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:

• The alternate interior angles theorem states that if two parallel lines are cut by a transversal, then alternate interior angles are congruent. The converse of this theorem is used to prove that two lines are parallel. But in this case, the angles are \(72^{\circ}\) and \(108^{\circ}\), which are supplementary (\(72 + 108=180\)).
• The same - side interior angles theorem states that if two parallel lines are cut by a transversal, then same - side interior angles are supplementary. 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.
• Here, the two angles (\(72^{\circ}\) and \(108^{\circ}\)) are same - side interior angles and they are supplementary. So we use the converse of the same - side interior angles theorem.

Answer:

converse of the same side interior angles theorem