Question

two parallel lines are cut by a transversal as shown below. suppose ( mangle8 = 81^{circ} ). find ( mangle1 ) and ( mangle3 ).

Explanation:

Step1: Find \(m\angle1\)

When two parallel lines are cut by a transversal, alternate interior angles are equal. \(\angle1\) and \(\angle8\) are alternate interior angles. So \(m\angle1 = m\angle8\). Given \(m\angle8 = 81^{\circ}\), then \(m\angle1=81^{\circ}\).

Step2: Find \(m\angle3\)

\(\angle3\) and \(\angle8\) are same - side interior angles. The sum of same - side interior angles is \(180^{\circ}\) (because the two lines are parallel). So \(m\angle3 + m\angle8=180^{\circ}\). Substitute \(m\angle8 = 81^{\circ}\) into the equation: \(m\angle3=180^{\circ}-m\angle8\). Then \(m\angle3 = 180^{\circ}-81^{\circ}=99^{\circ}\).

Answer:

\(m\angle1 = 81^{\circ}\) and \(m\angle3 = 99^{\circ}\)