Question

in the figure, $overrightarrow{cj}$ and $overrightarrow{cr}$ are opposite rays. $overrightarrow{cp}$ bisects $angle ncw$ and $overrightarrow{cl}$ bisects $angle jcn$. if $mangle pcw = 4b - 6$ and $mangle wcr = 3b + 2$, find $mangle wcr$.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{CP}$ bisects $\angle NCW$, then $m\angle NCP=m\angle PCW = 4b - 6$.

Step2: Consider linear - pair relationship

$\angle JCR$ is a straight - angle, so $m\angle JCN+m\angle NCW+m\angle WCR=180^{\circ}$. Also, since $\overrightarrow{CJ}$ and $\overrightarrow{CR}$ are opposite rays, the sum of angles around point $C$ that are part of the straight - line $JR$ is $180^{\circ}$. In terms of the given angles, $m\angle NCP + m\angle PCW+m\angle WCR=180^{\circ}$. Substituting $m\angle NCP = m\angle PCW = 4b - 6$ and $m\angle WCR = 3b + 2$ into the equation, we get $(4b - 6)+(4b - 6)+(3b + 2)=180$.

Step3: Simplify the equation

Combine like terms: $4b+4b + 3b-6-6 + 2=180$.
$11b-10 = 180$.

Step4: Solve for $b$

Add 10 to both sides of the equation: $11b=180 + 10=190$.
Then $b=\frac{190}{11}$.

Step5: Find $m\angle WCR$

Substitute $b=\frac{190}{11}$ into the expression for $m\angle WCR$: $m\angle WCR=3b + 2=3\times\frac{190}{11}+2=\frac{570}{11}+\frac{22}{11}=\frac{570 + 22}{11}=\frac{592}{11}\approx53.82$.

Answer:

$\frac{592}{11}$