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)

Question

Responses

LN<LM<MN
upper L upper N is less than upper L upper M is less than upper M upper N

ML>NL>MN
upper M upper L is greater than upper N upper L is greater than upper M upper N

MN>NL>LM
upper M upper N is greater than upper N upper L is greater than upper L upper M

LM<MN<NL

Answer 1

To find the side lengths in increasing order based on the angles of triangle \( \triangle LMN \), we first need to determine the measures of the angles \( \angle L \) and \( \angle M \).

Given: \[ \angle L = 4x + 10 \] \[ \angle M = 2x \] \[ \angle N = 44^\circ \]

Using the fact that the sum of the angles in a triangle is \( 180^\circ \), we can set up the equation: \[ \angle L + \angle M + \angle N = 180^\circ \] Substituting the values we have: \[ (4x + 10) + 2x + 44 = 180 \] Simplifying: \[ 6x + 54 = 180 \] Subtracting 54 from both sides: \[ 6x = 126 \] Dividing by 6: \[ x = 21 \]

Now substituting \( x \) back into the expressions for the angles: \[ \angle L = 4(21) + 10 = 84 + 10 = 94^\circ \] \[ \angle M = 2(21) = 42^\circ \] \[ \angle N = 44^\circ \]

Thus, the angle measures are:

• \( \angle L = 94^\circ \)
• \( \angle M = 42^\circ \)
• \( \angle N = 44^\circ \)

Now, we rank these angles:

• \( \angle M = 42^\circ \)
• \( \angle N = 44^\circ \)
• \( \angle L = 94^\circ \)

In a triangle, the side lengths are opposite to their respective angles. Therefore, the order from the smallest angle to the largest angle gives us the increasing order of the side lengths:

• \( \angle M \) (42°) → side \( LN \)
• \( \angle N \) (44°) → side \( LM \)
• \( \angle L \) (94°) → side \( MN \)

Thus, the side lengths in increasing order are: \[ LN < LM < MN \]

The correct response is: \[ LN < LM < MN \]