Question

which diagram shows lines that must be parallel lines cut by a transversal? (four diagrams with intersecting lines and angle measures: 89°, 91°; 89°, 91°; 91°, 91°; 91°, 91°)

Explanation:

Step1: Recall Parallel Line Theorems

For two lines cut by a transversal, if corresponding angles are equal, alternate - interior angles are equal, or consecutive interior angles are supplementary, the lines are parallel. Let's analyze each diagram:

Diagram 1:

The angles given are \(89^{\circ}\) and \(91^{\circ}\). These are not equal, and \(89 + 91=180\)? No, \(89 + 91 = 180\)? Wait, \(89+91 = 180\)? \(89 + 91=180\), but let's check the position. If they were consecutive interior angles, but the position here doesn't suggest that. Wait, maybe we need to check corresponding angles or alternate - interior angles.

Diagram 2:

Angles \(89^{\circ}\) and \(91^{\circ}\) are adjacent, not corresponding or alternate - interior in a way that would imply parallelism.

Diagram 3:

The two angles are both \(91^{\circ}\) and they are corresponding angles (assuming the transversal cuts the two lines). If corresponding angles are equal, then the two lines cut by the transversal are parallel.

Diagram 4:

The angle is \(91^{\circ}\) on one side, but the other angle's position and measure don't suggest equal corresponding or alternate - interior angles.

Step2: Confirm the Parallel Line Condition

For two lines cut by a transversal, the Corresponding Angles Postulate states that if two parallel lines are cut by a transversal, then corresponding angles are congruent. Conversely, if corresponding angles are congruent, the lines are parallel. In the third diagram, the two angles formed by the transversal and the two lines are equal (both \(91^{\circ}\)), so they are corresponding angles. By the converse of the Corresponding Angles Postulate, the two lines must be parallel.

Answer:

The third diagram (the one with two \(91^{\circ}\) angles)