Question

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

Explanation:

Step1: Recall parallel - line theorem

If two lines are cut by a transversal and the corresponding angles are congruent, or the alternate - interior angles are congruent, or the alternate - exterior angles are congruent, or the consecutive interior angles are supplementary, then the two lines are parallel.

Step2: Analyze the first diagram

In the first diagram, we have two angles marked as \(89^{\circ}\). These two angles are corresponding angles. Since corresponding angles are congruent (\(89^{\circ}=89^{\circ}\)), the lines \(m\) and \(r\) must be parallel lines cut by a transversal.

Step3: Analyze the second diagram

In the second diagram, the two angles marked as \(91^{\circ}\) are not in a position (corresponding, alternate - interior, alternate - exterior, or consecutive interior) that would guarantee the parallelism of the lines \(m\) and \(r\) just based on these angle measures.

Answer:

The first diagram shows lines that must be parallel lines cut by a transversal.