Question

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

Explanation:

Step1: Recall the converse of the same - side interior angles theorem

If two lines are cut by a transversal and the same - side interior angles are supplementary (\(180^{\circ}\)), then the two lines are parallel.

Step2: Analyze each diagram

• In the first diagram: \(89^{\circ}+91^{\circ}=180^{\circ}\), but these are not same - side interior angles.
• In the second diagram: \(89^{\circ}+91^{\circ}=180^{\circ}\), but these are not same - side interior angles.
• In the third diagram: \(89^{\circ}+89^{\circ}

eq180^{\circ}\)

• In the fourth diagram: The \(91^{\circ}\) angle and the \(91^{\circ}\) angle (using the vertical angle property, the angle adjacent to the \(91^{\circ}\) on the other line is also \(91^{\circ}\)) are same - side interior angles. And \(91^{\circ}+89^{\circ}=180^{\circ}\) (since \(89^{\circ}\) and \(91^{\circ}\) are supplementary).

Answer:

The fourth diagram.