1. **State the problem:** We need to find the arc measure of the minor arc \(\stackrel{\large{\frown}}{BC}\) in degrees on a circle centered at point P with points A, B, C, and D on the circle. Given that \(AC\) and \(BD\) are diameters, and angles \(\angle APD = 2K + 153^\circ\) and \(\angle BPC = 4K + 159^\circ\). 2. **Recall important facts:** - Since \(AC\) and \(BD\) are diameters, points A and C are opposite ends of a diameter, and points B and D are opposite ends of another diameter. - The angles \(\angle APD\) and \(\angle BPC\) are central angles subtending arcs \(AD\) and \(BC\) respectively. - The sum of central angles around point P is \(360^\circ\). 3. **Set up the equation:** \[ \angle APD + \angle BPC = 360^\circ \] Substitute the expressions: \[ (2K + 153) + (4K + 159) = 360 \] 4. **Simplify and solve for K:** \[ 2K + 153 + 4K + 159 = 360 \] \[ 6K + 312 = 360 \] \[ 6K = 360 - 312 \] \[ 6K = 48 \] \[ K = \frac{48}{6} \] \[ K = 8 \] 5. **Find \(\angle BPC\):** \[ \angle BPC = 4K + 159 = 4(8) + 159 = 32 + 159 = 191^\circ \] 6. **Interpret the result:** The central angle \(\angle BPC\) measures \(191^\circ\), which corresponds to the major arc \(BC\). 7. **Find the minor arc \(\stackrel{\large{\frown}}{BC}\):** Since the full circle is \(360^\circ\), the minor arc measure is: \[ 360^\circ - 191^\circ = 169^\circ \] **Final answer:** The arc measure of the minor arc \(\stackrel{\large{\frown}}{BC}\) is \(169^\circ\).