The missing reason in step 3 pertains to why the sum of the measures of angles [tex]\( \angle TRS \)[/tex] and [tex]\( \angle TRV \)[/tex] equals 180 degrees. This is because the angles are a linear pair, and the sum of the measures of a linear pair of angles is always 180 degrees. Hence, the correct reason is the linear pair postulate (or linear pair theorem).
Therefore, the detailed step-by-step reasoning should be:
\begin{tabular}{|c|c|c|c|}
\hline
& Statements & & Reasons \\
\hline
1 & [tex]$m \angle T R V = 60^{\circ};\ m \angle T R S = (4 x)^{\circ}$[/tex] & 1. & Given \\
\hline
2 & [tex]$\angle TRS$[/tex] and [tex]$\angle TRV$[/tex] are a linear pair & 2. & Definition of linear pair \\
\hline
3 & [tex]$m \angle T R S + m \angle T R V = 180^{\circ}$[/tex] & 3. & Linear pair postulate \\
\hline
4 & [tex]$60 + 4x = 180$[/tex] & 4. & Substitution property of equality \\
\hline
5 & [tex]$4x = 120$[/tex] & 5. & \begin{tabular}{l}
Subtraction property of equality \\
Division property of equality
\end{tabular} \\
\hline
& [tex]$x = 30$[/tex] & & \\
\hline
\end{tabular}
By stating that the angles [tex]\( \angle TRS \)[/tex] and [tex]\( \angle TRV \)[/tex] form a linear pair, and then asserting that their measures add up to 180 degrees by the linear pair postulate, we properly justify step 3.
