1. **Problem statement:** In the figure, AB \parallel CE and DF \parallel AC. Given angles are $\angle BAC = 71^\circ$, $\angle EFC = 80^\circ$, and $\angle BFE = 4x^\circ$. Find the values of $x$ and $y$. 2. **Relevant properties and formulas:** - When two lines are parallel, alternate interior angles are equal. - The sum of angles around a point is 360°. - The sum of angles in a triangle is 180°. 3. **Step 1: Analyze triangle BFE** Since DF \parallel AC and AB \parallel CE, angles at points B, E, F relate through parallel line properties. 4. **Step 2: Use the given angles** We know $\angle BAC = 71^\circ$ and $\angle EFC = 80^\circ$. 5. **Step 3: Find $x$** At point F, angles $\angle BFE = 4x^\circ$ and $\angle EFC = 80^\circ$ are adjacent. Since DF \parallel AC, $\angle BFE$ and $\angle BAC$ are corresponding angles, so $$4x = 71$$ Solving for $x$: $$x = \frac{71}{4} = 17.75$$ 6. **Step 4: Find $y$** Since AB \parallel CE, $\angle BAC = 71^\circ$ and $\angle ACE = y^\circ$ are alternate interior angles, so $$y = 71$$ **Final answers:** $$x = 17.75^\circ$$ $$y = 71^\circ$$