Question

in the figure, $overrightarrow{cj}$ and $overrightarrow{cr}$ are opposite rays. $overrightarrow{cp}$ bisects $angle ncw$ and $overrightarrow{cl}$ bisects $angle jcn$. if $mangle ncp = 5k - 8$ and $mangle ncw = 8k + 2$, find $mangle pcw$.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{CP}$ bisects $\angle NCW$, then $m\angle NCP=m\angle PCW$. Also, $m\angle NCW = 2m\angle NCP$.

Step2: Set up an equation

We know that $m\angle NCW = 8k + 2$ and $m\angle NCP=5k - 8$. Since $m\angle NCW = 2m\angle NCP$, we have the equation $8k + 2=2(5k - 8)$.

Step3: Solve the equation for $k$

Expand the right - hand side: $8k + 2 = 10k-16$.
Subtract $8k$ from both sides: $2=10k - 8k-16$.
Simplify: $2 = 2k-16$.
Add 16 to both sides: $2 + 16=2k$, so $18 = 2k$.
Divide both sides by 2: $k = 9$.

Step4: Find $m\angle NCP$

Substitute $k = 9$ into the expression for $m\angle NCP$: $m\angle NCP=5k - 8=5\times9 - 8=45 - 8 = 37$.

Step5: Find $m\angle PCW$

Since $m\angle PCW=m\angle NCP$, then $m\angle PCW = 37$.

Answer:

$37$