Question

angle side relationship
question
in \\( \delta \mathrm{jkl}, \mathrm{kl}=3, \mathrm{lj}=14 \\), and \\( \mathrm{jk}=13 \\). which list
has the angles of \\( \delta \mathrm{jkl} \\) in order from largest to
smallest?
answer
\\( \mathrm{m} \angle j, \mathrm{~m} \angle k, \mathrm{~m} \angle l \\)
\\( \mathrm{m} \angle l, \mathrm{~m} \angle k, \mathrm{~m} \angle j \\)
\\( \mathrm{m} \angle j, \mathrm{~m} \angle l, \mathrm{~m} \angle k \\)
\\( \mathrm{m} \angle l, \mathrm{~m} \angle j, \mathrm{~m} \angle k \\)
\\( \mathrm{m} \angle k, \mathrm{~m} \angle l, \mathrm{~m} \angle j \\)
\\( \mathrm{m} \angle k, \mathrm{~m} \angle j, \mathrm{~m} \angle l \\)

Explanation:

Step1: Recall the Angle-Side Relationship

In a triangle, the larger side is opposite the larger angle, and the smaller side is opposite the smaller angle. So we first identify the lengths of the sides and their opposite angles.
For $\triangle JKL$:
- Side $KL = 3$ is opposite angle $J$ (since in $\triangle JKL$, side opposite $\angle J$ is $KL$).
- Side $LJ = 14$ is opposite angle $K$ (side opposite $\angle K$ is $LJ$).
- Side $JK = 13$ is opposite angle $L$ (side opposite $\angle L$ is $JK$).

Step2: Order the Sides by Length

Now, order the side lengths from largest to smallest:
$LJ = 14$, $JK = 13$, $KL = 3$. So $14>13>3$, which means $LJ > JK > KL$.

Step3: Order the Angles by Their Opposite Sides

Since the larger side is opposite the larger angle, the angle opposite the largest side ($LJ = 14$) is the largest angle. The side $LJ$ is opposite $\angle K$, so $\angle K$ is the largest angle.
The next largest side is $JK = 13$, which is opposite $\angle L$, so $\angle L$ is the next largest angle.
The smallest side is $KL = 3$, which is opposite $\angle J$, so $\angle J$ is the smallest angle.

So the order of angles from largest to smallest is $m\angle K$, $m\angle L$, $m\angle J$? Wait, no, wait. Wait, let's re - check:

Wait, side opposite $\angle J$: in $\triangle JKL$, vertices are $J$, $K$, $L$. So side opposite $\angle J$ is $KL$ (connecting $K$ and $L$), side opposite $\angle K$ is $LJ$ (connecting $L$ and $J$), side opposite $\angle L$ is $JK$ (connecting $J$ and $K$).

So side lengths: $KL = 3$, $LJ = 14$, $JK = 13$. So $LJ$ (14) is opposite $\angle K$, $JK$ (13) is opposite $\angle L$, $KL$ (3) is opposite $\angle J$.

So the largest side is $LJ = 14$ (opposite $\angle K$), so $\angle K$ is the largest angle. Then the next largest side is $JK = 13$ (opposite $\angle L$), so $\angle L$ is the next largest. Then the smallest side is $KL = 3$ (opposite $\angle J$), so $\angle J$ is the smallest.

Wait, no, wait, I think I made a mistake. Wait, let's label the triangle properly. Let's denote the triangle with vertices $J$, $K$, $L$. So:

- Angle at $J$: $\angle J$, opposite side is $KL$ (length 3)
- Angle at $K$: $\angle K$, opposite side is $LJ$ (length 14)
- Angle at $L$: $\angle L$, opposite side is $JK$ (length 13)

So the lengths of the sides opposite the angles:

- Opposite $\angle J$: 3
- Opposite $\angle K$: 14
- Opposite $\angle L$: 13

So the order of side lengths (opposite angles) is 14 (opposite $\angle K$), 13 (opposite $\angle L$), 3 (opposite $\angle J$). So the angles opposite these sides will have the same order as the sides. So the largest angle is opposite the largest side (14), which is $\angle K$. Then the next largest angle is opposite the next largest side (13), which is $\angle L$. Then the smallest angle is opposite the smallest side (3), which is $\angle J$. So the order of angles from largest to smallest is $m\angle K$, $m\angle L$, $m\angle J$? Wait, but let's check the options. Wait, the options are:

1. $m\angle J$, $m\angle K$, $m\angle L$
2. $m\angle L$, $m\angle K$, $m\angle J$
3. $m\angle J$, $m\angle L$, $m\angle K$
4. $m\angle L$, $m\angle J$, $m\angle K$
5. $m\angle K$, $m\angle L$, $m\angle J$
6. $m\angle K$, $m\angle J$, $m\angle L$

Wait, no, I think I messed up the opposite angles. Let's use the standard notation: in $\triangle ABC$, side $a$ is opposite $\angle A$, side $b$ opposite $\angle B$, side $c$ opposite $\angle C$.

So in $\triangle JKL$:

- Let $\angle J$ be $\angle A$, then side $a$ (opposite $\angle A$) is $KL$.
- Let $\angle K$ be $\angle B$, then side $b$ (opposite $\angle B$) is $LJ$.
- Let $\angle L$ be $\angle C$, then side $c$ (opposite $\angle C$) is $JK$.

