Consider the following proof of the Triangle Angle Sum Theorem. Which statement has a mistake? How should the mistake be fixed?

Question

Given:  △ABC  with  DE←→∥AC¯¯¯¯¯¯¯¯

Prove:  m∠A+m∠ABC+m∠C=180°

Statements	Reasons
1.  △ABC  with  DE←→∥AC¯¯¯¯¯¯¯¯ 	1. Given
2.  ∠DBA≅∠B ;  ∠EBC≅∠C 	2.  DE←→∥AC¯¯¯¯¯¯¯¯  and alternate interior angles are congruent when lines are parallel
3.  m∠DBA=m∠A ;  m∠EBC=m∠C 	3. Definition of congruent angles 
4.  m∠DBA+m∠ABC+m∠EBC=180° 	4. Definition of a straight angle 
5.  m∠A+m∠ABC+m∠C=180° 	5. Substitution(1 point)
Responses

Statement 2 has a mistake. It should say ∠DBA≅∠B; ∠EBC≅∠A.
Statement 2 has a mistake. It should say  angle upper D upper B upper A congruent to angle upper B ;  angle upper E upper B upper C congruent to angle upper A .

Statement 4 has a mistake. It should say m∠A+m∠ABC+m∠C=180°.
Statement 4 has a mistake. It should say  m angle upper A plus m angle upper A upper B upper C plus m angle upper C equals 180 degrees .

Statement 2 has a mistake. It should say ∠DBA≅∠A; ∠EBC≅∠C.
Statement 2 has a mistake. It should say  angle upper D upper B upper A congruent to angle upper A ; angle upper E upper B upper C congruent to angle upper C .

Statement 4 has a mistake. It should say m∠DBA+m∠ABC+m∠ACB=180°.

GPT-4o mini · Answer

The mistake is in **Statement 2**. The correct conclusion should reflect that the angles formed by the transversal $DE$ with the parallel lines $AC$ and $BC$ yield the following relationships: - $\angle DBA \cong \angle A$ (not $\angle B$) - $\angle EBC \cong \angle C$ (not $\angle A$) The corrected statement should therefore read: **Statement 2 should say:** $\angle DBA \cong \angle A$; $\angle EBC \cong \angle C$. This indicates that the alternate interior angles are equal due to the fact that line $DE$ is parallel to line $AC$. The original proof incorrectly stated that $\angle DBA$ is congruent to $\angle B$, which is not correct based on the position of points $D$ and $E$. Thus, the correct choice would be: **Statement 2 has a mistake. It should say $\angle DBA \cong \angle A$; $\angle EBC \cong \angle C$.**