Question

which angles are adjacent angles? image of lines and points o, l, r, p, m, k, q, n options: ∠lmp and ∠qpr; ∠lmp and ∠nmk; ∠nmk and ∠lmk; ∠opm and ∠qpr

Explanation:

Step1: Recall adjacent angles definition

Adjacent angles share a common side and vertex, and don't overlap.

Step2: Analyze each option

• Option 1: $\angle LMP$ and $\angle QPR$ – Different vertices, no common side. Rejected.
• Option 2: $\angle LMP$ and $\angle NMK$ – Different vertices (M vs K? Wait, M is vertex. Wait, $\angle LMP$ (vertex M, sides ML, MP) and $\angle NMK$ (vertex M, sides MN, MK). Do they share a side? ML and MN are different, MP and MK are opposite rays? Wait, no. Wait, third option: $\angle NMK$ and $\angle LMK$ – Vertex M, common side MK, and sides MN and ML. So they share vertex M, common side MK, and form a linear pair? Wait, adjacent angles can be supplementary or not, but share side and vertex. $\angle NMK$ (MN, MK) and $\angle LMK$ (ML, MK) – common vertex M, common side MK, and non - overlapping. So they are adjacent. Let's check other options. $\angle OPM$ and $\angle QPR$: $\angle QPR$ is vertical to $\angle PMK$? Wait, $\angle OPM$ (OP, PM) and $\angle QPR$ (QP, PR). Wait, OP and QP are same line? Wait, OQ is a line, PR is a line. $\angle OPM$ and $\angle QPR$: vertex P, but sides OP (same as QP) and PM, and QP and PR. So they are vertical angles? No, vertical angles are opposite. Wait, $\angle OPM$ and $\angle QPR$: actually, $\angle QPR$ and $\angle PMK$ are vertical? Wait, maybe I made a mistake. Wait, the third option: $\angle NMK$ and $\angle LMK$ – vertex M, common side MK, and sides MN and ML. So they are adjacent. Let's confirm: adjacent angles have common vertex, common side, and their interiors don't overlap. $\angle NMK$ (between MN and MK) and $\angle LMK$ (between ML and MK) – common vertex M, common side MK, and the angles are next to each other. So they are adjacent.

Answer:

$\angle NMK$ and $\angle LMK$ (the option with $\angle NMK$ and $\angle LMK$)