1. **Problem Statement:** We are given a regular 10-sided polygon (decagon) labeled ABCDEFGHIJ. Two angles, $x$ at vertex J and $y$ at vertex B, are marked inside the polygon. We need to show that $x = y$. 2. **Key Formula:** The measure of each interior angle of a regular polygon with $n$ sides is given by: $$\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$$ 3. **Calculate the interior angle of the decagon:** For $n=10$, $$\text{Interior angle} = \frac{(10-2) \times 180^\circ}{10} = \frac{8 \times 180^\circ}{10} = 144^\circ$$ 4. **Properties of a regular polygon:** - All sides are equal. - All interior angles are equal. - The polygon is symmetric about its center. 5. **Reasoning about angles $x$ and $y$:** - Since the polygon is regular, vertices J and B are symmetric with respect to the center. - The angles $x$ and $y$ are formed by lines from vertex A to vertices J and B respectively. - Due to the symmetry and equal side lengths, the angles subtended at J and B by the same vertex A are equal. 6. **Conclusion:** Therefore, by symmetry and the properties of a regular decagon, $$x = y$$ This completes the proof that the two marked angles are equal.