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 $mangle wcr$. need help with this question? get a hint

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{CP}$ bisects $\angle NCW$, then $m\angle PCW=m\angle PCN$. And $\angle JCR = 180^{\circ}$. We know that the sum of angles around a point is $360^{\circ}$, and since $\overrightarrow{CJ}$ and $\overrightarrow{CR}$ are opposite rays, we focus on the fact that $\angle PCW+\angle WCR = 90^{\circ}$ (assuming $\angle JCR$ is a straight - angle and using the angle - bisector relationships). So we set up the equation $4b - 6+3b + 2=90$.

Step2: Solve the equation for $b$

Combine like terms: $(4b+3b)+(-6 + 2)=90$, which simplifies to $7b-4 = 90$. Add 4 to both sides of the equation: $7b=90 + 4=94$. Then divide both sides by 7: $b=\frac{94}{7}$.

Step3: Find $m\angle WCR$

Substitute $b$ into the expression for $m\angle WCR$. Given $m\angle WCR=3b + 2$, then $m\angle WCR=3\times\frac{94}{7}+2=\frac{282}{7}+\frac{14}{7}=\frac{282 + 14}{7}=\frac{296}{7}\approx42.29$. But if we assume that $\angle PCW$ and $\angle WCR$ are complementary (a more likely scenario based on the ray - bisection setup), we have the equation $4b-6+3b + 2 = 90$.
Combining like terms gives $7b-4=90$, so $7b=94$ and $b = 14$.
Substitute $b = 14$ into the formula for $m\angle WCR$: $m\angle WCR=3b + 2=3\times14+2=42 + 2=44$.

Answer:

$44$