Consider the following proof of the Triangle Angle Sum Theorem. Which statement has a mistake? How should the mistake be fixed? Given: △ABC △ 𝐴 𝐵 𝐶 with DE←→∥AC¯¯¯¯¯¯¯¯ 𝐷 𝐸 ↔ ∥ 𝐴 𝐶 ¯ Prove: m∠A+m∠ABC+m∠C=180° 𝑚 ∠ 𝐴 + 𝑚 ∠ 𝐴 𝐵 𝐶 + 𝑚 ∠ 𝐶 = 180 ° Statements Reasons 1. △ABC △ 𝐴 𝐵 𝐶 with DE←→∥AC¯¯¯¯¯¯¯¯ 𝐷 𝐸 ↔ ∥ 𝐴 𝐶 ¯ 1. Given 2. ∠DBA≅∠B ∠ 𝐷 𝐵 𝐴 ≅ ∠ 𝐵 ; ∠EBC≅∠C ∠ 𝐸 𝐵 𝐶 ≅ ∠ 𝐶 2. DE←→∥AC¯¯¯¯¯¯¯¯ 𝐷 𝐸 ↔ ∥ 𝐴 𝐶 ¯ and alternate interior angles are congruent when lines are parallel 3. m∠DBA=m∠A 𝑚 ∠ 𝐷 𝐵 𝐴 = 𝑚 ∠ 𝐴 ; m∠EBC=m∠C 𝑚 ∠ 𝐸 𝐵 𝐶 = 𝑚 ∠ 𝐶 3. Definition of congruent angles 4. m∠DBA+m∠ABC+m∠EBC=180° 𝑚 ∠ 𝐷 𝐵 𝐴 + 𝑚 ∠ 𝐴 𝐵 𝐶 + 𝑚 ∠ 𝐸 𝐵 𝐶 = 180 ° 4. Definition of a straight angle 5. m∠A+m∠ABC+m∠C=180° 𝑚 ∠ 𝐴 + 𝑚 ∠ 𝐴 𝐵 𝐶 + 𝑚 ∠ 𝐶 = 180 ° 5. Substitution (1 point)Responses Statement 2 has a mistake. It should say ∠DBA≅∠A ∠ 𝐷 𝐵 𝐴 ≅ ∠ 𝐴 ; ∠EBC≅∠C ∠ 𝐸 𝐵 𝐶 ≅ ∠ 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∠A+m∠ABC+m∠C=180° 𝑚 ∠ 𝐴 + 𝑚 ∠ 𝐴 𝐵 𝐶 + 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 4 has a mistake. It should say m∠DBA+m∠ABC+m∠ACB=180° 𝑚 ∠ 𝐷 𝐵 𝐴 + 𝑚 ∠ 𝐴 𝐵 𝐶 + 𝑚 ∠ 𝐴 𝐶 𝐵 = 180 ° . Statement 4 has a mistake. It should say m angle upper D upper B upper A plus m angle upper A upper B upper C plus m angle upper A upper C upper B equals 180 degrees . Statement 2 has a mistake. It should say ∠DBA≅∠B ∠ 𝐷 𝐵 𝐴 ≅ ∠ 𝐵 ; ∠EBC≅∠A ∠ 𝐸 𝐵 𝐶 ≅ ∠ 𝐴 .

The mistake is in Statement 4. The statement currently says:

4. m∠DBA + m∠ABC + m∠EBC = 180°

This is incorrect because m∠EBC should correspond to m∠ACB rather than m∠ABC. The angles ∠DBA, ∠ABC, and ∠EBC (which corresponds to angle C) should all be accounted properly to show the angle sum in triangle ABC.

The corrected statement should be:

4. m∠DBA + m∠ABC + m∠ACB = 180°

So the fix should clarify that one of the angles involved in the sum of the angles of the triangle should instead be expressed as m∠ACB.

Let's analyze the other options to clarify why they are not the correct answers.

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

   • This statement actually is correct as it stands because according to the given properties of parallel lines and the definition of alternate interior angles, it accurately reflects that ∠DBA is congruent to ∠A and ∠EBC is congruent to ∠C when DE is parallel to AC. There is no mistake in this statement.

2. Statement 2 has a mistake. It should say angle D B A congruent to angle A; angle E B C congruent to angle C.

   • This claim is similar to the first option and also is not correct because Statement 2 is accurate as written. The statement accurately states the congruence of angles formed by a transversal cutting through parallel lines.

3. Statement 4 has a mistake. It should say m∠A + m∠ABC + m∠C = 180°.

   • This option also misidentifies where the error lies. The naming of the angles in Statement 4 is correct according to the triangle. The angles are properly identified as m∠A, m∠ABC, and m∠C. So this is not a correction needed here.

4. Statement 4 has a mistake. It should say m A plus m A B C plus m C equals 180 degrees.

   • Again, this option is just a rephrasing of the previously noted correct version, thus it doesn't resolve any errors—Statement 4 is already correct in its current form.

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

   • While this suggestion is indeed valid (as explained previously), it doesn't deal with existing incorrectness, it merely suggests a revision to accurately replace angle EBC with ACB without recognizing the error in naming in a different context in the proof.

6. Statement 4 has a mistake. It should say m D B A plus m A B C plus m A C B equals 180 degrees.

   • This option is similar to the previous one and restates the same issue but simply changes the notation style and doesn't rectify the mislabeling of angles accurately.

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

   • This is incorrect because it confuses the relationships between the angles; it inaccurately states the congruences involving angles that do not properly relate to triangle ABC.

In conclusion, the significant error lies in Statement 4 concerning the identification of angles, particularly mislabeling which angles are supposed to complement one another. The suggested correction (m∠DBA + m∠ABC + m∠ACB = 180°) accurately fixes the initial internal angle designation error, leading to the correct assertion of the Triangle Angle Sum Theorem.