Sure! Let's complete the two-column proof based on the paragraph proof provided.
Here is the completed two-column proof with the correct statement and reason for line 5:
\begin{tabular}{|l|l|}
\hline
Statements & Reasons \\
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1. [tex]$\angle 2$[/tex] is supplementary to [tex]$\angle 3$[/tex] & 1. Linear pair theorem \\
\hline
2. [tex]$m \angle 2 + m \angle 3 = 180^{\circ}$[/tex] & 2. Definition of supplementary angles \\
\hline
3. [tex]$\angle 3 \cong \angle 4$[/tex] & 3. Given \\
\hline
4. [tex]$m \angle 3 = m \angle 4$[/tex] & 4. Definition of congruence \\
\hline
5. [tex]$m \angle 2 + m \angle 4 = 180^{\circ}$[/tex] & 5. Substitution property of equality \\
\hline
6. [tex]$\angle 2$[/tex] is supplementary to [tex]$\angle 4$[/tex] & 6. Definition of supplementary angles \\
\hline
7. [tex]$\angle 1$[/tex] is supplementary to [tex]$\angle 4$[/tex] & 7. Linear pair theorem \\
\hline
8. [tex]$\angle 1 \cong \angle 2$[/tex] & 8. Congruent supplements theorem \\
\hline
\end{tabular}
Explanation:
- For line 5, we substitute [tex]$m \angle 4$[/tex] for [tex]$m \angle 3$[/tex] in the equation [tex]$m \angle 2 + m \angle 3 = 180^{\circ}$[/tex], based on the fact that [tex]$m \angle 3 = m \angle 4$[/tex]. This uses the substitution property of equality.
- For line 6, the result directly follows from the definition of supplementary angles, given the equation [tex]$m \angle 2 + m \angle 4 = 180^{\circ}$[/tex].
- For line 8, the congruent supplements theorem states that if two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent.
