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$In /_\LMN, bar(LN) is extended through point N to point O,m/_MNO=(6x-13)^(@), m/_NLM=(x+7)^(@), and m/_LMN=(2x+4)^(@). Find m/_LMN. Answer:$

In $\triangle \mathrm{LMN}, \overline{L N}$ is extended through point $\mathrm{N}$ to point $\mathrm{O}, \mathrm{m} \angle M N O=(6 x-13)^{\circ}$ , $\mathrm{m} \angle N L M=(x+7)^{\circ}$ , and $\mathrm{m} \angle L M N=(2 x+4)^{\circ}$ . Find $\mathrm{m} \angle L M N$ . $\newline$ Answer:

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Math Problems

Algebra 2

Csc, sec, and cot of special angles

Full solution

Q. In

\triangle \mathrm{LMN}, \overline{L N}

is extended through point

\mathrm{N}

to point

\mathrm{O}, \mathrm{m} \angle M N O=(6 x-13)^{\circ}

\mathrm{m} \angle N L M=(x+7)^{\circ}

, and

\mathrm{m} \angle L M N=(2 x+4)^{\circ}

. Find

\mathrm{m} \angle L M N

\newline

Answer:

Set Up Equation: To find the measure of angle LMN, we need to use the fact that the sum of the angles in a triangle is $180$ degrees. We have the measures of the angles in terms of $x$ , so we can set up an equation to solve for $x$ .
Given Angle Measures: First, let's write down the given angle measures in terms of $x$ :
$m/\_MNO = (6x - 13)^\circ$
$m/\_NLM = (x + 7)^\circ$
$m/\_LMN = (2x + 4)^\circ$
Exterior Angle Theorem: Since $LN$ is extended through point $N$ to point $O$ , angle $\angle MNO$ is an exterior angle to triangle $LMN$ . According to the exterior angle theorem, the measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles. Therefore, $m\angle MNO = m\angle NLM + m\angle LMN$ .
Combine Like Terms: Now we can set up the equation based on the exterior angle theorem: $\newline$ $(6x - 13)^\circ = (x + 7)^\circ + (2x + 4)^\circ$
Solve for x: Combine like terms on the right side of the equation: $\newline$ $(6x - 13)^\circ = (3x + 11)^\circ$
Isolate Term with x: Now, solve for x by subtracting $3x$ from both sides: $\newline$ $6x - 13 - 3x = 3x + 11 - 3x$ $\newline$ $3x - 13 = 11$
Substitute $x$ Back: Add $13$ to both sides to isolate the term with $x$ : $3x - 13 + 13 = 11 + 13$ $3x = 24$
Calculate Value: Divide both sides by $3$ to solve for x: $\newline$ $\frac{3x}{3} = \frac{24}{3}$ $\newline$ $x = 8$
Calculate Value: Divide both sides by $3$ to solve for $x$ :
$\frac{3x}{3} = \frac{24}{3}$
$x = 8$ Now that we have the value of $x$ , we can find the measure of angle LMN by substituting $x$ back into the expression for $m/\_LMN$ :
$m/\_LMN = (2x + 4)^\circ$
$m/\_LMN = (2(8) + 4)^\circ$
Calculate Value: Divide both sides by $3$ to solve for $x$ :
$\frac{3x}{3} = \frac{24}{3}$
$x = 8$ Now that we have the value of $x$ , we can find the measure of angle LMN by substituting $x$ back into the expression for $m/\_LMN$ :
$m/\_LMN = (2x + 4)^\circ$
$m/\_LMN = (2(8) + 4)^\circ$ Calculate the value:
$m/\_LMN = (16 + 4)^\circ$
$m/\_LMN = 20^\circ$