To determine which side of triangle LMN would be listed first in decreasing order based on angles L, M, and N, we first need to find the measures of all three angles.
Given:
- Angle L = 54 degrees
- Angle M = 2x
- Angle N = 3x - 4 degrees
Since the sum of the angles in a triangle is always 180 degrees, we can set up the following equation:
\[ L + M + N = 180 \]
Substituting the known values:
\[ 54 + 2x + (3x - 4) = 180 \]
Now, simplify the equation:
\[ 54 + 2x + 3x - 4 = 180 \] \[ 50 + 5x = 180 \] \[ 5x = 180 - 50 \] \[ 5x = 130 \] \[ x = 26 \]
Now we can find the measures of angles M and N:
Angle M: \[ M = 2x = 2(26) = 52 \text{ degrees} \]
Angle N: \[ N = 3x - 4 = 3(26) - 4 = 78 - 4 = 74 \text{ degrees} \]
Now we have:
- Angle L = 54 degrees
- Angle M = 52 degrees
- Angle N = 74 degrees
Now let's determine the order based on angle sizes:
- Angle N (74 degrees) is the largest,
- Angle L (54 degrees) is the second largest,
- Angle M (52 degrees) is the smallest.
In a triangle, the side opposite the largest angle is the longest. Thus, in triangle LMN:
- Side opposite Angle N (which is side LM) is the longest,
- Side opposite Angle L (which is side MN) is the second longest,
- Side opposite Angle M (which is side NL) is the shortest.
In decreasing order of length:
- LM (opposite Angle N)
- MN (opposite Angle L)
- NL (opposite Angle M)
Thus, the side listed first in decreasing order of length is LM.