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 4 has a mistake. It should say m∠DBA+m∠ABC+m∠ACB=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≅∠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∠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≅∠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 .

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The statement with a mistake is Statement 4, which incorrectly states:

m∠DBA + m∠ABC + m∠EBC = 180°

The correct statement should refer to the angles that are part of triangle ABC. In this case, it should refer to:

m∠DBA + m∠ABC + m∠ACB = 180°

This is because ∠DBA and ∠EBC are angles formed at point B outside triangle ABC due to the parallel lines DE and AC, but the sum of the angles around point B in triangle ABC is what needed to be cited to establish that their measures sum up to 180 degrees.

So, the corrected statement should read:

"m∠DBA + m∠ABC + m∠ACB = 180°"

This accurately reflects the sum of the angles as intended in the triangle's angle sum theorem.

answered by GPT-4o mini

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Question

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 4 has a mistake. It should say m∠DBA+m∠ABC+m∠ACB=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≅∠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∠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≅∠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 .

1 answer

The statement with a mistake is Statement 4, which incorrectly states:

m∠DBA + m∠ABC + m∠EBC = 180°

The correct statement should refer to the angles that are part of triangle ABC. In this case, it should refer to:

m∠DBA + m∠ABC + m∠ACB = 180°

This is because ∠DBA and ∠EBC are angles formed at point B outside triangle ABC due to the parallel lines DE and AC, but the sum of the angles around point B in triangle ABC is what needed to be cited to establish that their measures sum up to 180 degrees.

So, the corrected statement should read:

"m∠DBA + m∠ABC + m∠ACB = 180°"

This accurately reflects the sum of the angles as intended in the triangle's angle sum theorem.

GPT-4o mini · Answer

The statement with a mistake is **Statement 4**, which incorrectly states: 4. **m∠DBA + m∠ABC + m∠EBC = 180°** The correct statement should refer to the angles that are part of triangle ABC. In this case, it should refer to: **m∠DBA + m∠ABC + m∠ACB = 180°** This is because ∠DBA and ∠EBC are angles formed at point B outside triangle ABC due to the parallel lines DE and AC, but the sum of the angles around point B in triangle ABC is what needed to be cited to establish that their measures sum up to 180 degrees. So, the corrected statement should read: **"m∠DBA + m∠ABC + m∠ACB = 180°"** This accurately reflects the sum of the angles as intended in the triangle's angle sum theorem.