consider $\triangle$lnm. which statements are true for triangle lnm? check all that apply. the side opposite $\angle$l is $\overline{nm}$. the side opposite $\angle$n is $\overline{ml}$. the hypotenuse is $\overline{nm}$. the hypotenuse is $\overline{ln}$. the side adjacent $\angle$l is $\overline{nm}$. the side adjacent $\angle$n is $\overline{ml}$.

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# Brief Explanations:
1. **For "The side opposite ∠L is $\overline{NM}$":** In a triangle, the side opposite an angle is the one not connected to the angle's vertex. ∠L is at vertex L, so the side opposite is $\overline{NM}$ (connects M and N, not L) – true.
2. **For "The side opposite ∠N is $\overline{ML}$":** ∠N is at vertex N, so the side opposite is $\overline{ML}$ (connects M and L, not N) – true.
3. **For "The hypotenuse is $\overline{NM}$":** In a right triangle, the hypotenuse is the side opposite the right angle. The right angle is at M, so the hypotenuse should be opposite (∠M), which is $\overline{LN}$, not $\overline{NM}$ – false.
4. **For "The hypotenuse is $\overline{LN}$":** As the right angle is at M, the hypotenuse is the side opposite (∠M), which is $\overline{LN}$ (the longest side in a right triangle, opposite the right angle) – true.
5. **For "The side adjacent ∠L is $\overline{NM}$":** Adjacent sides form the angle with the angle's vertex. ∠L is at L, with sides $\overline{ML}$ and $\overline{LN}$. Wait, no – in right triangle at M, ∠L is at L, so adjacent sides to ∠L are $\overline{ML}$ (one leg) and $\overline{LN}$ is hypotenuse? Wait, no, correction: In right triangle $ \triangle LNM $ with right angle at M, ∠L is at L, so the sides forming ∠L are $\overline{ML}$ (horizontal leg) and $\overline{LN}$ (hypotenuse)? Wait, no, adjacent side to ∠L: adjacent sides are the two sides that meet at ∠L. ∠L is at L, so sides $\overline{ML}$ (from M to L) and $\overline{LN}$ (from L to N). Wait, but the option says $\overline{NM}$ – no, $\overline{NM}$ is from N to M. Wait, maybe I messed up. Wait, the triangle: vertices L, N, M, right angle at M. So sides: $ \overline{ML} $ (horizontal, M to L), $ \overline{NM} $ (vertical, N to M), $ \overline{LN} $ (hypotenuse, L to N). So ∠L is at L, between $ \overline{ML} $ (M to L) and $ \overline{LN} $ (L to N). So the adjacent side to ∠L would be $ \overline{ML} $? Wait, no, the other adjacent side? Wait, no, in trigonometry, for an acute angle in a right triangle, the adjacent side is the leg that is part of the angle (not the hypotenuse). Wait, ∠L is acute (since right angle at M). So for ∠L, the adjacent side is the leg that is adjacent (forms the angle with ∠L), which is $ \overline{ML} $, and the opposite side is $ \overline{NM} $. Wait, but the option says "The side adjacent ∠L is $\overline{NM}$" – that would be false? Wait, no, maybe I made a mistake. Wait, let's re-express:

