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)
Responses

LN<LM<MN


ML>NL>MN


MN>NL>LM


LM<MN<NL

Answer 1

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**