Examine the paragraph proof. Which theorem does it offer proof for?

Question

Segments JK and HI are parallel, segment LO intersects segment JK at point N, segment LO intersects segment HI at point M, and points N and M are between points L and O on segment LO.

Prove: ∠JNL ≅ ∠HMN

According to the given information, segment JK is parallel to segment HI and points L, N, M, and O all lie on the same line. The measure of ∠LNM 180° by the definition of a straight angle. Because ∠JNL and ∠JNM are adjacent to one another, the Angle Addition Postulate allows the measure of ∠JNL and ∠JNM to equal the measure of ∠LNM. Through the Substitution Property of Equality, the measure of ∠JNL plus the measure of ∠JNM equals 180°. Since ∠JNM and ∠HMN are same-side interior angles, the measure of ∠JNM plus the measure of ∠HMN equals 180°. Using substitution again, the measure of ∠JNL plus the measure of ∠JNM equals the measure of ∠JNM plus the measure of ∠HMN. Finally, the Subtraction Property of Equality allows the measure of ∠JNM to be subtracted from both sides of the equation. The result is that the measure of ∠JNL is the same as the measure of ∠HMN. Because their angle measures are equal, the angles themselves are congruent by the definition of congruency.

Alternate Interior Angles Theorem
Corresponding Angles Theorem
Vertical Angles Theorem
Same-Side Interior Angles Theorem
Question 10(Multiple Choice Worth 1 points)

Answer 1

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.