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$In /_\LMN, bar(LN) is extended through point N to point O,m/_LMN=(x+12)^(@), m/_MNO=(5x+10)^(@), and m/_NLM=(2x+12)^(@). Find m/_MNO. Answer:$

In $\triangle \mathrm{LMN}, \overline{L N}$ is extended through point $\mathrm{N}$ to point $\mathrm{O}, \mathrm{m} \angle L M N=(x+12)^{\circ}$ , $\mathrm{m} \angle M N O=(5 x+10)^{\circ}$ , and $\mathrm{m} \angle N L M=(2 x+12)^{\circ}$ . Find $\mathrm{m} \angle M N O$ . $\newline$ Answer:

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Algebra 1

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Q. In

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

is extended through point

\mathrm{N}

to point

\mathrm{O}, \mathrm{m} \angle L M N=(x+12)^{\circ}

\mathrm{m} \angle M N O=(5 x+10)^{\circ}

, and

\mathrm{m} \angle N L M=(2 x+12)^{\circ}

. Find

\mathrm{m} \angle M N O

\newline

Answer:

Identify Exterior Angle Theorem: To find the measure of angle $MNO$ , we need to use the fact that the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. $\newline$ So, $m\angle MNO = m\angle LMN + m\angle NLM$ .
Substitute Given Expressions: Substitute the given expressions for $m/_{\text{LMN}}$ and $m/_{\text{NLM}}$ into the equation. $m/_{\text{MNO}} = (x + 12) + (2x + 12)$ .
Combine Like Terms: Combine like terms to simplify the equation. $\frac{m}{\_MNO} = x + 12 + 2x + 12 = 3x + 24$ .
Set Up Equation: Now we have an expression for $m/_{\text{MNO}}$ in terms of $x$ , but we also have the measure of angle MNO given as $(5x + 10)$ . Since both expressions represent the same angle, we can set them equal to each other. $3x + 24 = 5x + 10.$
Solve for x: Solve for x by subtracting $3x$ from both sides of the equation. $\newline$ $3x + 24 - 3x = 5x + 10 - 3x$ $\newline$ $24 = 2x + 10.$
Isolate x Term: Subtract $10$ from both sides to isolate the term with $x$ . $\newline$ $24 - 10 = 2x + 10 - 10$ $\newline$ $14 = 2x$ .
Find Value of x: Divide both sides by $2$ to solve for x. $\newline$ $\frac{14}{2} = \frac{2x}{2}$ $\newline$ $7 = x$ .
Calculate Angle MNO: Now that we have the value of $x$ , we can find the measure of angle MNO by substituting $x$ back into the expression for $m/_{\text{MNO}}$ . $\newline$ $m/_{\text{MNO}} = 5x + 10$ $\newline$ $m/_{\text{MNO}} = 5(7) + 10$ $\newline$ $m/_{\text{MNO}} = 35 + 10$ $\newline$ $m/_{\text{MNO}} = 45.$