Question

1. a transversal intersects two lines. which condition would always make the two lines parallel? 1) vertical angles are congruent. 2) alternate interior angles are congruent. 3) corresponding angles are supplementary. 4) same - side interior angles are complementary.

Explanation:

• Vertical angles: Vertical angles are always congruent, whether the two lines are parallel or not. So this condition does not ensure parallelism.
• Alternate interior angles: By the converse of the alternate - interior angles theorem, if two lines are cut by a transversal and the alternate interior angles are congruent, then the two lines are parallel.
• Corresponding angles: If corresponding angles are supplementary (\(\angle1+\angle2 = 180^{\circ}\)), it does not follow the rules of parallel - line theorems. For parallel lines, corresponding angles are congruent (\(\angle1=\angle2\)).
• Same - side interior angles: For parallel lines, same - side interior angles are supplementary (\(\angle1+\angle2=180^{\circ}\)), not complementary (\(\angle1+\angle2 = 90^{\circ}\)).

Answer:

1. Alternate interior angles are congruent.