1. **Problem statement:** We have a circle with center O and a vertical tangent line touching the circle on the right. Inside the circle, there is an angle of 31° formed at the top left. We need to find the unknown angles $x$ and $y$ marked in the figure. 2. **Key facts and formulas:** - The radius of a circle is perpendicular to the tangent line at the point of tangency. So, the angle between the radius and the tangent line is 90°. - The angle $x$ is formed between the radius $\overline{O}$ to the tangent point and the vertical tangent line. - The angle inside the circle given is 31°, which is part of a right triangle formed by the radius and the tangent. - The angle $y$ is an exterior angle to the triangle formed outside the circle adjacent to the tangent line. 3. **Find angle $x$:** Since the radius is perpendicular to the tangent line, $$x = 90^\circ$$ 4. **Find angle $y$:** The angle $y$ is supplementary to angle $x$ because they form a straight line along the tangent. So, $$y + x = 180^\circ$$ Substitute $x = 90^\circ$: $$y + 90^\circ = 180^\circ$$ $$y = 180^\circ - 90^\circ = 90^\circ$$ 5. **Summary:** - Angle $x = 90^\circ$ - Angle $y = 90^\circ$ Both unknown angles are right angles due to the properties of the tangent and radius.