Question

select all the pairs of adjacent angles.
∠lmo and ∠nmj ∠lmj and ∠kjm
∠kjm and ∠ijm ∠lmj and ∠nmj

Explanation:

Step1: Recall adjacent angles definition

Adjacent angles share a common side and vertex, no overlap.

Step2: Analyze each pair

- $\angle LMO$ and $\angle NMJ$: No common side/vertex? Check graph. $M$ is vertex? Wait, $\angle LMO$ has vertex $M$, $\angle NMJ$ has vertex $M$? Wait, no, $\angle LMO$: sides $LM$, $MO$; $\angle NMJ$: sides $NM$, $MJ$. Do they share a side? No. So not adjacent.
- $\angle LMJ$ and $\angle KJM$: $\angle LMJ$ (sides $LM$, $MJ$), $\angle KJM$ (sides $KJ$, $JM$). Common side $JM$, common vertex $J$? Wait, vertex $J$? $\angle LMJ$ vertex $M$? Wait, no, labels: $J$ and $M$ are points. Wait, $\angle LMJ$: vertex $M$? Wait, maybe I misread. Wait, $\angle LMJ$: points $L$, $M$, $J$: vertex $M$. $\angle KJM$: points $K$, $J$, $M$: vertex $J$. Different vertices? No, wait, $J$ and $M$ are connected. Wait, maybe better: $\angle LMJ$ (sides $LM$, $MJ$), $\angle KJM$ (sides $KJ$, $JM$). Common side $MJ$ (or $JM$), common vertex $M$? Wait, $J$ and $M$: $\angle KJM$ is at $J$? Wait, no, angle at $J$: $\angle KJM$: $K - J - M$, so vertex $J$. $\angle LMJ$: $L - M - J$, vertex $M$. Different vertices, so no common vertex. Not adjacent.
- $\angle KJM$ and $\angle IJM$: $\angle KJM$ (sides $KJ$, $JM$), $\angle IJM$ (sides $IJ$, $JM$). Common side $JM$, common vertex $J$, no overlap. So adjacent.
- $\angle LMJ$ and $\angle NMJ$: $\angle LMJ$ (sides $LM$, $MJ$), $\angle NMJ$ (sides $NM$, $MJ$). Common side $MJ$, common vertex $M$, no overlap. So adjacent. Wait, earlier mistake: $\angle LMJ$ and $\angle NMJ$: vertex $M$, common side $MJ$, sides $LM$ and $NM$ (no overlap). So they are adjacent. Wait, also $\angle KJM$ and $\angle IJM$: common side $JM$, vertex $J$, sides $KJ$ and $IJ$ (no overlap). So let's recheck:

Wait, the options: Wait, the original options (after the graph) are:

1. $\angle LMO$ and $\angle NMJ$ – no
2. $\angle LMJ$ and $\angle KJM$ – no
3. $\angle KJM$ and $\angle IJM$ – yes (common side $JM$, vertex $J$, sides $KJ$ and $IJ$)
4. $\angle LMJ$ and $\angle NMJ$ – yes (common side $MJ$, vertex $M$, sides $LM$ and $NM$)

Wait, maybe I misread the options. Wait, the user's options (after the graph) are:

- $\angle LMO$ and $\angle NMJ$
- $\angle LMJ$ and $\angle KJM$
- $\angle KJM$ and $\angle IJM$
- $\angle LMJ$ and $\angle NMJ$

So correct pairs: $\angle KJM$ and $\angle IJM$ (share $JM$, vertex $J$), $\angle LMJ$ and $\angle NMJ$ (share $MJ$, vertex $M$). Wait, but let's confirm with the graph. Lines: $IJ$ and $LN$ are parallel? $HJ$ and $MO$ are transversals? So $\angle KJM$ and $\angle IJM$: at point $J$, on line $IJ$, with $KJ$ and $IJ$ as sides, sharing $JM$. $\angle LMJ$ and $\angle NMJ$: at point $M$, on line $LN$, with $LM$ and $NM$ as sides, sharing $MJ$.

So the correct pairs are $\angle KJM$ and $\angle IJM$, $\angle LMJ$ and $\angle NMJ$. Wait, but the options given (the checkboxes) – maybe the user's options are:

Wait, the user's image shows:

Options (with checkboxes):

1. $\angle LMO$ and $\angle NMJ$
2. $\angle LMJ$ and $\angle KJM$
3. $\angle KJM$ and $\angle IJM$
4. $\angle LMJ$ and $\angle NMJ$

So we need to select the adjacent ones.

Adjacent angles: share a common side and vertex, and their interiors do not overlap.

- $\angle LMO$ and $\angle NMJ$: Different vertices (M and...), no common side. Not adjacent.
- $\angle LMJ$ and $\angle KJM$: Different vertices (M and J), no common side (sides LM, MJ vs KJ, JM – different vertices, no common side). Not adjacent.
- $\angle KJM$ and $\angle IJM$: Common side JM, common vertex J, sides KJ and IJ (no overlap). Adjacent.
- $\angle LMJ$ and $\angle NMJ$: Common side MJ, common vertex M, sides LM and NM (no overlap). Adjacent.

Answer:

$\angle KJM$ and $\angle IJM$, $\angle LMJ$ and $\angle NMJ$ (the pairs: $\boldsymbol{\angle KJM \text{ and } \angle IJM}$, $\boldsymbol{\angle LMJ \text{ and } \angle NMJ}$)