Sure, let me break down the steps and provide the missing reasoning for step 3.
Here's the complete solution with detailed reasoning:
1. \( m \angle TRV = 60^{\circ} ; m \angle TRS = (4x)^{\circ} \)
- Reason: Given
2. \( \angle TRS \) and \( \angle TRV \) are a linear pair
- Reason: Definition of linear pair
3. \( m \angle TRS + m \angle TRV = 180^{\circ} \)
- Reason: Linear pairs of angles are supplementary (definition of supplementary angles)
4. \( 60 + 4x = 180 \)
- Reason: Substitution property of equality (substituting the given measures of the angles into the equation)
5. \( 4x = 120 \)
- Reason: Subtraction property of equality (subtracting 60 from both sides)
6. \( x = 30 \)
- Reason: Division property of equality (dividing both sides by 4)
Here is the completed table with the missing reason in step 3:
\begin{tabular}{ll|ll}
\multicolumn{1}{c|}{ Statements } & & \multicolumn{1}{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. & linear pairs of angles are supplementary \\
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}
The missing reason in step 3 is "linear pairs of angles are supplementary".
