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What is the missing reason in step 3?

| Statements | Reasons |
|--------------------------------------------|-----------------------------------|
| 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. Angle addition postulate |
| 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 |

A. Substitution property of equality
B. Angle addition postulate
C. Subtraction property of equality
D. Addition property of equality

Sagot :

GinnyAnswer
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.
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