1. **State the problem:** We need to find the values of angles $x$ and $y$ in the given geometric figure with points A, B, C, D, E, and F. 2. **Identify known angles and relationships:** - Angle at E between AE and CE is $93^\circ$. - Angle at F between EF and FD is $125^\circ$. - Angles $x$ and $y$ are at points C and A respectively. 3. **Use the straight line angle sum rule:** Since CE extends through E to F and then to D, angles on a straight line sum to $180^\circ$. At point F, angle $125^\circ$ is given, so the adjacent angle on the straight line is: $$180^\circ - 125^\circ = 55^\circ$$ 4. **Use the triangle angle sum rule:** In triangle ABE, the sum of angles is $180^\circ$. Given angle $y$ at A and angle $93^\circ$ at E, the third angle at B is: $$180^\circ - y - 93^\circ = 87^\circ - y$$ 5. **Use the triangle angle sum rule in triangle ACE:** Angles at C ($x$), A ($y$), and E ($93^\circ$) sum to $180^\circ$: $$x + y + 93^\circ = 180^\circ$$ Simplify: $$x + y = 87^\circ$$ 6. **Use the linear pair at point C:** Angles $x$ and the adjacent angle on line AC and CE sum to $180^\circ$. Since the adjacent angle is $55^\circ$ (from step 3), then: $$x + 55^\circ = 180^\circ$$ Simplify: $$x = 125^\circ$$ 7. **Substitute $x$ back into the equation from step 5:** $$125^\circ + y = 87^\circ$$ Subtract $125^\circ$ from both sides: $$y = 87^\circ - 125^\circ = -38^\circ$$ Since angle $y$ cannot be negative, re-examine step 6: the adjacent angle at C is not $55^\circ$ but rather the angle formed by the lines AC and CE, which is $x$. The $55^\circ$ angle is at F, not C. 8. **Correct approach:** At point E, angles around point E sum to $360^\circ$. Given $93^\circ$ at E and $x$ at C, the angle adjacent to $x$ on line CE is $180^\circ - x$. 9. **Use the exterior angle theorem at point F:** Angle $125^\circ$ at F is an exterior angle to triangle CEF, so: $$125^\circ = x + 93^\circ$$ Solve for $x$: $$x = 125^\circ - 93^\circ = 32^\circ$$ 10. **Use the triangle angle sum in triangle ABE:** $$y + 93^\circ + (angle\ at\ B) = 180^\circ$$ Assuming angle at B is $x = 32^\circ$ (since B and C are connected), then: $$y + 93^\circ + 32^\circ = 180^\circ$$ Simplify: $$y = 180^\circ - 125^\circ = 55^\circ$$ **Final answers:** $$x = 32^\circ$$ $$y = 55^\circ$$