Question

in the figure, $overrightarrow{cj}$ and $overrightarrow{cr}$ are opposite rays. $overrightarrow{cp}$ bisects $angle ncw$ and $overrightarrow{cl}$ bisects $angle jcn$. if $mangle jcn = 8r - 2$ and $mangle lcn = 3r + 15$, find $mangle jcl$.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{CL}$ bisects $\angle JCN$, then $m\angle JCN = 2m\angle LCN$.
We know that $m\angle JCN=8r - 2$ and $m\angle LCN = 3r+15$. So, $8r - 2=2(3r + 15)$.

Step2: Solve the equation for $r$

Expand the right - hand side: $8r - 2=6r + 30$.
Subtract $6r$ from both sides: $8r-6r - 2=6r-6r + 30$, which gives $2r-2 = 30$.
Add 2 to both sides: $2r-2 + 2=30 + 2$, so $2r=32$.
Divide both sides by 2: $r = 16$.

Step3: Find $m\angle LCN$

Substitute $r = 16$ into the formula for $m\angle LCN$: $m\angle LCN=3r + 15=3\times16+15=48 + 15=63$.

Step4: Find $m\angle JCL$

Since $\overrightarrow{CL}$ bisects $\angle JCN$, $m\angle JCL=m\angle LCN$. So $m\angle JCL = 63$.

Answer:

$63$