1. **Problem statement:** Given two parallel lines $\ell \parallel m$, and $\overrightarrow{DE}$ bisects $\angle CDF$, find the value of $y$. 2. **Understanding the problem:** Since $\ell \parallel m$, alternate interior angles and corresponding angles are equal. 3. **Given:** $\angle C = 130^\circ$ (on line $\ell$), $\overrightarrow{DE}$ bisects $\angle CDF$, and we want to find $y = \angle E$. 4. **Step 1: Identify angles related by parallel lines.** Since $\ell \parallel m$, $\angle C$ and $\angle CDF$ are supplementary because $\angle C$ is on $\ell$ and $\angle CDF$ is on $m$ with transversal $CD$. 5. **Calculate $\angle CDF$: ** $$\angle CDF = 180^\circ - 130^\circ = 50^\circ$$ 6. **Step 2: Use the angle bisector property.** Since $\overrightarrow{DE}$ bisects $\angle CDF$, it divides $\angle CDF$ into two equal angles: $$\angle CDE = \angle EDF = \frac{1}{2} \times 50^\circ = 25^\circ$$ 7. **Step 3: Find $y$.** $y$ is the angle at $E$ adjacent to $\angle EDF$ on line $\ell$. Since $\ell \parallel m$, $\angle EDF$ and $y$ are alternate interior angles, so: $$y = \angle EDF = 25^\circ$$ **Final answer:** $$y = 25$$