To prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex], we can complete the statements with the appropriate reasons in the following manner:
1. Points [tex]\(A, B\)[/tex], and [tex]\(C\)[/tex] form a triangle. & [tex]\( \text{given} \)[/tex]
2. Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex]. & [tex]\( \text{definition of parallel lines} \)[/tex]
3. [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex]. & [tex]\( \text{corresponding angles are congruent} \)[/tex]
4. [tex]\(m\angle 1 = m\angle 4\)[/tex] and [tex]\(m\angle 3 = m\angle 5\)[/tex]. & [tex]\(\text{measures of congruent angles are equal}\)[/tex]
5. [tex]\(m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ\)[/tex]. & [tex]\(\text{sum of angles on a straight line}\)[/tex]
6. [tex]\(m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ\)[/tex]. & [tex]\(\text{angle addition and definition of a straight line}\)[/tex]
