Question

in $\triangle jkl$, $kl = 11$, $lj = 10$, and $jk = 13$. which statement about the angles of $\triangle jkl$ must be true?
answer
$\circ$ $m\angle k > m\angle l > m\angle j$ $\circ$ $m\angle j > m\angle k > m\angle l$
$\circ$ $m\angle l > m\angle k > m\angle j$ $\circ$ $m\angle k > m\angle j > m\angle l$
$\circ$ $m\angle l > m\angle j > m\angle k$ $\circ$ $m\angle j > m\angle l > m\angle k$

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger the length of a side, the larger the measure of the angle opposite that side. This is a fundamental property of triangles in geometry.

Step2: Identify the sides and their opposite angles

• In $\triangle JKL$, side $KL = 11$, and the angle opposite to $KL$ is $\angle J$ (because in $\triangle JKL$, side $KL$ is opposite vertex $J$).
• Side $LJ=10$, and the angle opposite to $LJ$ is $\angle K$ (side $LJ$ is opposite vertex $K$).
• Side $JK = 13$, and the angle opposite to $JK$ is $\angle L$ (side $JK$ is opposite vertex $L$).

Step3: Compare the lengths of the sides

We are given the lengths of the sides: $JK = 13$, $KL=11$, and $LJ = 10$. So, we can order the sides from longest to shortest: $JK>LK > LJ$.

Step4: Order the angles based on the side - angle relationship

Since the larger side is opposite the larger angle, the angle opposite the longest side ($JK$) will be the largest angle, and the angle opposite the shortest side ($LJ$) will be the smallest angle.

• The angle opposite $JK$ (which is $\angle L$) is the largest.
• The angle opposite $KL$ (which is $\angle J$) is the middle - sized angle.
• The angle opposite $LJ$ (which is $\angle K$) is the smallest.

Wait, no, let's re - check:
Wait, side $KL = 11$, opposite angle $J$; side $LJ=10$, opposite angle $K$; side $JK = 13$, opposite angle $L$.
So side lengths: $JK = 13$ (longest), $KL = 11$ (middle), $LJ=10$ (shortest).
So angles: angle opposite $JK$ is $\angle L$, angle opposite $KL$ is $\angle J$, angle opposite $LJ$ is $\angle K$.
So the order of angles from largest to smallest is $\angle L>\angle J>\angle K$? Wait, no, wait:
Wait, side $JK = 13$ (opposite $\angle L$), side $KL = 11$ (opposite $\angle J$), side $LJ=10$ (opposite $\angle K$).
Since $13>11 > 10$, then the angles opposite them: $\angle L>\angle J>\angle K$? Wait, no, wait, let's do it again.
Wait, in triangle $JKL$:

• Vertex $J$: opposite side is $KL = 11$
• Vertex $K$: opposite side is $LJ=10$
• Vertex $L$: opposite side is $JK = 13$

So the lengths of the sides opposite the angles:

• Opposite $\angle J$: $KL = 11$
• Opposite $\angle K$: $LJ=10$
• Opposite $\angle L$: $JK = 13$

Since $13>11>10$, the angles opposite these sides will have the same order. So $\angle L>\angle J>\angle K$? Wait, no, wait, the angle opposite the longer side is larger. So:

Since $JK$ (length 13) is the longest side, the angle opposite to it ($\angle L$) is the largest angle.

$KL$ (length 11) is the middle - length side, so the angle opposite to it ($\angle J$) is the middle - sized angle.

$LJ$ (length 10) is the shortest side, so the angle opposite to it ($\angle K$) is the smallest angle.

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

Wait, but let's check the options again. The options are:

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

So the correct one is $m\angle L>m\angle J>m\angle K$.

Answer:

$m\angle L>m\angle J>m\angle K$ (the option with this statement)