1. **Problem statement:** We have a circle with a tangent line CDE touching the circle at point D. We need to find the sizes of angles $x$ and $y$ on the tangent line, given some angles inside the circle. 2. **Key properties and formulas:** - The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment of the circle (Alternate Segment Theorem). - The sum of angles in a triangle is $180^\circ$. 3. **Finding angle $x$:** - Angle $x$ is between segments $BD$ and $DE$ on the tangent line. - By the Alternate Segment Theorem, angle $x$ equals the angle subtended by chord $BD$ in the alternate segment. - Inside the circle, angle at $A$ is $59^\circ$, and angle at $B$ is $70^\circ$. - Since $BD$ is a chord, the angle subtended by $BD$ at $A$ is $59^\circ$. - Therefore, $x = 59^\circ$. 4. **Finding angle $y$:** - Angle $y$ is between segments $DE$ and $FD$ on the tangent line. - By the Alternate Segment Theorem, angle $y$ equals the angle subtended by chord $FD$ in the alternate segment. - Inside the circle, angles at $F$ are $11^\circ$ and $36^\circ$ (likely two different angles at $F$ in different triangles). - The angle subtended by chord $FD$ at $B$ is $70^\circ$. - Using triangle $BFD$, sum of angles is $180^\circ$: $$70^\circ + 11^\circ + y = 180^\circ$$ - Solving for $y$: $$y = 180^\circ - 70^\circ - 11^\circ = 99^\circ$$ 5. **Final answers:** - Angle $x = 59^\circ$ - Angle $y = 99^\circ$