Question

Consider the following proof of the Triangle Angle Sum Theorem. Which statement has a mistake?Given:  △ABC  with  DE←→∥AC¯¯¯¯¯¯¯¯ Prove:  m∠A+m∠ABC+m∠C=180° Statements Reasons1.  △ABC  with  DE←→∥AC¯¯¯¯¯¯¯¯ 1. Given2.  ∠DBA≅∠A ;  ∠EBC≅∠C 2.  DE←→∥AC¯¯¯¯¯¯¯¯  and alternate interior angles are congruent when lines are parallel3. m∠DBA=m∠A ;  m∠EBC=m∠C 3. Definition of congruent angles 4.  m∠DBA+m∠ABC+m∠ACB=180° 4. Definition of a straight angle 5.  m∠A+m∠ABC+m∠C=180° 5. Substitution 

Answer 1

The mistake in the proof lies in statement 4.

Here's the reasoning:

1. Statement 4 asserts that \( m∠DBA + m∠ABC + m∠ACB = 180° \) using the reason "Definition of a straight angle." However, this statement is incorrect because the angles \( \angle DBA \), \( \angle ABC \), and \( \angle ACB \) do not necessarily form a straight angle.

Instead, the correct angle to consider here must involve the angles at point \( B \) (where \( A \) is an angle of triangle \( ABC \) and \( C \) is another angle of triangle \( ABC \)). For triangle \( ABC \), the sum of the interior angles is always 180°, but \( \angle DBA \), \( \angle ABC \), and \( \angle ACB \) (not \( AC \)) do not correspond to the angles of triangle \( ABC \).

A corrected version of the statement might involve recognizing that \( \angle DBA \), \( \angle ABC \), and \( \angle CBA \) could be summed correctly if relevant relationships are maintained. Essentially, the statement should clarify which angles are actually summing to 180 degrees, possibly indicating that \( \angle ABC \) is being considered along with the angles formed by the transversal \( DB \) and line \( DE \).

So, the incorrect statement is specifically statement 4.