Let's find the missing reason in step 3 of the given proof.
Here are the statements and reasons provided up to step 3:
1. Given:
[tex]\[
m \angle TRV = 60^{\circ} \quad; \quad m \angle TRS = (4x)^{\circ}
\][/tex]
2. Reason:
[tex]\[
\angle TRS \text{ and } \angle TRV \text{ are a linear pair} \quad \text{(definition of linear pair)}
\][/tex]
3. Statement:
[tex]\[
m \angle TRS + m \angle TRV = 180^{\circ}
\][/tex]
The missing reason in step 3 is the principle that allows us to combine the measures of angles to equal 180 degrees when they form a linear pair.
The correct reason for this step is the angle addition postulate, which states that if two angles form a linear pair, their measures add up to 180 degrees.
Therefore, the completed proof looks like this:
[tex]\[
\begin{array}{ll|ll}
\multicolumn{1}{c|}{\text{Statements}} & \multicolumn{2}{c}{\text{Reasons}} \\
\hline
1. & m \angle TRV = 60^{\circ} ; m \angle TRS = (4x)^{\circ} & 1. & \text{Given} \\
2. & \angle TRS \text{ and } \angle TRV \text{ are a linear pair} & 2. & \text{Definition of linear pair} \\
3. & m \angle TRS + m \angle TRV = 180^{\circ} & 3. & \text{Angle addition postulate} \\
4. & 60 + 4x = 180 & 4. & \text{Substitution property of equality} \\
5. & 4x = 120 & 5. & \text{Subtraction property of equality} \\
6. & x = 30 & 6. & \text{Division property of equality}
\end{array}
\][/tex]
