1. **Problem statement:** Find the angles $x$ and $y$ in the first figure where two parallel lines are intersected by a transversal, forming angles $72^\circ$, $y$, $x$, and $60^\circ$. 2. **Relevant rules:** When two parallel lines are cut by a transversal, alternate interior angles are equal, and adjacent angles on a straight line sum to $180^\circ$. 3. **Step 1:** Since $72^\circ$ and $x$ are alternate interior angles, we have: $$x = 72^\circ$$ 4. **Step 2:** Angles $x$ and $y$ are on a straight line, so: $$x + y = 180^\circ$$ 5. **Step 3:** Substitute $x = 72^\circ$: $$72^\circ + y = 180^\circ$$ 6. **Step 4:** Solve for $y$: $$y = 180^\circ - 72^\circ = 108^\circ$$ **Final answers:** $$x = 72^\circ$$ $$y = 108^\circ$$