1. **Problem statement:** We need to find the value of the angle $x$ formed at the center $O$ of a regular octagon by two lines extending from $O$ to two adjacent vertices. 2. **Key fact:** A regular octagon has 8 equal sides and 8 equal central angles formed at the center by lines to adjacent vertices. 3. **Formula:** The sum of all central angles around point $O$ is $360^\circ$. 4. Since the octagon is regular, each central angle is equal, so each angle $x$ is given by: $$x = \frac{360^\circ}{8}$$ 5. Calculate: $$x = 45^\circ$$ 6. **Answer:** The value of $x$ is $45^\circ$. This means the angle between two adjacent vertices at the center of a regular octagon is $45$ degrees.