Question

A triangle is labeled △LMN, and the angles can be represented by the following expressions: ∠L=4x+10, ∠M=2x, and ∠N=44°. Use the angle measures to list the side lengths in increasing order.(1 point)ResponsesLN<LM<MNupper L upper N is less than upper L upper M is less than upper M upper NMN>NL>LMupper M upper N is greater than upper N upper L is greater than upper L upper MML>NL>MNupper M upper L is greater than upper N upper L is greater than upper M upper NLM<MN<NL

Answer 1

To solve for the angles of triangle \( \triangle LMN \) using the given expressions, we first note that the sum of the angles in a triangle is always \( 180^\circ \). Therefore, we set up the equation:

\[ ∠L + ∠M + ∠N = 180^\circ \]

Substituting the expressions for the angles, we have:

\[ (4x + 10) + 2x + 44° = 180° \]

Combining the terms gives:

\[ 6x + 54° = 180° \]

Now, we isolate \( x \):

\[ 6x = 180° - 54° \] \[ 6x = 126° \] \[ x = 21° \]

Now that we have the value for \( x \), we can find the measures of the angles:

\[ ∠L = 4x + 10 = 4(21) + 10 = 84 + 10 = 94° \]

\[ ∠M = 2x = 2(21) = 42° \]

\[ ∠N = 44° \]

Now we have the measures of the angles:

• \( ∠L = 94° \)
• \( ∠M = 42° \)
• \( ∠N = 44° \)

To list the sides in increasing order, we use the fact that the larger the angle, the longer the opposite side. Therefore, we have:

• \( ∠M = 42° \) corresponds to side \( LN \)
• \( ∠N = 44° \) corresponds to side \( LM \)
• \( ∠L = 94° \) corresponds to side \( MN \)

This gives us the following order of the sides based on the angle measures:

\[ LN < LM < MN \]

Thus, the answer is:

LN < LM < MN.