To fill in line 4 of the two-column proof, let's examine the statements and reasons given in the paragraph proof.
So far, we have:
1. [tex]$C E = C D + D E$[/tex] (segment addition)
2. [tex]$C D = E F$[/tex] (given)
3. [tex]$D F = C D + D E$[/tex] (substitution property of equality)
From the paragraph proof, we know that since [tex]$C E$[/tex] and [tex]$D F$[/tex] equal the same quantity ([tex]$C D + D E$[/tex]), we can state that [tex]$C E = D F$[/tex] using the transitive property of equality.
Thus, line 4 should state:
- Statement: [tex]$C E = D F$[/tex]
- Reason: Transitive property of equality
Therefore, placing this into the two-column proof:
\begin{tabular}{|c|c|}
\hline Statements & Reasons \\
\hline \begin{tabular}{l}
[tex]$C E=C D+D E$[/tex] \\
[tex]$D F=E F+D E$[/tex]
\end{tabular} & 1. segment addition \\
\hline 2. [tex]$C D=E F$[/tex] & 2. given \\
\hline 3. [tex]$D F=C D+D E$[/tex] & 3. substitution property of equality \\
\hline 4. [tex]$C E=D F$[/tex] & 4. transitive property of equality \\
\hline 5. [tex]$A B=C E$[/tex] & 5. given \\
\hline 6. [tex]$A B=D F$[/tex] & 6. transitive property of equality \\
\hline
\end{tabular}
