Question

in the figure, $overrightarrow{cj}$ and $overrightarrow{cr}$ are opposite rays. $overrightarrow{cp}$ bisects $angle ncw$ and $overrightarrow{cl}$ bisects $angle jcn$. if $mangle pcw = 7g$ and $mangle wcr = 5g + 14$, find $mangle wcr$.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{CP}$ bisects $\angle NCW$, then $m\angle PCW=m\angle PCN = 7g$.

Step2: Set up an equation based on linear - pair

$\angle JCR$ is a straight - angle, so $m\angle PCW + m\angle PCN+m\angle WCR=180^{\circ}$. Substituting the values, we have $7g + 7g+(5g + 14)=180$.

Step3: Combine like terms

Combining the $g$ terms on the left - hand side gives $(7g+7g + 5g)+14 = 180$, which simplifies to $19g+14 = 180$.

Step4: Solve for $g$

Subtract 14 from both sides: $19g=180 - 14=166$. Then $g=\frac{166}{19}=8.736842\approx8.74$.

Step5: Find $m\angle WCR$

Substitute $g$ into the expression for $m\angle WCR$. Since $m\angle WCR = 5g+14$, then $m\angle WCR=5\times8.74 + 14=43.7+14 = 57.7$.

Answer:

$57.7$