Certainly! Let's go through the step-by-step solution to understand the missing reason in step 3.
1. [tex]\( m \angle TRV = 60^\circ ; m \angle TRS = (4x)^\circ \)[/tex]
- Given: These angles and their expressions are provided in the problem statement.
2. [tex]\( \angle TRS \)[/tex] and [tex]\( \angle TRV \)[/tex] are a linear pair
- Reason: This statement is based on the definition of a linear pair. Linear pairs are pairs of adjacent angles formed when two lines intersect that are supplementary.
3. [tex]\( m \angle TRS + m \angle TRV = 180^\circ \)[/tex]
- Missing Reason: The measure of angles that form a linear pair are always supplementary (i.e., their measures add up to 180 degrees). This is based on the Linear Pair Postulate, which states that if two angles form a linear pair, then they are supplementary.
4. [tex]\( 60 + 4x = 180 \)[/tex]
- Reason: This follows from the substitution property of equality, where we replace [tex]\( m \angle TRV \)[/tex] with 60 and [tex]\( m \angle TRS \)[/tex] with [tex]\( 4x \)[/tex].
5. [tex]\( 4x = 120 \)[/tex]
- Reason: By applying the subtraction property of equality, we subtract 60 from both sides of the equation.
6. [tex]\( x = 30 \)[/tex]
- Reason: Finally, using the division property of equality, we divide both sides of the equation by 4 to solve for [tex]\( x \)[/tex].
So, the complete reasoning for step 3 is:
```
3. [tex]\( m \angle TRS + m \angle TRV = 180^\circ \)[/tex]
- Reason: Linear Pair Postulate
```
This postulate states that if two angles form a linear pair, their measures add up to 180 degrees.
