Question

write a two - column proof for the following. given: m∠2 = 30 m∠1 = 2m∠2 prove: m∠3 + m∠4 = 90 (do not include the degree symbol in your answers.) statements reasons 1) m∠2 = 30 1) given 2) m∠1 = 2m∠2 2) given 3) m∠1 = 2(□) 3) substitution

Explanation:

Step1: Find the measure of ∠1

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

Step2: Use linear - pair property

\(\angle1\) and \(\angle2\) form a linear - pair with \(\angle3\) and \(\angle4\). The sum of angles around a point is 360 degrees, and the vertical angles are equal. Also, \(\angle1\) and \(\angle3\) are vertical angles, and \(\angle2\) and \(\angle4\) are vertical angles. Since \(\angle1\) and \(\angle2\) are adjacent and \(\angle3\) and \(\angle4\) are adjacent, and \(\angle1 + \angle2+\angle3+\angle4 = 180\) (a straight - line). Because \(\angle1 = 60\) and \(\angle2 = 30\), then \(m\angle3=m\angle1 = 60\) and \(m\angle4=m\angle2 = 30\) (vertical angles are equal).

Step3: Calculate \(m\angle3 + m\angle4\)

\(m\angle3+m\angle4=60 + 30=90\).

Answer:

The two - column proof is completed as follows:

Statements	Reasons
2) \(m\angle1 = 2m\angle2\)	2) Given
3) \(m\angle1=2\times30\)	3) Substitution
4) \(m\angle1 = 60\)	4) Simplification
5) \(\angle1\) and \(\angle3\) are vertical angles, \(\angle2\) and \(\angle4\) are vertical angles	5) Definition of vertical angles
6) \(m\angle3=m\angle1\), \(m\angle4=m\angle2\)	6) Vertical angles are equal
7) \(m\angle3 = 60\), \(m\angle4 = 30\)	7) Substitution (from steps 4 and 1)
8) \(m\angle3 + m\angle4=90\)	8) Addition