1. **Stating the problem:** We are given a circle with center U and points P, Q, R, S, T on the circumference. The measures of arcs are given as $mPQ=71^\circ$, $mSR=161^\circ$, and $mQRT=19^\circ$. We need to find $mPSR$ and $mPS$. 2. **Understanding the problem:** The circle is divided into arcs by points P, Q, R, S, and T. The sum of all arcs around the circle is $360^\circ$. 3. **Calculate the missing arcs:** - The arcs given are $mPQ=71^\circ$, $mSR=161^\circ$, $mQRT=19^\circ$, and the last arc $mPSR$ is unknown. - Since $mQRT$ is $19^\circ$, and the arcs $mPQ$, $mSR$, and $mPSR$ cover the rest of the circle, we use the total $360^\circ$ to find $mPSR$. 4. **Formula:** $$mPQ + mSR + mQRT + mPSR = 360^\circ$$ 5. **Substitute known values:** $$71 + 161 + 19 + mPSR = 360$$ 6. **Simplify:** $$251 + mPSR = 360$$ 7. **Solve for $mPSR$:** $$mPSR = 360 - 251 = 109^\circ$$ 8. **Find $mPS$:** - $mPS$ is the measure of the arc from P to S passing through Q and R. - This arc includes $mPQ$, $mQRT$, and $mRS$ (which is $mSR$ reversed, so $mSR=161^\circ$ is the arc from S to R, so $mRS$ is also $161^\circ$ since arcs are measured in the direction given). - However, since $mSR=161^\circ$ is given, and $mQRT=19^\circ$, and $mPQ=71^\circ$, the arc $mPS$ is the sum of $mPQ + mQRT + mRS$. 9. **Calculate $mPS$:** $$mPS = mPQ + mQRT + mRS = 71 + 19 + 161 = 251^\circ$$ **Final answers:** - $mPSR = 109^\circ$ - $mPS = 251^\circ$