Final Answer:
The theorem that explains why
) is the Converse of the Alternate Interior Angles Theorem.
Step-by-step explanation:
The Converse of the Alternate Interior Angles Theorem states that if two lines are cut by a transversal, and the alternate interior angles are congruent, then the lines are parallel. In the given scenario,
are parallel lines, and the angles
and
are alternate interior angles. Therefore, according to the Converse of the Alternate Interior Angles Theorem, when alternate interior angles are congruent, the lines are parallel. This explains why
as corresponding segments of parallel lines are equal.
Understanding the relationship between angles and parallel lines is crucial in geometry. The Converse of the Alternate Interior Angles Theorem provides a powerful tool for establishing parallelism based on the congruence of alternate interior angles. In this case, it allows us to deduce that
and
) are parallel and, consequently, that the corresponding segments
and
are equal in length.
In summary, the application of the Converse of the Alternate Interior Angles Theorem in this context elucidates why
This theorem plays a significant role in geometric proofs involving parallel lines and the relationships between angles formed by intersecting lines.