So side lengths: $KL = 3$ (side $a$), $LJ = 14$ (side $b$), $JK = 13$ (side $c$).

So side $b = 14$ (opposite $\angle K$), side $c = 13$ (opposite $\angle L$), side $a = 3$ (opposite $\angle J$).

So the order of sides: $b>c>a$ (14 > 13>3). So the order of angles (since angle opposite longer side is larger) is $\angle B>\angle C>\angle A$, which is $\angle K>\angle L>\angle J$. So the measure of angles from largest to smallest is $m\angle K$, $m\angle L$, $m\angle J$? Wait, no, $\angle B$ is $\angle K$, $\angle C$ is $\angle L$, $\angle A$ is $\angle J$. So $\angle K$ (opposite 14) is largest, $\angle L$ (opposite 13) is next, $\angle J$ (opposite 3) is smallest. So the order is $m\angle K$, $m\angle L$, $m\angle J$? But looking at the options, option 5 is $m\angle K$, $m\angle L$, $m\angle J$? Wait, no, let's re - check the options:

Wait the options are:

- $m\angle J$, $m\angle K$, $m\angle L$

- $m\angle L$, $m\angle K$, $m\angle J$

- $m\angle J$, $m\angle L$, $m\angle K$

- $m\angle L$, $m\angle J$, $m\angle K$

- $m\angle K$, $m\angle L$, $m\angle J$

- $m\angle K$, $m\angle J$, $m\angle L$

Wait, I think I made a mistake in identifying the opposite angles. Let's do it again:

In $\triangle JKL$:

- Vertex $J$: the sides adjacent to $J$ are $LJ$ and $JK$, so the side opposite $J$ is $KL$.

- Vertex $K$: the sides adjacent to $K$ are $JK$ and $KL$, so the side opposite $K$ is $LJ$.

- Vertex $L$: the sides adjacent to $L$ are $KL$ and $LJ$, so the side opposite $L$ is $JK$.

So:

- Side opposite $\angle J$: $KL = 3$

- Side opposite $\angle K$: $LJ = 14$

- Side opposite $\angle L$: $JK = 13$

Now, order the sides from longest to shortest: $LJ = 14$, $JK = 13$, $KL = 3$. So $14>13>3$.

Since the larger the side, the larger the angle opposite it, the angle opposite the longest side ($LJ = 14$) is the largest angle. The side $LJ$ is opposite $\angle K$, so $\angle K$ is the largest angle.

The next longest side is $JK = 13$, which is opposite $\angle L$, so $\angle L$ is the next largest angle.

The shortest side is $KL = 3$, which is opposite $\angle J$, so $\angle J$ is the smallest angle.

So the order of angles from largest to smallest is $m\angle K$, $m\angle L$, $m\angle J$? Wait, no, wait: $\angle K$ (opposite 14) is largest, $\angle L$ (opposite 13) is next, $\angle J$ (opposite 3) is smallest. So the order is $m\angle K$, $m\angle L$, $m\angle J$? But let's check the options. Option 5 is $m\angle K$, $m\angle L$, $m\angle J$? Wait, no, option 5 is $m\angle K$, $m\angle L$, $m\angle J$? Wait the fifth option is "m∠K, m∠L, m∠J" and the sixth is "m∠K, m∠J, m∠L". Wait, I think I messed up. Wait, no, let's re - arrange:

Wait, if $LJ = 14$ (opposite $\angle K$), $JK = 13$ (opposite $\angle L$), $KL = 3$ (opposite $\angle J$). So the angles opposite these sides:

- $\angle K$ is opposite 14 (largest side) → largest angle.

- $\angle L$ is opposite 13 (second largest side) → second largest angle.

- $\angle J$ is opposite 3 (smallest side) → smallest angle.

So the order from largest to smallest is $m\angle K$, $m\angle L$, $m\angle J$. So the correct option is the fifth one: $m\angle K$, $m\angle L$, $m\angle J$? Wait no, wait the fifth option is "m∠K, m∠L, m∠J" and the sixth is "m∠K, m∠J, m∠L". Wait, no, let's check the original problem's options again:

The options are:

1. $m\angle J$, $m\angle K$, $m\angle L$

2. $m\angle L$, $m\angle K$, $m\angle J$

3. $m\angle J$, $m\angle L$, $m\angle K$

4. $m\angle L$, $m\angle J$, $m\angle K$

5. $m\angle K$, $m\angle L$, $m\angle J$

6. $m\angle K$, $m\angle J$, $m\angle L$

Wait, I think I made a mistake in the opposite angles. Let's use a different approach. Let's list the sides and their opposite angles:

- Side $KL = 3$: opposite $\angle J$

- Side $LJ = 14$: opposite $\angle K$

- Side $JK = 13$: opposite $\angle L$

So the lengths: $LJ = 14$ (longest), $JK = 13$ (middle), $KL = 3$ (shortest).

So the angles opposite:

- $\angle K$ (opposite 14) → largest angle

- $\angle L$ (opposite 13) → middle angle

- $\angle J$ (opposite 3) → smallest angle

So the order of angles from largest to smallest is $m\angle K$, $m\angle L$, $m\angle J$, which is option 5.

Answer:

1. $m\angle K$, $m\angle L$, $m\angle J$