If two lines are cut by a transversal and a pair of alternate interior angles are congruent, what can be concluded?

If two lines are cut by a transversal and a pair of alternate interior angles are congruent, what can be concluded?

• A. The lines are parallel.
• B. The lines are perpendicular.
• C. The lines are the same line.
• D. The lines are skew.

Answer: (A)

The key idea is that equal alternate interior angles occur exactly when the two lines are parallel. Alternate interior angles are the angles located between the two lines on opposite sides of the transversal. If one pair of those angles are congruent, the lines cannot converge or meet; they must run in the same direction, so they are parallel. If the lines were not parallel, they'd intersect and the angles formed would not be equal in that alternating pair. The other options don’t fit: being perpendicular would require a 90-degree relationship, which isn’t specified by the equality of the angles; having the lines be the same line isn’t a situation with two distinct lines; and skew lines occur only in three dimensions, not in a single plane. Therefore, the lines are parallel.

The key idea is that equal alternate interior angles occur exactly when the two lines are parallel. Alternate interior angles are the angles located between the two lines on opposite sides of the transversal. If one pair of those angles are congruent, the lines cannot converge or meet; they must run in the same direction, so they are parallel. If the lines were not parallel, they'd intersect and the angles formed would not be equal in that alternating pair. The other options don’t fit: being perpendicular would require a 90-degree relationship, which isn’t specified by the equality of the angles; having the lines be the same line isn’t a situation with two distinct lines; and skew lines occur only in three dimensions, not in a single plane. Therefore, the lines are parallel.