Question

two parallel lines are cut by a transversal as shown below. suppose ( mangle6 = 78^circ ). find ( mangle1 ) and ( mangle4 ).

Explanation:

Step1: Find \(m\angle 1\)

When two parallel lines are cut by a transversal, \(\angle 1\) and \(\angle 6\) are alternate - exterior angles. Alternate - exterior angles are congruent. So \(m\angle 1=m\angle 6\).
Since \(m\angle 6 = 78^{\circ}\), then \(m\angle 1=78^{\circ}\).

Step2: Find \(m\angle 4\)

\(\angle 4\) and \(\angle 6\) are same - side interior angles. Same - side interior angles are supplementary, that is \(m\angle 4+m\angle 6 = 180^{\circ}\).
Substitute \(m\angle 6=78^{\circ}\) into the equation: \(m\angle 4=180^{\circ}-m\angle 6\).
\(m\angle 4=180^{\circ}- 78^{\circ}=102^{\circ}\).

Answer:

\(m\angle 1 = 78^{\circ}\), \(m\angle 4=102^{\circ}\)