To show that the sum of the interior angles of [tex]$\triangle ABC$[/tex] is [tex]$180^\circ$[/tex], we need to complete the proof with the correct statements and reasons.
Here's the step-by-step completion:
\begin{tabular}{|l|l|}
\hline
Statement & Reason \\
\hline
Points [tex]$A, B,$[/tex] and [tex]$C$[/tex] form a triangle. & given \\
\hline
Let [tex]$\overline{DE}$[/tex] be a line passing through [tex]$B$[/tex] and parallel to [tex]$\overline{AC}$[/tex]. & definition of parallel lines \\
\hline
[tex]$\angle 3 = \angle 5$[/tex] and [tex]$\angle 1 = \angle 4$[/tex]. & alternate interior angles theorem \\
\hline
[tex]$m \angle 1 = m \angle 4$[/tex] and [tex]$m \angle 3 = m \angle 5$[/tex]. & alternate interior angles are equal \\
\hline
[tex]$m \angle 4 + m \angle 2 + m \angle 5 = 180^\circ$[/tex]. & angle addition and definition of a straight line \\
\hline
[tex]$m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ$[/tex]. & substitution \\
\hline
\end{tabular}
This structured approach, both with statements and reasons, proves that the sum of the interior angles of [tex]$\triangle ABC$[/tex] is [tex]$180^\circ$[/tex].
