The Alternate Interior Angles Theorem is a key concept that arises when a transversal intersects two parallel lines. According to this theorem, the alternate interior angles formed are congruent. This theorem holds significant value in geometry, as it helps us determine the relationship between angles in parallel lines.

To understand this theorem, consider two parallel lines intersected by a transversal. The alternate interior angles are the pairs of angles that are on the inside of the parallel lines, but on opposite sides of the transversal.

For example, let's consider two parallel lines, line AB and line CD, intersected by a transversal line EF. If we designate angle 1 and angle 2 as alternate interior angles, the theorem states that angle 1 is congruent to angle 2.

To prove this theorem, we can use the fact that corresponding angles are congruent when a transversal intersects two parallel lines. Let angles 1 and 3 be corresponding angles and angles 2 and 4 be corresponding angles. By the Corresponding Angles Theorem, angle 1 is congruent to angle 3, and angle 2 is congruent to angle 4. Since angle 3 and angle 4 are vertical angles, they are also congruent. Therefore, we can conclude that angle 1 is congruent to angle 2, which validates the Alternate Interior Angles Theorem.