Question

given that $overrightarrow{op}$ bisects $angle moe$. then $mangle mop=(10 - 3x)^{circ}$ and $mangle poe=(12x + 5)^{circ}$. find the $mangle moe.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{OP}$ bisects $\angle MOE$, then $m\angle MOP=m\angle POE$. So we set up the equation $10 - 3x=12x + 5$.

Step2: Solve the equation for $x$

Add $3x$ to both sides: $10=12x + 5+3x$, which simplifies to $10 = 15x+5$. Then subtract 5 from both sides: $10 - 5=15x$, so $5 = 15x$. Divide both sides by 15: $x=\frac{5}{15}=\frac{1}{3}$.

Step3: Find $m\angle MOP$ or $m\angle POE$

Substitute $x = \frac{1}{3}$ into $m\angle MOP=10 - 3x$. Then $m\angle MOP=10-3\times\frac{1}{3}=10 - 1 = 9^{\circ}$.

Step4: Calculate $m\angle MOE$

Since $m\angle MOE=m\angle MOP + m\angle POE$ and $m\angle MOP=m\angle POE$, then $m\angle MOE = 2m\angle MOP$. So $m\angle MOE=2\times9^{\circ}=18^{\circ}$.

Answer:

$18^{\circ}$