Let's analyze the given statement and the corresponding reasons to find the missing reason in step 3.
Firstly, consider the problem setup:
1. Statements: [tex]\( m \angle TRV = 60^{\circ} \)[/tex] and [tex]\( m \angle TRS = (4x)^{\circ} \)[/tex]
- Reasons: Given.
2. Statements: [tex]\( \angle TRS \)[/tex] and [tex]\( \angle TRV \)[/tex] are a linear pair.
- Reasons: Definition of a linear pair.
Next, we need to find the missing reason for step 3:
3. Statements: [tex]\( m \angle TRS + m \angle TRV = 180^{\circ} \)[/tex]
- Reasons: ?
When two angles form a linear pair, their measures add up to 180 degrees. This is because a linear pair is formed by adjacent angles that are supplementary (i.e., they add up to [tex]\(180^{\circ}\)[/tex]).
This leads us to the step-by-step reasoning for the missing reason:
- A linear pair of angles has this property due to the Angle Addition Postulate. This postulate states that the measure of an angle formed by two adjacent angles is the sum of the measures of the two angles. Therefore, when [tex]\( \angle TRS \)[/tex] and [tex]\( \angle TRV \)[/tex] are stated to be a linear pair, their measures add up to [tex]\( 180^{\circ} \)[/tex] due to this postulate.
So, the missing reason for step 3 is "angle addition postulate".
Summarizing the solution:
3. Statements: [tex]\( m \angle TRS + m \angle TRV = 180^{\circ} \)[/tex]
- Reasons: Angle Addition Postulate
