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$In /_\MNO, bar(MO) is extended through point O to point P,m/_MNO=(2x+17)^(@), m/_OMN=(3x+18)^(@), and m/_NOP=(8x-13)^(@). Find m/_MNO. Answer:$

In $\triangle \mathrm{MNO}, \overline{M O}$ is extended through point $\mathrm{O}$ to point $\mathrm{P}, \mathrm{m} \angle M N O=(2 x+17)^{\circ}$ , $\mathrm{m} \angle O M N=(3 x+18)^{\circ}$ , and $\mathrm{m} \angle N O P=(8 x-13)^{\circ}$ . Find $\mathrm{m} \angle M N O$ . $\newline$ Answer:

Full solution

Q. In

\triangle \mathrm{MNO}, \overline{M O}

is extended through point

\mathrm{O}

to point

\mathrm{P}, \mathrm{m} \angle M N O=(2 x+17)^{\circ}

\mathrm{m} \angle O M N=(3 x+18)^{\circ}

, and

\mathrm{m} \angle N O P=(8 x-13)^{\circ}

. Find

\mathrm{m} \angle M N O

\newline

Answer:

Understand Angles Relationship: Understand the relationship between the angles in the problem. $\newline$ In triangle $MNO$ , the sum of the interior angles is $180$ degrees. The exterior angle $NOP$ is equal to the sum of the two non-adjacent interior angles ( $MNO$ and $OMN$ ).
Set Up Equation: Set up the equation based on the angle sum property. $\newline$ $m/\_MNO + m/\_OMN + m/\_NOP = 180$ degrees (interior angles of a triangle) $\newline$ $(2x + 17) + (3x + 18) + (8x - 13) = 180$
Combine Terms and Solve: Combine like terms and solve for $x$ . $2x + 3x + 8x + 17 + 18 - 13 = 180$ $13x + 22 = 180$
Subtract to Find $x$ : Subtract $22$ from both sides of the equation. $\newline$ $13x = 180 - 22$ $\newline$ $13x = 158$
Divide to Find x: Divide both sides by $13$ to find the value of $x$ . $\newline$ $x = \frac{158}{13}$ $\newline$ $x = 12.1538461538$ (This is an approximation; we will use the exact value for further calculations.)
Substitute $x$ into Expression: Substitute the value of $x$ back into the expression for $m/\_MNO$ .
$m/\_MNO = 2x + 17$
$m/\_MNO = 2(12.1538461538) + 17$
$m/\_MNO = 24.3076923077 + 17$
$m/\_MNO = 41.3076923077$ (This is an approximation; we will round to the nearest whole number.)
Round to Nearest Whole Number: Round the value of $m/_{MNO}$ to the nearest whole number. $\newline$ $m/_{MNO} \approx 41$ degrees