12 list the side lengths from shortest to longest.

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Accepted Answer

# Explanation:
## Step1: Find the third angle
The sum of angles in a triangle is $180^\circ$. So the third angle (opposite side $m$) is $180 - 68 - 97 - 15$? Wait, no, wait. Wait, the triangle has angles $15^\circ$, $68^\circ$, and let's calculate the third angle. Wait, $180 - 15 - 68 = 97^\circ$. Wait, the angles are $15^\circ$, $68^\circ$, $97^\circ$. So the sides opposite these angles: side $n$ is opposite $68^\circ$? Wait, no, let's label the triangle. Let's say the angles are: angle $A = 15^\circ$, angle $B = 68^\circ$, angle $C = 97^\circ$. Then side opposite angle $A$ (15°) is $m$? Wait, no, looking at the diagram: the side labeled $m$ is between $68^\circ$ and $97^\circ$, so the angle opposite $m$ is $15^\circ$. The side labeled $n$ is between $15^\circ$ and $97^\circ$, so the angle opposite $n$ is $68^\circ$. The side labeled $o$ is between $15^\circ$ and $68^\circ$, so the angle opposite $o$ is $97^\circ$.

## Step2: Relate angles to side lengths
In a triangle, the larger the angle, the longer the side opposite it. So we order the angles from smallest to largest: $15^\circ < 68^\circ < 97^\circ$. Therefore, the sides opposite these angles (which are $m$, $n$, $o$ respectively) will be in the same order. Wait, angle $15^\circ$ is opposite side $m$? Wait, no, let's correct: angle $15^\circ$ is at the top, so the side opposite angle $15^\circ$ is $m$ (the side between $68^\circ$ and $97^\circ$). Angle $68^\circ$ is at the left, so the side opposite angle $68^\circ$ is $n$ (the side from the left angle to the top angle). Angle $97^\circ$ is at the bottom, so the side opposite angle $97^\circ$ is $o$ (the side from the top angle to the bottom angle). Wait, no, maybe I mixed up. Let's re-express:

- Angle $15^\circ$: opposite side $m$ (since $m$ is between the other two angles)
- Angle $68^\circ$: opposite side $o$? Wait, no, the diagram: the side labeled $n$ is the longest side? Wait, no, let's use the angle-side relationship: in a triangle, longer side is opposite larger angle.

So angles: $15^\circ$ (smallest), $68^\circ$, $97^\circ$ (largest). So the sides opposite these angles:

- Opposite $15^\circ$: side $m$ (wait, no, maybe the labels are: the side with length $m$ is opposite $15^\circ$, side $n$ opposite $68^\circ$, side $o$ opposite $97^\circ$. Wait, no, let's check the diagram again. The triangle has three sides: $m$, $n$, $o$. The angles are $15^\circ$ (top), $68^\circ$ (left), $97^\circ$ (bottom). So:

- Side $m$ is between left angle ($68^\circ$) and bottom angle ($97^\circ$), so it's opposite the top angle ($15^\circ$).
- Side $n$ is between top angle ($15^\circ$) and bottom angle ($97^\circ$), so it's opposite the left angle ($68^\circ$).
- Side $o$ is between top angle ($15^\circ$) and left angle ($68^\circ$), so it's opposite the bottom angle ($97^\circ$).

Therefore, side opposite $15^\circ$ is $m$, opposite $68^\circ$ is $n$, opposite $97^\circ$ is $o$.

Since $15^\circ < 68^\circ < 97^\circ$, the sides opposite them ( $m$, $n$, $o$) will be in the order $m < n < o$? Wait, no: wait, the larger the angle, the longer the side opposite. So angle $15^\circ$ (smallest) → side $m$ (shortest), angle $68^\circ$ → side $n$ (longer than $m$), angle $97^\circ$ (largest) → side $o$ (longest). Wait, no, that can't be. Wait, no: if angle is larger, side opposite is longer. So angle $15^\circ$ (small) → side opposite (m) is short. Angle $68^\circ$ (medium) → side opposite (n) is medium. Angle $97^\circ$ (large) → side opposite (o) is long. So the order from shortest to longest is $m < n < o$? Wait, no, wait, maybe I mixed up the sides. Wait, let's re-express:

Wait, the angle of $15^\circ$ is at the top. The side opposite to $15^\circ$ is the side between the $68^\circ$ and $97^\circ$ angles, which is labeled $m$. The angle of $68^\circ$ is at the left. The side opposite to $68^\circ$ is the side between the $15^\circ$ and $97^\circ$ angles, labeled $n$. The angle of $97^\circ$ is at the bottom. The side opposite to $97^\circ$ is the side between the $15^\circ$ and $68^\circ$ angles, labeled $o$.

So:

- Angle $15^\circ$ → opposite side $m$
- Angle $68^\circ$ → opposite side $n$
- Angle $97^\circ$ → opposite side $o$

Since $15^\circ < 68^\circ < 97^\circ$, the sides opposite them are in the same order: $m < n < o$? Wait, no, that would mean $m$ is shortest, $n$ middle, $o$ longest. But let's check the angle sizes: $15$ is smallest, $68$ is next, $97$ is largest. So the side opposite $15$ (m) is shortest, side opposite $68$ (n) is longer than m, side opposite $97$ (o) is longest. So the order from shortest to longest is $m < n < o$? Wait, but maybe I got the sides wrong. Wait, maybe the side labeled $m$ is opposite $97^\circ$? No, the diagram: the side $m$ is between $68^\circ$ and $97^\circ$, so it's opposite the $15^\circ$ angle. Let's confirm with the angle sum: $15 + 68 + 97 = 180$, which is correct. So the angles are $15^\circ$, $68^\circ$, $97^\circ$. So the sides opposite are $m$ (15°), $n$ (68°), $o$ (97°). Therefore, since $15 < 68 < 97$, the sides are $m < n < o$. Wait, but maybe the labels are different. Wait, maybe the side labeled $n$ is opposite $15^\circ$, side $m$ opposite $68^\circ$, side $o$ opposite $97^\circ$. Let's re-examine the diagram. The side labeled $n$ is the longest side? Wait, no, the angle of $97^\circ$ is the largest, so the side opposite it (o) should be the longest. The angle of $15^\circ$ is the smallest, so the side opposite it (m) should be the shortest. Then the middle angle is $68^\circ$, so the side opposite (n) is middle length. So the order is $m < n < o$.

# Answer:
$m < n < o$

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QUESTION IMAGE

Question

Explanation:

Step1: Find the third angle

Step2: Relate angles to side lengths

Answer: