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 DB←→
𝐷
𝐵
↔
parallel to AC¯¯¯¯¯¯¯¯
𝐴
𝐶
¯
and CB¯¯¯¯¯¯¯¯
𝐶
𝐵
¯
extended to make CE←→
𝐶
𝐸
↔


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 DB←→
𝐷
𝐵
↔
parallel to AC¯¯¯¯¯¯¯¯
𝐴
𝐶
¯
and CB¯¯¯¯¯¯¯¯
𝐶
𝐵
¯
extended to make CE←→
𝐶
𝐸
↔
1. Given
2. ∠A≅∠ABD
∠
𝐴
≅
∠
𝐴
𝐵
𝐷
2. DB←→∥AC¯¯¯¯¯¯¯¯
𝐷
𝐵
↔
∥
𝐴
𝐶
¯
and alternate interior angles are congruent when lines are parallel
3. ∠C≅∠DBC
∠
𝐶
≅
∠
𝐷
𝐵
𝐶
3. DB←→∥AC¯¯¯¯¯¯¯¯
𝐷
𝐵
↔
∥
𝐴
𝐶
¯
and alternate interior angles are congruent when lines are parallel
4. m∠A=m∠ABD; m∠C=m∠DBE
𝑚
∠
𝐴
=
𝑚
∠
𝐴
𝐵
𝐷
;

𝑚
∠
𝐶
=
𝑚
∠
𝐷
𝐵
𝐸
4. Definiti

Answer 1

The mistake in the proof lies in **Statement 4**, where it incorrectly states that \( m∠C = m∠DBE \).

The correct relationship is that \( m∠C = m∠DBC \) because \( ∠C \) and \( ∠DBC \) are also alternate interior angles created by line \( DB \) being parallel to line \( AC \) with transversal \( BC \).

### Correction:

Change the statement in Statement 4 from:
- \( m∠C=m∠DBE \)

To:
- \( m∠C=m∠DBC \)

The corrected statement should read:
4. \( m∠A = m∠ABD; \, m∠C = m∠DBC \)
**Reason:** Definition of congruent angles (alternating interior angles).

By including this correction, the overall reasoning leading to the conclusion of the Triangle Angle Sum Theorem becomes correct, ensuring that the sum \( m∠A + m∠ABC + m∠C = 180° \) is valid.