Question

two parallel lines are cut by a transversal as shown below. suppose ( mangle3 = 106^{circ}). find ( mangle5) and ( mangle8).

Explanation:

Step1: Find \(m\angle5\)

When two parallel lines are cut by a transversal, \(\angle3\) and \(\angle5\) are alternate - interior angles. Alternate - interior angles are congruent. So \(m\angle5=m\angle3\).
Since \(m\angle3 = 106^{\circ}\), then \(m\angle5=106^{\circ}\).

Step2: Find \(m\angle8\)

\(\angle3\) and \(\angle4\) form a linear pair. So \(m\angle3 + m\angle4=180^{\circ}\). But we can also use the property of corresponding angles. \(\angle3\) and \(\angle7\) are corresponding angles (\(m\angle3=m\angle7 = 106^{\circ}\)). And \(\angle7\) and \(\angle8\) form a linear pair (\(m\angle7+m\angle8 = 180^{\circ}\)).
Another way: \(\angle3\) and \(\angle8\) are same - side exterior and interior angles. But more simply, \(\angle5\) and \(\angle8\) are supplementary (they form a linear pair). Since \(m\angle5 = 106^{\circ}\), then \(m\angle8=180^{\circ}-m\angle5\).
Substitute \(m\angle5 = 106^{\circ}\) into the formula: \(m\angle8=180 - 106=74^{\circ}\).

Answer:

\(m\angle5 = 106^{\circ}\), \(m\angle8 = 74^{\circ}\)