Let's analyze the steps provided and determine the missing reason for step 3.
1. The statement is [tex]\( m \angle TRV = 60^\circ ; m \angle TRS = (4x)^\circ \)[/tex], and the reason is "given."
2. The statement is [tex]\( \angle TRS \)[/tex] and [tex]\( \angle TRV \)[/tex] are a linear pair, and the reason is "definition of linear pair."
Now, for step 3:
3. The statement is [tex]\( m \angle TRS + m \angle TRV = 180^\circ \)[/tex].
When we have a linear pair of angles, the sum of their measures is always 180 degrees. This is based on the angle addition postulate, which states that the sum of angles in a linear pair is equal to 180 degrees.
Thus, the missing reason for step 3 is:
[tex]\[ \text{Reason: angle addition postulate} \][/tex]
Here's the completed step-by-step solution:
\begin{tabular}{ll|ll}
\multicolumn{1}{c|}{Statements} & \multicolumn{2}{c}{Reasons} \\
\hline
1. [tex]\(m \angle TRV = 60^\circ ; m \angle TRS = (4x)^\circ\)[/tex] & 1. & given \\
2. [tex]\(\angle TRS\)[/tex] and [tex]\(\angle TRV\)[/tex] are a linear pair & 2. & definition of linear pair \\
3. [tex]\(m \angle TRS + m \angle TRV = 180^\circ\)[/tex] & 3. & angle addition postulate \\
4. [tex]\(60 + 4x = 180\)[/tex] & 4. & substitution property of equality \\
5. [tex]\(4x = 120\)[/tex] & 5. & subtraction property of equality \\
6. [tex]\(x = 30\)[/tex] & 6. & division property of equality \\
\end{tabular}
