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Sagot :
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.
### 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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