Question

4
which statement about different pairs of angles could be used to prove two lines are parallel?
a. if any two alternate interior angles are not congruent, then the lines are parallel.
b. if any two consecutive interior angles are complementary, then the lines are parallel.
c. if any two alternate exterior angles are complementary, then the lines are parallel.
d. if any two corresponding angles are congruent, then the lines are parallel.

Explanation:

To determine which statement proves two lines are parallel, we recall the angle - related parallel line theorems:

• For option A: The theorem states that if alternate interior angles are congruent, then the lines are parallel. If alternate interior angles are not congruent, the lines are not parallel. So A is incorrect.
• For option B: The theorem for consecutive interior angles is that if consecutive interior angles are supplementary (sum to \(180^{\circ}\)), then the lines are parallel, not complementary (sum to \(90^{\circ}\)). So B is incorrect.
• For option C: The theorem for alternate exterior angles is that if alternate exterior angles are congruent, then the lines are parallel, not complementary. So C is incorrect.
• For option D: The Corresponding Angles Postulate (or Theorem) states that if two corresponding angles are congruent, then the two lines cut by a transversal are parallel. So D is correct.

Answer:

D. If any two corresponding angles are congruent, then the lines are parallel.