option C: He overlooks the importance of angles in determining congruence. To show that two isosceles triangles are congruent, you cannot rely only on congruent corresponding sides. Although two sides being congruent is an important aspect, the angle between those sides must also be congruent.

For instance, suppose we have two isosceles triangles, △ABC and △DEF, with sides AB ≅ DE and BC ≅ EF. If we do not know anything about the angles, we cannot be certain that AC is congruent to DF. This is because the base angles opposite the congruent sides AB and DE must also be congruent to guarantee that the triangles are congruent. This property is stipulated in the congruence criteria for triangles, such as SAS (Side-Angle-Side), where two sides and the included angle must be congruent, or ASA (Angle-Side-Angle), where two angles and the included side must be congruent.