what is the arc measure of minor arc $\\widehat{bc}$ in degrees?

Explanation:

Step1: Identify vertical angles

The angles $\angle BPC$ and $\angle APD$ are vertical angles formed by intersecting lines $AC$ and $BD$.

Step2: Set up the equation

Vertical angles are equal in measure, so we equate the given expressions. $$4k + 159 = 2k + 153$$

Step3: Solve for $k$

Subtract $2k$ and $159$ from both sides to isolate the variable. $$2k = -6 \implies k = -3$$

Step4: Calculate the central angle measure

Substitute $k = -3$ into the expression for $m\angle BPC$. $$m\angle BPC = 4(-3) + 159 = -12 + 159 = 147^{\circ}$$

Step5: Determine the arc measure

The measure of a minor arc is equal to its central angle. $$m\widehat{BC} = m\angle BPC = 147^{\circ}$$

Answer:

147^{\circ}