Question

two parallel lines are cut by a transversal as shown below. suppose ( m angle 7 = 142 ^ { circ } ). find ( m angle 2 ) and ( m angle 4 ).

Explanation:

Step1: Find \(m\angle2\)

Since \(\angle2\) and \(\angle7\) are alternate - exterior angles. When two parallel lines are cut by a transversal, alternate - exterior angles are congruent.
So \(m\angle2 = m\angle7\)
Given \(m\angle7 = 142^{\circ}\), then \(m\angle2=142^{\circ}\)

Step2: Find \(m\angle4\)

Since \(\angle4\) and \(\angle7\) are supplementary (they form a linear pair with \(\angle5\) and \(\angle5\) and \(\angle7\) are vertical angles, and \(\angle4\) and \(\angle5\) are same - side interior angles. Also, \(\angle4\) and \(\angle2\) are supplementary as they are adjacent angles on a straight line).
We know that \(m\angle4 + m\angle2=180^{\circ}\) (linear pair). Substitute \(m\angle2 = 142^{\circ}\) into the equation:
\(m\angle4=180^{\circ}-m\angle2\)
\(m\angle4 = 180^{\circ}-142^{\circ}\)
\(m\angle4 = 38^{\circ}\)

Answer:

\(m\angle2 = 142^{\circ}\)
\(m\angle4 = 38^{\circ}\)