To determine which side of triangle \( \triangle LMN \) is the longest, we first need to find the measures of all the angles.
We know that the sum of the angles in a triangle is 180°. Given the angles:
\[ \angle L = 54° \] \[ \angle M = 2x \] \[ \angle N = 3x - 4 \]
The equation for the sum of the angles is:
\[ \angle L + \angle M + \angle N = 180° \]
Substituting the angles, we get:
\[ 54° + 2x + (3x - 4°) = 180° \]
Simplifying this:
\[ 54° + 2x + 3x - 4° = 180° \] \[ 50° + 5x = 180° \] \[ 5x = 180° - 50° \] \[ 5x = 130° \] \[ x = 26° \]
Now we can find the measures of angles \( M \) and \( N \):
\[ \angle M = 2x = 2(26°) = 52° \] \[ \angle N = 3x - 4 = 3(26°) - 4° = 78° - 4° = 74° \]
Now we have the measures of all the angles:
- \( \angle L = 54° \)
- \( \angle M = 52° \)
- \( \angle N = 74° \)
The side opposite to the largest angle will be the longest side. In this case:
- \( \angle L = 54° \) is opposite side \( MN \)
- \( \angle M = 52° \) is opposite side \( NL \)
- \( \angle N = 74° \) is opposite side \( LM \)
Since \( \angle N \) (74°) is the largest angle, side \( LM \) will be the longest side.
Thus, the side listed first in decreasing order would be \( LM \).
Option #1: LM
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