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Given: [tex]$\angle 1$[/tex] and [tex]$\angle 2$[/tex] are vertical angles, [tex]$m \angle 1=5(x+12)$[/tex], and [tex]$m \angle 2=145$[/tex]

Prove: [tex]$x=17$[/tex]

Statement:
1. [tex]$\angle 1$[/tex] and [tex]$\angle 2$[/tex] are vertical angles.
2. [tex]$m \angle 1 = m \angle 2$[/tex]
3. [tex]$m \angle 1 = 5(x+12)$[/tex]
4. [tex]$m \angle 1 = 5x + 60$[/tex]
5. [tex]$m \angle 2 = 145$[/tex]
6. [tex]$5x + 60 = 145$[/tex]
7. [tex]$5x = 85$[/tex]
8. [tex]$x = 17$[/tex]

Reason:
1. Given
2. Definition of Vertical Angles
3. Given
4. Distributive Property
5. Given
6. Substitution
7. Subtraction Property of Equality
8. Division Property of Equality

Select the reason that best supports Statement 4 in the given proof.
A. Subtraction Property of Equality
B. Distributive Property
C. Division Property of Equality
D. Transitive Property



Answer :

GinnyAnswer
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.

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