Question

which theorem correctly justifies why the lines m and n are parallel when cut by transversal k?
○ converse of the corresponding angles theorem
○ converse of the alternate interior angles theorem
○ converse of the same side interior angles theorem
○ converse of the alternate exterior angles theorem

Explanation:

1. Recall the definitions of each converse theorem:

• Corresponding Angles Converse: If corresponding angles are equal, lines are parallel.
• Alternate Interior Angles Converse: If alternate interior angles are equal, lines are parallel.
• Same - Side Interior Angles Converse: If same - side interior angles are supplementary, lines are parallel.
• Alternate Exterior Angles Converse: If alternate exterior angles are equal, lines are parallel.

1. Analyze the angles in the diagram: The two \(50^{\circ}\) angles are corresponding angles (they are in the same relative position with respect to the transversal \(k\) and the two lines \(m\) and \(n\)). When corresponding angles are equal, by the converse of the corresponding angles theorem, the lines \(m\) and \(n\) are parallel. The alternate interior angles would be on the inside of the two lines, the same - side interior angles would be supplementary (sum to \(180^{\circ}\)) for parallel lines, and alternate exterior angles are on the outside. Since the given equal angles are corresponding, the converse of the corresponding angles theorem applies.

Answer:

converse of the corresponding angles theorem