Sure! Let's go through the problem step-by-step to identify the missing reason in step 3.
### Step-by-Step Solution
1. Statement 1: [tex]\(m \angle TRV = 60^{\circ} ; m \angle TRS = (4x)^{\circ}
- Reason: Given
- This is a given statement where the measure of angle \( \angle TRV \)[/tex] is [tex]\( 60^{\circ} \)[/tex] and the measure of angle [tex]\( \angle TRS \)[/tex] is expressed as [tex]\( 4x \)[/tex].
2. Statement 2: [tex]\( \angle TRS \)[/tex] and [tex]\( \angle TRV \)[/tex] are a linear pair
- Reason: Definition of linear pair
- By definition, two angles are a linear pair if they are adjacent (share a common side) and their non-common sides form a straight line.
3. Statement 3: [tex]\( m \angle TRS + m \angle TRV = 180 \)[/tex]
- Reason: Angle addition postulate
- The missing reason here is the angle addition postulate, which states that if two angles form a linear pair (which means they are supplementary), their measures add up to [tex]\( 180^\circ \)[/tex].
4. Statement 4: [tex]\( 60 + 4x = 180 \)[/tex]
- Reason: Substitution property of equality
- Here, we substitute the values [tex]\( m \angle TRV \)[/tex] and [tex]\( m \angle TRS \)[/tex] with [tex]\( 60 \)[/tex] and [tex]\( 4x \)[/tex] respectively in the equation from step 3.
5. Statement 5: [tex]\( 4x = 120 \)[/tex]
- Reason: Subtraction property of equality
- We subtract [tex]\( 60 \)[/tex] from both sides of the equation [tex]\( 60 + 4x = 180 \)[/tex] to isolate the term [tex]\( 4x \)[/tex].
6. Statement 6: [tex]\( x = 30 \)[/tex]
- Reason: Division property of equality
- Finally, we divide both sides of the equation [tex]\( 4x = 120 \)[/tex] by [tex]\( 4 \)[/tex] to solve for [tex]\( x \)[/tex].
### Conclusion
So, the missing reason in step 3 is the angle addition postulate.
