Question

cj and cr are opposite rays; cp bisects ∠ncw and cl bisects ∠jcn. if ( mangle ncp = 5k - 8 ) and ( mangle ncw = 8k + 2 ), find ( mangle pcw ).

Explanation:

Step1: Analyze Angle Bisectors

Since \( \overrightarrow{CP} \) bisects \( \angle NCW \), we know that \( m\angle NCP = m\angle PCW \). Also, \( m\angle NCW = m\angle NCP + m\angle PCW = 2m\angle NCP \) (because \( m\angle NCP = m\angle PCW \)).

Given \( m\angle NCP = 5k - 8 \) and \( m\angle NCW = 8k + 2 \), substitute into the equation \( m\angle NCW = 2m\angle NCP \):
\[
8k + 2 = 2(5k - 8)
\]

Step2: Solve for \( k \)

Expand the right - hand side:
\[
8k + 2 = 10k - 16
\]
Subtract \( 8k \) from both sides:
\[
2 = 2k - 16
\]
Add 16 to both sides:
\[
2 + 16 = 2k
\]
\[
18 = 2k
\]
Divide both sides by 2:
\[
k=\frac{18}{2}=9
\]

Step3: Find \( m\angle NCP \)

Substitute \( k = 9 \) into \( m\angle NCP = 5k - 8 \):
\[
m\angle NCP=5\times9 - 8=45 - 8 = 37
\]

Step4: Find \( m\angle PCW \)

Since \( \overrightarrow{CP} \) bisects \( \angle NCW \), \( m\angle PCW=m\angle NCP \). So \( m\angle PCW = 37 \)

Answer:

\( m\angle PCW = 37 \)