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°StatementsReasons1. △ABC with DE←→∥AC¯¯¯¯¯¯¯¯1. Given2. ∠DBA≅∠B; ∠EBC≅∠C2. DE←→∥AC¯¯¯¯¯¯¯¯ and alternate interior angles are congruent when lines are parallel3. m∠DBA=m∠A; m∠EBC=m∠C3. 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)ResponsesStatement 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 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 .

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The mistake in the proof is in Statement 2. It incorrectly states:

∠DBA≅∠B; ∠EBC≅∠C

The correct relationships should be:

∠DBA≅∠A; ∠EBC≅∠C

This is because when two lines are parallel (DE and AC), the angles formed with a transversal (here, line BC) are such that the angle ∠DBA relates to angle ∠A and angle ∠EBC relates to angle ∠C, due to the properties of alternate interior angles.

So the correction for Statement 2 should read:

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

Or, in more explicit terms:

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.

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°StatementsReasons1. △ABC with DE←→∥AC¯¯¯¯¯¯¯¯1. Given2. ∠DBA≅∠B; ∠EBC≅∠C2. DE←→∥AC¯¯¯¯¯¯¯¯ and alternate interior angles are congruent when lines are parallel3. m∠DBA=m∠A; m∠EBC=m∠C3. 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)ResponsesStatement 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 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 .

1 answer

The mistake in the proof is in Statement 2. It incorrectly states:

∠DBA≅∠B; ∠EBC≅∠C

The correct relationships should be:

∠DBA≅∠A; ∠EBC≅∠C

This is because when two lines are parallel (DE and AC), the angles formed with a transversal (here, line BC) are such that the angle ∠DBA relates to angle ∠A and angle ∠EBC relates to angle ∠C, due to the properties of alternate interior angles.

So the correction for Statement 2 should read:

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

Or, in more explicit terms:

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.

GPT-4o mini · Answer

The mistake in the proof is in **Statement 2**. It incorrectly states: - ∠DBA≅∠B; ∠EBC≅∠C The correct relationships should be: - ∠DBA≅∠A; ∠EBC≅∠C This is because when two lines are parallel (DE and AC), the angles formed with a transversal (here, line BC) are such that the angle ∠DBA relates to angle ∠A and angle ∠EBC relates to angle ∠C, due to the properties of alternate interior angles. So the correction for Statement 2 should read: - Statement 2 has a mistake. It should say ∠DBA≅∠A; ∠EBC≅∠C. Or, in more explicit terms: - 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.