- ∠L: vertex L, sides: $ \overline{ML} $ (from M to L) and $ \overline{LN} $ (from L to N). So the sides adjacent to ∠L are $ \overline{ML} $ (one leg) and $ \overline{LN} $ is hypotenuse? No, adjacent sides are the two sides that meet at the angle. So ∠L is formed by $ \overline{ML} $ (M-L) and $ \overline{LN} $ (L-N). So the adjacent side (other than hypotenuse) would be $ \overline{ML} $, but the option says $ \overline{NM} $. Wait, $ \overline{NM} $ is from N to M, which is vertical leg. So maybe the option is wrong? Wait, no, maybe I mislabeled the triangle. Wait, the triangle has right angle at M, so M is the right angle, so sides: $ \overline{ML} $ (horizontal, M to L), $ \overline{NM} $ (vertical, N to M), $ \overline{LN} $ (hypotenuse, L to N). So ∠L is at L, between $ \overline{ML} $ (M to L) and $ \overline{LN} $ (L to N). So the adjacent side to ∠L is $ \overline{ML} $ (the leg that is part of ∠L, not the hypotenuse), and the opposite side is $ \overline{NM} $ (the leg not part of ∠L). Wait, but the option says "The side adjacent ∠L is $\overline{NM}$" – that would be false. Wait, maybe the question has a typo, or I'm misinterpreting. Wait, maybe the adjacent side to ∠L is $ \overline{ML} $, but the option says $ \overline{NM} $ – no, that's opposite. Wait, maybe the original triangle: M is right angle, so ∠M = 90°, so vertices L, N, M with right angle at M. So sides: $ \overline{ML} $ (length a), $ \overline{NM} $ (length b), $ \overline{LN} $ (length c, hypotenuse). So ∠L: at L, between $ \overline{ML} $ (a) and $ \overline{LN} $ (c). So adjacent side to ∠L is $ \overline{ML} $ (one leg), opposite side is $ \overline{NM} $ (the other leg). So the option "The side adjacent ∠L is $\overline{NM}$" is false. Wait, but maybe I got adjacent and opposite mixed. Wait, no: in angle ∠L, the sides are: - One side is $ \overline{ML} $ (from M to L) – this is adjacent (forms the angle with ∠L, along with $ \overline{LN} $). - The other side is $ \overline{LN} $ (hypotenuse). Wait, no, adjacent sides are the two sides that meet at the angle. So ∠L is at L, so the two sides meeting at L are $ \overline{ML} $ (M to L) and $ \overline{LN} $ (L to N). So the adjacent side to ∠L is $ \overline{ML} $ (the leg), and the opposite side is $ \overline{NM} $ (the leg not meeting at L). So the option "The side adjacent ∠L is $\overline{NM}$" is false. Wait, but maybe the question considers adjacent as the other leg? No, adjacent is the leg that is part of the angle. So this option is false. Wait, but maybe I made a mistake. Let's check the next option: "The side adjacent ∠N is $\overline{ML}$". ∠N is at N, between $ \overline{NM} $ (N to M) and $ \overline{LN} $ (L to N). So the adjacent side to ∠N is $ \overline{NM} $ (the leg), and the opposite side is $ \overline{ML} $ (the leg not meeting at N). Wait, the option says "The side adjacent ∠N is $\overline{ML}$" – that would be false, because adjacent side to ∠N is $ \overline{NM} $, and opposite is $ \overline{ML} $. Wait, but maybe the question has a different labeling. Wait, maybe the triangle is labeled with M at the top, L at the right, N at the bottom, right angle at M. So $ \overline{ML} $ is horizontal (right), $ \overline{NM} $ is vertical (down), $ \overline{LN} $ is hypotenuse (from L to N). So ∠L is at L (right), between $ \overline{ML} $ (left to L) and $ \overline{LN} $ (L to N). So adjacent side to ∠L is $ \overline{ML} $ (horizontal leg), opposite side is $ \overline{NM} $ (vertical leg). So "The side adjacent ∠L is $\overline{NM}$" – no, that's opposite. So that option is false. Then "The side adjacent ∠N is $\overline{ML}$" – ∠N is at N (bottom), between $ \overline{NM} $ (up to M) and $ \overline{LN} $ (N to L). So adjacent side to ∠N is $ \overline{NM} $ (vertical leg), opposite side is $ \overline{ML} $ (horizontal leg). So the option says adjacent is $ \overline{ML} $ – that's opposite, so false. Wait, but maybe the question's options have some correct ones. Let's re-express:

- Side opposite ∠L: ∠L is at L, so the side not connected to L is $ \overline{NM} $ (connects N and M, not L) – correct. So first option is true.
- Side opposite ∠N: ∠N is at N, so the side not connected to N is $ \overline{ML} $ (connects M and L, not N) – correct. So second option is true.
- Hypotenuse is $ \overline{NM} $: Hypotenuse is opposite right angle (at M), so hypotenuse is $ \overline{LN} $, not $ \overline{NM} $ (which is a leg) – false.
- Hypotenuse is $ \overline{LN} $: Correct, as it's opposite right angle at M – true.
- Side adjacent ∠L is $ \overline{NM} $: Adjacent side to ∠L should be $ \overline{ML} $ (the leg forming ∠L with hypotenuse), not $ \overline{NM} $ (opposite) – false.
- Side adjacent ∠N is $ \overline{ML} $: Adjacent side to ∠N should be $ \overline{NM} $ (the leg forming ∠N with hypotenuse), not $ \overline{ML} $ (opposite) – false.

Wait, but maybe I messed up adjacent sides. Let's recall: in a right triangle, for an acute angle, the adjacent side is the leg that is part of the angle (i.e., one of the two sides that form the angle, excluding the hypotenuse). So for ∠L (acute angle at L), the two sides forming ∠L are $ \overline{ML} $ (leg) and $ \overline{LN} $ (hypotenuse). So the adjacent side is $ \overline{ML} $, and the opposite side is $ \overline{NM} $ (the other leg). So "The side adjacent ∠L is $\overline{NM}$" is false. For ∠N: the two sides forming ∠N are $ \overline{NM} $ (leg) and $ \overline{LN} $ (hypotenuse). So adjacent side is $ \overline{NM} $, opposite side is $ \overline{ML} $. So "The side adjacent ∠N is $\overline{ML}$" is false.

So the true statements are:

- The side opposite ∠L is $\overline{NM}$.
- The side opposite ∠N is $\overline{ML}$.
- The hypotenuse is $\overline{LN}$.

Wait, but let's confirm with standard right triangle notation. In $ \triangle ABC $ with right angle at C, side opposite ∠A is BC, side opposite ∠B is AC, hypotenuse is AB. So applying that: in $ \triangle LNM $ with right angle at M, side opposite ∠L is NM (since ∠L is at L, opposite side is NM), side opposite ∠N is ML (∠N at N, opposite side is ML), hypotenuse is LN (opposite right angle at M). So yes, those three are true. The other three are false.

# Answer:
- The side opposite $ \angle L $ is $ \overline{NM} $.
- The side opposite $ \angle N $ is $ \overline{ML} $.
- The hypotenuse is $ \overline{LN} $.

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