Question

which diagram shows lines that must be parallel lines cut by a transversal?

Explanation:

Step1: Recall the property of same - side interior angles

If two lines are cut by a transversal, and the sum of same - side interior angles is \(180^{\circ}\), then the two lines are parallel.
For the first diagram, \(115^{\circ}+65^{\circ}=180^{\circ}\).

Step2: Analyze other cases (not shown in the first diagram)

In the second diagram (the one with \(115^{\circ}\) and \(65^{\circ}\) adjacent on one line), these angles are not same - side interior angles for a potential pair of parallel lines cut by a transversal in the way that would imply parallelism.

Answer:

The first diagram (the one with \(115^{\circ}\) and \(65^{\circ}\) as same - side interior angles) shows lines that must be parallel lines cut by a transversal.