Question

Use the image to answer the question.

An illustration shows a triangle with vertices labeled clockwise as upper A, upper B, and upper C, starting at the lower left vertex. A line with arrows at both ends is drawn above the triangle.

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

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°.

Answer 1

The mistake in the provided proof is in Statement 2. The correct angles should state that ∠DBA is congruent to ∠A, and ∠EBC is congruent to ∠C due to the parallel lines and alternate interior angles.

So, the correct statement should be:

Statement 2 has a mistake. It should say ∠DBA≅∠A; ∠EBC≅∠C.