Use the image to answer the question. 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 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

Question

Use the image to answer the question. 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 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

Answer 1

The mistake is in Statement 4. It incorrectly states the angles on a straight line. Since \( DE \) is parallel to \( AC \), the correct equation should be:

\[ m∠DBA + m∠ABC + m∠ACB = 180° \]

Thus, the correction should specify that \( ∠ACB \) is the corresponding angle instead of \( ∠EBC \).

So the correct answer is:

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