1. **State the problem:** We have a regular 9-sided polygon (nonagon) with interior angles of 140 degrees and exterior angles of 40 degrees. We want to find the angle $x$ formed at point $K$ on the straight line through points $O$, $D$, and $K$. 2. **Recall the properties of regular polygons:** - The sum of exterior angles of any polygon is always $360^\circ$. - Each exterior angle of a regular polygon with $n$ sides is $\frac{360}{n}$. - Each interior angle is $180^\circ -$ exterior angle. 3. **Calculate the exterior angle:** $$\text{Exterior angle} = \frac{360}{9} = 40^\circ$$ 4. **Calculate the interior angle:** $$\text{Interior angle} = 180^\circ - 40^\circ = 140^\circ$$ 5. **Analyze the angle at point $K$:** - Since $O$, $D$, and $K$ are collinear, the angle at $K$ on line $ODK$ is supplementary to the interior angle at vertex $D$. - The interior angle at $D$ is $140^\circ$, so the angle $x$ at $K$ is: $$x = 180^\circ - 140^\circ = 40^\circ$$ 6. **Final answer:** $$\boxed{40^\circ}$$