The proof given examines the relationship between angles formed when a transversal intersects two parallel lines. The conclusion that ∠JNL is congruent to ∠HMN is based on the properties of angles created by a transversal crossing parallel lines.
Since ∠JNL and ∠HMN are formed by the transversal LO intersecting the parallel lines JK and HI, and they are shown to have measures that relate to supplementary angles (same-side interior angles), the proof directly supports the conclusion that ∠JNL is congruent to ∠HMN.
Therefore, the theorem this proof offers is the Alternate Interior Angles Theorem, as the angles involved are defined by their positions in relation to the parallel lines and the transversal.
So the answer is:
Alternate Interior Angles Theorem.