Question

given the information in the diagram, which theorem best justifies why lines e and f must be parallel?
○ alternate interior angles theorem
○ same side interior angles theorem
○ converse of the alternate interior angles theorem
○ converse of the same side interior angles theorem

Explanation:

1. Recall the properties of same - side interior angles: Same - side interior angles are two angles that lie between two lines and on the same side of a transversal. The sum of same - side interior angles that are supplementary (their sum is \(180^{\circ}\)) implies that the two lines are parallel (this is the converse of the same - side interior angles theorem).
2. Calculate the sum of the given angles: The given angles are \(72^{\circ}\) and \(108^{\circ}\). Their sum is \(72^{\circ}+ 108^{\circ}=180^{\circ}\), which means they are supplementary same - side interior angles.
3. Analyze the converse of the same - side interior angles theorem: The converse of the same - side interior angles theorem states that if two lines are cut by a transversal and the same - side interior angles are supplementary, then the two lines are parallel. Since the sum of the two angles (\(72^{\circ}\) and \(108^{\circ}\)) is \(180^{\circ}\) (supplementary), by the converse of the same - side interior angles theorem, lines \(e\) and \(f\) must be parallel.

• The alternate interior angles theorem and its converse deal with angles that are equal (alternate interior angles), not supplementary. The same - side interior angles theorem (not the converse) states that if two parallel lines are cut by a transversal, then same - side interior angles are supplementary, which is the reverse of what we need here. We need the converse (to prove lines are parallel from angle relationships), and since the angles are supplementary same - side interior angles, the converse of the same - side interior angles theorem applies.

Answer:

converse of the same side interior angles theorem