Question

prove: ∠jnm ≅ ∠nmi

according to the given information in the image, $overline{jk} \parallel \overline{hi}$ while ∠jnm and ∠lnk are vertical angles. ∠jnm and ∠lnk are congruent by the vertical angles theorem. because ∠lnk and ∠nmi are corresponding angles, they are congruent according to the corresponding angles theorem. finally, ∠jnm is congruent to ∠nmi by the transitive property of equality.

alternate interior angles theorem

corresponding angles theorem

vertical angles theorem

same - side interior angles theorem

Explanation:

To determine the correct theorem, we analyze the reasoning:

• Vertical Angles Theorem: States vertical angles (like ∠JNM and ∠LNK) are congruent.
• Corresponding Angles Theorem: States corresponding angles (like ∠LNK and ∠NMI, since \(\overline{JK} \parallel \overline{HI}\)) are congruent.
• Transitive Property: If \(a \cong b\) and \(b \cong c\), then \(a \cong c\) (here, ∠JNM ≅ ∠LNK and ∠LNK ≅ ∠NMI, so ∠JNM ≅ ∠NMI).

The question asks to identify the theorem used for ∠LNK and ∠NMI. Since ∠LNK and ∠NMI are corresponding angles (formed by a transversal intersecting parallel lines \(\overline{JK}\) and \(\overline{HI}\)), the Corresponding Angles Theorem applies.

Answer:

Corresponding Angles Theorem