Question

name
per
proofs with parallel lines
1.
given: ( p parallel q, m angle 3 = 105 ^ { circ } )
prove: ( m angle 2 = 75 ^ { circ } )
2.
given: ( p parallel q, m angle 1 = 125 ^ { circ } )
prove: ( m angle 2 = 55 ^ { circ } )

Explanation:

Step1: Find \(m\angle1\)

Since \(\angle3\) and \(\angle1\) are supplementary (linear - pair), \(m\angle3 + m\angle1=180^{\circ}\). Given \(m\angle3 = 105^{\circ}\), then \(105^{\circ}+m\angle1 = 180^{\circ}\), so \(m\angle1=180^{\circ}- 105^{\circ}=75^{\circ}\).

Step2: Use parallel - line property

Because \(p\parallel q\), \(\angle1\) and \(\angle2\) are alternate - interior angles. By the Alternate - Interior Angles Theorem, \(m\angle1=m\angle2\). Since \(m\angle1 = 75^{\circ}\), then \(m\angle2 = 75^{\circ}\).

Step1: Find \(m\angle3\)

Since \(\angle1\) and \(\angle3\) are vertical angles. By the Vertical Angles Theorem, \(m\angle1=m\angle3\). Given \(m\angle1 = 125^{\circ}\), so \(m\angle3=125^{\circ}\).

Step2: Find \(m\angle2\)

Since \(\angle3\) and \(\angle2\) are supplementary (linear - pair), \(m\angle3 + m\angle2=180^{\circ}\). Substitute \(m\angle3 = 125^{\circ}\) into the equation: \(125^{\circ}+m\angle2=180^{\circ}\). Then \(m\angle2=180^{\circ}-125^{\circ}=55^{\circ}\).

Answer:

Statements	Reasons
\(m\angle3+m\angle1 = 180^{\circ}\)	Linear - pair Postulate
\(m\angle1=180^{\circ}-105^{\circ}=75^{\circ}\)	Subtraction Property of Equality
\(m\angle1=m\angle2\)	Alternate - Interior Angles Theorem (\(p\parallel q\))
\(m\angle2 = 75^{\circ}\)	Substitution Property of Equality