Let's analyze each step in the given problem and provide a detailed explanation for finding the missing reason in step 3.
### Step-by-Step Solution:
1. Statement 1: [tex]\(m \angle TRV = 60^\circ ; m \angle TRS = (4x)^\circ\)[/tex]
Reason 1: This is provided as given data.
2. Statement 2: [tex]\(\angle TRS\)[/tex] and [tex]\(\angle TRV\)[/tex] are a linear pair
Reason 2: This is based on the definition of a linear pair. Two angles are a linear pair if they are adjacent (share a common arm) and their non-common sides are opposite rays (they form a straight line).
3. Statement 3: [tex]\(m \angle TRS + m \angle TRV = 180^\circ\)[/tex]
Reason 3: The missing reason here is crucial. For this statement, we need to explain why the measures of these two angles add up to 180 degrees.
Missing Reason: Two angles that form a linear pair are supplementary, which implies their measures add up to 180 degrees. These angles are adjacent and their outer sides form a straight line, making the sum of their measures equal to 180 degrees. This is known as the angle addition postulate.
4. Statement 4: [tex]\(60 + 4x = 180\)[/tex]
Reason 4: We use the substitution property of equality. Here, we substitute the given values of the angle measures into the equation formed in step 3.
5. Statement 5: [tex]\(4x = 120\)[/tex]
Reason 5: We apply the subtraction property of equality. By subtracting 60 from both sides of the equation [tex]\(60 + 4x = 180\)[/tex], we obtain [tex]\(4x = 120\)[/tex].
6. Statement 6: [tex]\(x = 30\)[/tex]
Reason 6: We use the division property of equality. By dividing both sides of the equation [tex]\(4x = 120\)[/tex] by 4, we solve for [tex]\(x\)[/tex] and get [tex]\(x = 30\)[/tex].
### Conclusion:
Thus, the missing reason in step 3, which justifies why the measures of [tex]\(\angle TRS\)[/tex] and [tex]\(\angle TRV\)[/tex] add up to 180 degrees, is the angle addition postulate.
