Question

C is the incenter of isosceles triangle ABD with the vertex angle ∠ABD. Does the following proof correctly justify that triangles ABC and DBC are congruent?

A) There is an error in line 1; segment BC should be an angle bisector.
B) The proof is correct.
C) There is an error in line 3; segments AB and BC are congruent.
D) There is an error in line 5; the ASA Postulate should be used.

Answer 1

The following proof C) There is an error in line 3; segments AB and BC are congruent is correctly justify that triangles ABC and DBC are congruent.

Step-by-step explanation:

The given proof incorrectly states that segments AB and BC are congruent in line 3. However, in an isosceles triangle, the sides opposite the equal angles are congruent, not the legs themselves. Therefore, the correct statement should be that segments AC and BC are congruent. This error impacts the logical flow of the proof and the subsequent congruence statement.

In an isosceles triangle ABD with the vertex angle ∠ABD, the sides opposite the equal angles, namely segments AC and BC, are congruent. This congruence can be justified using the SAS (Side-Angle-Side) Postulate. The correct sequence would involve establishing that ∠ABC is congruent to ∠DBC (angle B is shared) and then stating that segments AC and BC are congruent.

In summary, the error lies in line 3, where the proof incorrectly asserts that segments AB and BC are congruent. The accurate statement is that segments AC and BC are congruent in an isosceles triangle, leading to the correct application of the SAS Postulate for proving the congruence of triangles ABC and DBC.

So, the correct answer is C) There is an error in line 3; segments AB and BC are congruent.