Question

two parallel lines are cut by a transversal as shown below. suppose ( m angle 1 = 39 ^ { circ } ). find ( m angle 6 ) and ( m angle 7 ).

Explanation:

Step1: Find \(m\angle6\)

When two parallel lines are cut by a transversal, \(\angle1\) and \(\angle3\) are vertical angles, so \(m\angle1 = m\angle3=39^{\circ}\). Also, \(\angle3\) and \(\angle6\) are same - side interior angles. But wait, no! \(\angle3\) and \(\angle5\) are alternate interior angles, and \(\angle5\) and \(\angle6\) are supplementary. Wait, another approach: \(\angle1\) and \(\angle5\) are corresponding angles (since lines are parallel), so \(m\angle5 = 39^{\circ}\). And \(\angle5\) and \(\angle6\) are supplementary (\(\angle5+\angle6 = 180^{\circ}\)), so \(m\angle6=180 - 39=141^{\circ}\).

Step2: Find \(m\angle7\)

\(\angle6\) and \(\angle7\) are vertical angles. Since vertical angles are equal, \(m\angle7=m\angle6 - 180\) (no, wait). Wait, \(\angle3\) and \(\angle7\) are corresponding angles. Since \(m\angle3 = 39^{\circ}\) (from Step1, \(\angle1\) and \(\angle3\) are vertical angles), and because of the parallel lines and transversal, \(m\angle7 = 39^{\circ}\) (corresponding angles are equal).

Answer:

\(m\angle6 = 141^{\circ}\)
\(m\angle7 = 39^{\circ}\)