To prove that the sum of the interior angles of [tex]\(\triangle ABC\)[/tex] is [tex]\(180^\circ\)[/tex], let's complete the statements logically.
- Points [tex]\(A, B\)[/tex], and [tex]\(C\)[/tex] form a triangle. & given
- Let [tex]\(\overline{DE}\)[/tex] be a line passing through [tex]\(B\)[/tex] and parallel to [tex]\(\overline{AC}\)[/tex] & definition of parallel lines
- [tex]\(\angle 3 \cong \angle 5\)[/tex] and [tex]\(\angle 1 \cong \angle 4\)[/tex] & alternate interior angles
- [tex]\(m \angle 1 = m \angle 4\)[/tex] and [tex]\(m \angle 3 = m \angle 5\)[/tex] & congruent angles have equal measures
- [tex]\(m \angle 4 + m \angle 2 + m \angle 5 = 180^\circ\)[/tex] & angles forming a straight line sum to [tex]\(180^\circ\)[/tex]
- [tex]\(m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ\)[/tex] & angle addition and definition of a straight line
Thereby, the table, completed with correct reasons, should look like this:
\begin{tabular}{|l|l|}
\hline Statement & Reason \\
\hline Points [tex]$A, B$[/tex], and [tex]$C$[/tex] form a triangle. & given \\
\hline Let [tex]$\overline{D E}$[/tex] be a line passing through [tex]$B$[/tex] and parallel to [tex]$\overline{A C}$[/tex] & definition of parallel lines \\
\hline[tex]$\angle 3 \cong \angle 5$[/tex] and [tex]$\angle 1 \cong \angle 4$[/tex] & alternate interior angles \\
\hline [tex]$m \angle 1= m \angle 4$[/tex] and [tex]$m \angle 3= m \angle 5$[/tex] & congruent angles have equal measures \\
\hline [tex]$m \angle 4+ m \angle 2+ m \angle 5=180^{\circ}$[/tex] & angles forming a straight line sum to [tex]$180^\circ$[/tex] \\
\hline [tex]$m \angle 1+ m \angle 2+ m \angle 3=180^{\circ}$[/tex] & angle addition and definition of a straight line \\
\hline
\end{tabular}
This detailed step-by-step reasoning demonstrates that the sum of the interior angles in [tex]\(\triangle ABC\)[/tex] is indeed [tex]\(180^\circ\)[/tex].
