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What is the missing reason in step 3?

[tex]\[
\begin{array}{l|ll}
\multicolumn{1}{c|}{\text{Statements}} & \multicolumn{1}{c}{\text{Reasons}} \\
\hline
1. \, m \angle TRV = 60^{\circ}, \, m \angle TRS = (4x)^{\circ} & 1. \, \text{Given} \\
2. \, \angle TRS \text{ and } \angle TRV \text{ are a linear pair} & 2. \, \text{Definition of linear pair} \\
3. \, m \angle TRS + m \angle TRV = 180^{\circ} & 3. \, \text{Angle addition postulate} \\
4. \, 60 + 4x = 180 & 4. \, \text{Substitution property of equality} \\
5. \, 4x = 120 & 5. \, \text{Subtraction property of equality} \\
6. \, x = 30 & 6. \, \text{Division property of equality} \\
\end{array}
\][/tex]

Sagot :

GinnyAnswer
To provide a detailed, step-by-step solution to address the missing reason in step 3, let's analyze the statements and reasons provided in the question:

### Given Statements and Reasons
1. Statement: [tex]\( m \angle TRV = 60^\circ, \, m \angle TRS = (4x)^\circ \)[/tex]
Reason: Given.

2. Statement: [tex]\( \angle TRS \)[/tex] and [tex]\( \angle TRV \)[/tex] are a linear pair.
Reason: Definition of a linear pair.

3. Statement: [tex]\( m \angle TRS + m \angle TRV = 180^\circ \)[/tex]
Reason: ?

4. Statement: [tex]\( 60 + 4x = 180 \)[/tex]
Reason: Substitution property of equality.

5. Statement: [tex]\( 4x = 120 \)[/tex]
Reason: Subtraction property of equality.

6. Statement: [tex]\( x = 30 \)[/tex]
Reason: Division property of equality.

### Detailed Solution and Explanation

#### Step 1:
- Statement: [tex]\( m \angle TRV = 60^\circ, \, m \angle TRS = (4x)^\circ \)[/tex]
- Reason: Given.

This step sets up the measures of the two angles involved.

#### Step 2:
- Statement: [tex]\( \angle TRS \)[/tex] and [tex]\( \angle TRV \)[/tex] are a linear pair.
- Reason: Definition of a linear pair.

This statement defines the relationship between the two angles. A linear pair of angles are adjacent, forming a straight line at their shared vertex, making the sum of their measures 180 degrees.

#### Step 3:
- Statement: [tex]\( m \angle TRS + m \angle TRV = 180^\circ \)[/tex]
- Reason: Linear Pair Postulate.

The sum of the measures of a linear pair of angles is 180 degrees. This property is known as the Linear Pair Postulate.

#### Step 4:
- Statement: [tex]\( 60 + 4x = 180 \)[/tex]
- Reason: Substitution property of equality.

In this step, we substitute the given measures [tex]\( m \angle TRV = 60^\circ \)[/tex] and [tex]\( m \angle TRS = 4x \)[/tex] into the equation from step 3.

#### Step 5:
- Statement: [tex]\( 4x = 120 \)[/tex]
- Reason: Subtraction property of equality.

To isolate the term with the variable [tex]\( x \)[/tex], we subtract 60 from both sides of the equation:

[tex]\[ 60 + 4x - 60 = 180 - 60 \implies 4x = 120. \][/tex]

#### Step 6:
- Statement: [tex]\( x = 30 \)[/tex]
- Reason: Division property of equality.

Finally, we solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 4:

[tex]\[ 4x \div 4 = 120 \div 4 \implies x = 30. \][/tex]

### Conclusion
The missing reason in step 3 is the Linear Pair Postulate, which states that the sum of the measures of a linear pair of angles is always 180 degrees.
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