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.