To determine the order of side lengths in triangle \( \triangle LMN \) using the angles given, we first need to find the value of \( x \) and the angles.
The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can set up the following equation:
\[ \angle L + \angle M + \angle N = 180^\circ \]
Substituting the expressions provided:
\[ (4x + 10) + (2x) + 44 = 180 \]
Combining like terms:
\[ 6x + 54 = 180 \]
Next, we solve for \( x \):
\[ 6x = 180 - 54 \] \[ 6x = 126 \] \[ x = 21 \]
Now we can find the measures of the angles:
\[ \angle L = 4x + 10 = 4(21) + 10 = 84 + 10 = 94^\circ \] \[ \angle M = 2x = 2(21) = 42^\circ \] \[ \angle N = 44^\circ \]
The measures of the angles are:
- \( \angle L = 94^\circ \)
- \( \angle M = 42^\circ \)
- \( \angle N = 44^\circ \)
The relationship between the angles and the sides in a triangle is that the larger the angle, the longer the opposite side. Based on the angles, we can order them:
- \( \angle M < \angle N < \angle L \)
This means the corresponding sides are ordered as:
- Side opposite \( \angle M \) (which is side \( LN \)) < Side opposite \( \angle N \) (which is side \( LM \)) < Side opposite \( \angle L \) (which is side \( MN \))
Therefore, the order of the sides in increasing length is:
\[ LN < LM < MN \]
This corresponds to the response option:
LN < LM < MN