1. **State the problem:** We are given a straight horizontal line intersected by two diagonal lines forming several angles. The angles given are $y^\circ$ and $55^\circ$ on the left side, and $x^\circ$, $50^\circ$, and another $50^\circ$ on the right side. We need to find the values of $x$ and $y$. 2. **Recall important rules:** - Angles on a straight line sum to $180^\circ$. - Vertically opposite angles are equal. 3. **Analyze the left side:** The angles $y^\circ$ and $55^\circ$ are adjacent on a straight line, so: $$y + 55 = 180$$ 4. **Solve for $y$:** $$y = 180 - 55 = 125$$ 5. **Analyze the right side:** There are two $50^\circ$ angles and an unknown $x^\circ$ angle around the intersection. Since the two $50^\circ$ angles are adjacent and the line is straight, their sum plus $x$ must be $180^\circ$: $$x + 50 + 50 = 180$$ 6. **Simplify and solve for $x$:** $$x + 100 = 180$$ $$x = 180 - 100 = 80$$ 7. **Final answers:** $$x = 80^\circ, \quad y = 125^\circ$$