Question

two parallel lines are cut by a transversal as shown below. suppose ( m angle 2 = 121 ^ { circ } ). find ( m angle 5 ) and ( m angle 7 ). ( m angle 5 = ) ( m angle 7 = )

Explanation:

Step1: Find \(m\angle5\)

\(\angle2\) and \(\angle5\) are same - side interior angles. But \(\angle2\) and \(\angle3\) are supplementary (\(\angle2+\angle3 = 180^{\circ}\)) since they form a linear pair. Also, \(\angle3\) and \(\angle5\) are alternate - interior angles (because the two lines are parallel).
First, find \(\angle3\): \(\angle3=180^{\circ}-\angle2\). Given \(\angle2 = 121^{\circ}\), so \(\angle3=180 - 121=59^{\circ}\). Then, since \(\angle3\) and \(\angle5\) are alternate - interior angles (parallel lines cut by a transversal), \(m\angle5=m\angle3 = 59^{\circ}\).

Step2: Find \(m\angle7\)

\(\angle5\) and \(\angle7\) are vertical angles. Vertical angles are equal. So \(m\angle7=m\angle5\). Since \(m\angle5 = 59^{\circ}\), then \(m\angle7 = 59^{\circ}\).

Answer:

\(m\angle5 = 59^{\circ}\), \(m\angle7 = 59^{\circ}\)