Question

two parallel lines are cut by a transversal as shown below. suppose ( mangle4 = 97^{circ} ). find ( mangle5 ) and ( mangle7 ).

Explanation:

Step1: Find \(m\angle5\)

When two parallel lines are cut by a transversal, \(\angle4\) and \(\angle5\) are same - side interior angles.
By the same - side interior angles theorem, \(m\angle4 + m\angle5=180^{\circ}\).
Given \(m\angle4 = 97^{\circ}\), then \(m\angle5=180^{\circ}-m\angle4\).
Substitute \(m\angle4 = 97^{\circ}\) into the formula: \(m\angle5 = 180^{\circ}-97^{\circ}=83^{\circ}\).

Step2: Find \(m\angle7\)

\(\angle5\) and \(\angle7\) are vertical angles.
By the vertical angles theorem, \(m\angle7=m\angle5\).
Since \(m\angle5 = 83^{\circ}\), then \(m\angle7 = 83^{\circ}\).

Answer:

\(m\angle5 = 83^{\circ}\), \(m\angle7 = 83^{\circ}\)