Question

part 6 of 7
write a two - column proof for the following.
given: m∠2 = 30
m∠1 = 2m∠2
prove: m∠3 + m∠4 = 90
statements

1. m∠2 = 30
2. m∠1 = 2m∠2
3. m∠1 = 2(30)
4. m∠1 = 60
5. m∠3 = 30
6. m∠3 + m∠4 = 30 + m∠4
7. m∠4 = 60
8. m∠3 + m∠4 = 30+□

reasons

1. given
2. given
3. substitution
4. simplification
5. vertical angles theorem
6. addition property of equality
7. vertical angles theorem
8. substitution

Explanation:

Step1: Substitute value of ∠2 into ∠1 formula

Since \(m\angle2 = 30\) and \(m\angle1=2m\angle2\), then \(m\angle1 = 2\times30=60\).

Step2: Use vertical - angles theorem for ∠3

\(\angle2\) and \(\angle3\) are vertical angles. By the Vertical Angles Theorem, \(m\angle3=m\angle2 = 30\).

Step3: Use vertical - angles theorem for ∠4

\(\angle1\) and \(\angle4\) are vertical angles. By the Vertical Angles Theorem, \(m\angle4=m\angle1 = 60\).

Step4: Calculate \(m\angle3 + m\angle4\)

Substitute \(m\angle3 = 30\) and \(m\angle4 = 60\) into \(m\angle3 + m\angle4\), we get \(m\angle3 + m\angle4=30 + 60=90\).

Answer:

The two - column proof is completed as follows:

Statements	Reasons
2) \(m\angle1 = 2m\angle2\)	2) Given
3) \(m\angle1=2\times(30)\)	3) Substitution
4) \(m\angle1 = 60\)	4) Simplification
5) \(m\angle3 = 30\)	5) Vertical Angles Theorem
6) \(m\angle3 + m\angle4=30 + m\angle4\)	6) Addition Property of Equality
7) \(m\angle4 = 60\)	7) Vertical Angles Theorem
8) \(m\angle3 + m\angle4=30+60\)	8) Substitution
9) \(m\angle3 + m\angle4 = 90\)	9) Simplification