Questions: In the figure, CJ and CR are opposite rays. CP bisects angle NCW and CL bisects angle JCN. If m angle JCN=8r-2 and m angle LCN=3r+15, find m angle JCL.

In the figure, CJ and CR are opposite rays. CP bisects angle NCW and CL bisects angle JCN.

If m angle JCN=8r-2 and m angle LCN=3r+15, find m angle JCL.

Transcript text: In the figure, $\overrightarrow{C J}$ and $\overrightarrow{C R}$ are opposite rays. $\overrightarrow{C P}$ bisects $\angle N C W$ and $\overrightarrow{C L}$ bisects $\angle J C N$. If $m \angle J C N=8 r-2$ and $m \angle L C N=3 r+15$, find $m \angle J C L$.

Solution

Solution Steps

Step 1: Identify Given Information

\( \overrightarrow{CJ} \) and \( \overrightarrow{CR} \) are opposite rays.
\( \overrightarrow{CP} \) bisects \( \angle NCW \).
\( \overrightarrow{CL} \) bisects \( \angle JCN \).
\( m\angle JCN = 8r - 2 \).
\( m\angle LCN = 3r + 15 \).

Step 2: Determine Relationship Between Angles

Since \( \overrightarrow{CL} \) bisects \( \angle JCN \), we have: \[ m\angle JCL = \frac{1}{2} m\angle JCN \]

Step 3: Substitute Given Values

Substitute \( m\angle JCN = 8r - 2 \) into the equation: \[ m\angle JCL = \frac{1}{2} (8r - 2) \]

Step 4: Simplify the Expression

Simplify the expression to find \( m\angle JCL \): \[ m\angle JCL = \frac{1}{2} (8r - 2) \] \[ m\angle JCL = 4r - 1 \]

Final Answer

\[ m\angle JCL = 4r - 1 \]

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