1. **Stating the problem:** Find the values of $x$ and $y$ in the given polygon where some interior and exterior angles are labeled, including an interior angle of $130^\circ$, an adjacent exterior angle of $(x+30)^\circ$, an interior angle of $60^\circ$, and an opposite exterior angle of $150^\circ$. 2. **Relevant formulas and rules:** - The sum of an interior angle and its adjacent exterior angle is always $180^\circ$ because they form a linear pair. - Opposite angles formed by intersecting lines are equal. 3. **Step-by-step solution:** - For the vertex with interior angle $130^\circ$ and adjacent exterior angle $(x+30)^\circ$: $$130 + (x+30) = 180$$ Simplify: $$130 + x + 30 = 180$$ $$x + 160 = 180$$ $$x = 180 - 160 = 20$$ - For the vertex with interior angle $60^\circ$ and opposite exterior angle $150^\circ$: Since opposite angles are equal, $$y = 150$$ 4. **Final answers:** - $x = 20$ - $y = 150$ These values satisfy the angle relationships in the figure.