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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

5 answers

To determine the order of the side lengths in triangle △LMN, we first need to find the measures of the angles ∠L, ∠M, and ∠N.

We know that the sum of the angles in a triangle is 180°, so we can set up the equation:

\[ \angle L + \angle M + \angle N = 180° \]

Substituting the given expressions, we have:

\[ (4x + 10) + (2x) + 44° = 180° \]

Combine like terms:

\[ 6x + 54° = 180° \]

Now, solve for \(x\):

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

Now we can find the measures of the angles:

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

  2. ∠M: \[ \angle M = 2x = 2(21) = 42° \]

  3. ∠N: \[ \angle N = 44° \]

So we have:

  • ∠L = 94°
  • ∠M = 42°
  • ∠N = 44°

Next, we can determine the relationship between the angles and the sides. According to the properties of triangles, the side lengths are inversely proportional to the angles opposite them (larger angle ≠ longer side).

Thus, the hierarchy of angles is:

  • ∠M (42°) < ∠N (44°) < ∠L (94°)

Correspondingly, the sides opposite these angles will have the following order (using the labels of the triangle):

  • Side opposite ∠M (LM)
  • Side opposite ∠N (LN)
  • Side opposite ∠L (MN)

Thus, the order of the sides in increasing length is:

\[ LM < LN < MN \]

Since none of your provided options exactly match this conclusion, it is likely that the intended format of the answer may be included in responses if errors in interpretation were present, but the closest in structure would be:

Responses: LM < LN < MN (though not explicitly listed within the options given).

you sure? whats the answer

Let's carefully analyze the angles and their corresponding sides again:

We have the angles:

  • ∠L = 94°
  • ∠M = 42°
  • ∠N = 44°

According to the properties of triangles, the side lengths are determined based on the angles opposite to them:

  • The side opposite the smallest angle is the shortest side.
  • The side opposite the largest angle is the longest side.

From our angles:

  • The smallest angle is ∠M (42°), so the side opposite this angle (which is side LN) is the shortest.
  • The next smallest angle is ∠N (44°), so the side opposite this angle (which is side LM) is next in length.
  • The largest angle is ∠L (94°), so the side opposite this angle (which is side MN) is the longest.

Thus, in order of increasing side lengths:

  • \( LN < LM < MN \)

Looking at the response options given initially:

  • The correct answer is not listed verbatim. However, since \(LN < LM < MN\) does not match any of the provided options, it appears there was a misunderstanding.

The correct arrangement based on our calculations is: LN < LM < MN

Please verify with your answer choices, as it seems an option representing that order specifically was missing. Otherwise, based on the relationships derived from angle measures, that should be the confirmed order of the side lengths.

what is the order of the angles in INCREASING order

The angles we calculated for triangle \( \triangle LMN \) are:

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

To list the angles in increasing order, we look for the smallest to the largest:

  1. \( \angle M = 42^\circ \)
  2. \( \angle N = 44^\circ \)
  3. \( \angle L = 94^\circ \)

Therefore, the order of the angles in increasing order is:

\( \angle M < \angle N < \angle L \)

This represents the angles in ascending size.