Question

which statement proves ∠3 ≅ ∠5?
diagram: two horizontal lines ( m ) (top) and ( n ) (bottom), intersected by a transversal ( p ). angles labeled: ( angle 1, angle 2, angle 3, angle 4 ) at ( m ); ( angle 5, angle 6, angle 7, angle 8 ) at ( n ).
options:

• alternate interior angles theorem
• alternate exterior angles theorem
• corresponding angles postulate
• vertical angles theorem

Explanation:

To determine which theorem proves \( \angle 3 \cong \angle 5 \), we analyze the positions of the angles. \( \angle 3 \) and \( \angle 5 \) are on opposite sides of the transversal \( p \) and inside the two parallel lines \( m \) and \( n \). The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then alternate interior angles are congruent.

• The Alternate Exterior Angles Theorem applies to exterior angles, so it's not relevant here.
• The Corresponding Angles Postulate applies to angles in the same position relative to the transversal and parallel lines, which is not the case for \( \angle 3 \) and \( \angle 5 \).
• The Vertical Angles Theorem applies to angles opposite each other when two lines intersect, which also does not describe \( \angle 3 \) and \( \angle 5 \).

Thus, the Alternate Interior Angles Theorem is the correct choice.

Answer:

A. Alternate Interior Angles Theorem