1. **State the problem:** We are given angles $m\angle QOS = 53$, $m\angle POR = 56$, and $m\angle POQ = 22$. We need to find $m\angle ROS$. 2. **Understand the setup:** The rays $P$, $Q$, $R$, and $S$ originate from point $O$. The angles are formed between these rays. 3. **Use angle addition:** Since $\angle QOS$ and $\angle POR$ are given, and $\angle POQ$ is between $P$ and $Q$, we can find $\angle ROS$ by considering the full circle around $O$ which sums to $360$ degrees. 4. **Calculate intermediate angles:** - $\angle QOR = \angle QOS - \angle ROS$ (unknown yet) - $\angle POR = 56$ - $\angle POQ = 22$ 5. **Sum of angles around point $O$:** $$m\angle POQ + m\angle QOR + m\angle ROS + m\angle POR = 360$$ 6. **Express $m\angle QOR$ in terms of known angles:** Since $m\angle QOS = 53$, and $m\angle ROS$ is what we want, then $$m\angle QOR = m\angle QOS - m\angle ROS = 53 - m\angle ROS$$ 7. **Substitute into the sum:** $$22 + (53 - m\angle ROS) + m\angle ROS + 56 = 360$$ 8. **Simplify:** $$22 + 53 - m\angle ROS + m\angle ROS + 56 = 360$$ $$22 + 53 + 56 = 360$$ $$131 = 360$$ This is a contradiction, so let's reconsider the angle relationships. 9. **Re-examine the problem:** The rays are arranged as $P$, $Q$, $R$, $S$ counterclockwise. The angles given are: - $m\angle QOS = 53$ (between $Q$ and $S$) - $m\angle POR = 56$ (between $P$ and $R$) - $m\angle POQ = 22$ (between $P$ and $Q$) We want $m\angle ROS$ (between $R$ and $S$). 10. **Sum of angles around $O$:** $$m\angle POQ + m\angle QOR + m\angle ROS + m\angle SOP = 360$$ Note $m\angle SOP = m\angle QOS + m\angle ROS$ because $S$ is between $Q$ and $P$. 11. **Calculate $m\angle QOR$:** Since $m\angle POR = 56$ and $m\angle POQ = 22$, then $$m\angle QOR = m\angle POR - m\angle POQ = 56 - 22 = 34$$ 12. **Calculate $m\angle ROS$:** Since $m\angle QOS = 53$, and $m\angle QOS = m\angle QOR + m\angle ROS$, then $$53 = 34 + m\angle ROS$$ $$m\angle ROS = 53 - 34 = 19$$ **Final answer:** $$\boxed{19}$$