1. **Stating the problem:** We are given a circle with points S, T, U, P, Q, R on its circumference and a point O inside the circle. We know the angle at T formed by points S, T, U is 35° and need to find the values of angles $x$ and $y$ at point O formed by points S, P and S, R respectively. 2. **Relevant formulas and rules:** - The angle subtended by an arc at the center of the circle is twice the angle subtended at the circumference on the same arc. - Angles in the same segment of a circle are equal. - The sum of angles around point O is 360°. 3. **Step 1: Use the given angle at T:** The angle $\angle STU = 35^\circ$ is an inscribed angle. 4. **Step 2: Find the central angle subtended by the same arc SU:** By the rule, the central angle $\angle SOU = 2 \times 35^\circ = 70^\circ$. 5. **Step 3: Identify angle $x$:** Angle $x$ is $\angle SOP$ at the center. If $P$ lies on the arc SU, then $x$ corresponds to the central angle subtended by arc SP or PU. Without loss of generality, if $x$ corresponds to $\angle SOP = 70^\circ$ (same as $\angle SOU$), then $x = 70^\circ$. 6. **Step 4: Find angle $y$:** Angle $y$ is $\angle SOR$. If $R$ lies on the circle such that $\angle SOR$ complements $\angle SOP$ to 180° (straight line through O), then $$y = 180^\circ - x = 180^\circ - 70^\circ = 110^\circ.$$ 7. **Final answers:** $$x = 70^\circ, \quad y = 110^\circ.$$