1. **State the problem:** We are given angles formed by rotating vectors \(\overrightarrow{AB}\) and \(\overrightarrow{CD}\) to create angles \(\angle XPA\) and \(\angle XPC\) with measures 110° and 30° respectively. We need to find the sums: - \(m \angle CPX + m \angle XPB\) - \(m \angle CPX + m \angle APC\) - \(m \angle APC + m \angle APD\) 2. **Understand the angles:** - \(\angle XPA = 110^\circ\) - \(\angle XPC = 30^\circ\) Since \(P\) is the vertex, and \(X\) is a point on the circle, the angles around point \(P\) sum to 360°. 3. **Find \(m \angle CPX\) and \(m \angle XPB\):** - \(\angle CPX\) is adjacent to \(\angle XPC\), so \(m \angle CPX = 180^\circ - m \angle XPC = 180^\circ - 30^\circ = 150^\circ\) - \(\angle XPB\) is adjacent to \(\angle XPA\), so \(m \angle XPB = 180^\circ - m \angle XPA = 180^\circ - 110^\circ = 70^\circ\) 4. **Calculate the sums:** - \(m \angle CPX + m \angle XPB = 150^\circ + 70^\circ = 220^\circ\) but since the problem states the sum should be less than or equal to 180°, this suggests these angles are not adjacent or we need to consider the smaller angles formed. 5. **Re-examine the problem:** - Since \(m \angle CPX + m \angle XPB\) must be \(\leq 180^\circ\), the smaller angles adjacent to \(\angle XPC\) and \(\angle XPA\) are: \[m \angle CPX = 30^\circ, \quad m \angle XPB = 110^\circ\] - So the sum is \(30^\circ + 110^\circ = 140^\circ\) 6. **Next sums:** - \(m \angle CPX + m \angle APC\): Since \(\angle APC\) is vertically opposite to \(\angle XPB\), \(m \angle APC = m \angle XPB = 110^\circ\) So, \[m \angle CPX + m \angle APC = 30^\circ + 110^\circ = 140^\circ\] - \(m \angle APC + m \angle APD\): Assuming \(\angle APD\) is adjacent to \(\angle XPA\) and supplementary, \[m \angle APD = 180^\circ - m \angle XPA = 180^\circ - 110^\circ = 70^\circ\] So, \[m \angle APC + m \angle APD = 110^\circ + 70^\circ = 180^\circ\] 7. **Final answers:** - \(m \angle CPX + m \angle XPB = 140^\circ\) - \(m \angle CPX + m \angle APC = 140^\circ\) - \(m \angle APC + m \angle APD = 180^\circ\) --- For the second set of rotations: - \(\angle XPA = 70^\circ\) - \(\angle XPC = 90^\circ\) The problem does not ask for sums here explicitly, so we stop here.