Answer :
To prove that \( x = 17 \) using the given statements and reasons, we need to select the appropriate reason that supports Statement 4. Here is a step-by-step detailed solution:
1. \(\angle 1\) and \(\angle 2\) are vertical angles.
- Reason: Given
2. \( m \angle 1 = m \angle 2 \)
- Reason: Definition of Vertical Angles
3. \( m \angle 1 = 5(x + 12) \)
- Reason: Given
4. \( m \angle 1 = 5x + 60 \)
- Reason: Distributive Property
5. \( m \angle 2 = 145 \)
- Reason: Given
6. \( 5x + 60 = 145 \)
- Reason: Substitution (substituting the expressions for \( m \angle 1 \) and \( m \angle 2 \))
7. \( 5x = 85 \)
- Reason: Subtraction Property of Equality (subtracting 60 from both sides)
8. \( x = 17 \)
- Reason: Division Property of Equality (dividing both sides by 5)
So, the complete proof is as follows:
1. \(\angle 1\) and \(\angle 2\) are vertical angles.
- Reason: Given
2. \( m \angle 1 = m \angle 2 \)
- Reason: Definition of Vertical Angles
3. \( m \angle 1 = 5(x + 12) \)
- Reason: Given
4. \( m \angle 1 = 5x + 60 \)
- Reason: Distributive Property
5. \( m \angle 2 = 145 \)
- Reason: Given
6. \( 5x + 60 = 145 \)
- Reason: Substitution
7. \( 5x = 85 \)
- Reason: Subtraction Property of Equality
8. \( x = 17 \)
- Reason: Division Property of Equality
Thus, the correct reason that best supports Statement 4 is: Distributive Property.
1. \(\angle 1\) and \(\angle 2\) are vertical angles.
- Reason: Given
2. \( m \angle 1 = m \angle 2 \)
- Reason: Definition of Vertical Angles
3. \( m \angle 1 = 5(x + 12) \)
- Reason: Given
4. \( m \angle 1 = 5x + 60 \)
- Reason: Distributive Property
5. \( m \angle 2 = 145 \)
- Reason: Given
6. \( 5x + 60 = 145 \)
- Reason: Substitution (substituting the expressions for \( m \angle 1 \) and \( m \angle 2 \))
7. \( 5x = 85 \)
- Reason: Subtraction Property of Equality (subtracting 60 from both sides)
8. \( x = 17 \)
- Reason: Division Property of Equality (dividing both sides by 5)
So, the complete proof is as follows:
1. \(\angle 1\) and \(\angle 2\) are vertical angles.
- Reason: Given
2. \( m \angle 1 = m \angle 2 \)
- Reason: Definition of Vertical Angles
3. \( m \angle 1 = 5(x + 12) \)
- Reason: Given
4. \( m \angle 1 = 5x + 60 \)
- Reason: Distributive Property
5. \( m \angle 2 = 145 \)
- Reason: Given
6. \( 5x + 60 = 145 \)
- Reason: Substitution
7. \( 5x = 85 \)
- Reason: Subtraction Property of Equality
8. \( x = 17 \)
- Reason: Division Property of Equality
Thus, the correct reason that best supports Statement 4 is: Distributive Property.