1. **Problem:** Find the measures of angles $\angle ABC$, $\angle EBD$, and $\angle ABE$ given that $\angle ABC = 50^\circ$. 2. **Given:** $\angle ABC = 50^\circ$. 3. **Step 1:** Since $\angle ABC$ is given as $50^\circ$, we have: $$m\angle ABC = 50^\circ$$ 4. **Step 2:** Angles $\angle EBD$ and $\angle ABE$ are adjacent to $\angle ABC$ and form a straight line at point B. The sum of angles on a straight line is $180^\circ$. 5. **Step 3:** Therefore, $$m\angle ABC + m\angle EBD = 180^\circ$$ $$50^\circ + m\angle EBD = 180^\circ$$ 6. **Step 4:** Solve for $m\angle EBD$: $$m\angle EBD = 180^\circ - 50^\circ = 130^\circ$$ 7. **Step 5:** Similarly, $\angle ABE$ and $\angle ABC$ are vertical angles (opposite angles formed by two intersecting lines), so they are equal: $$m\angle ABE = m\angle ABC = 50^\circ$$ **Final answers:** $$m\angle ABC = 50^\circ$$ $$m\angle EBD = 130^\circ$$ $$m\angle ABE = 50^\circ$$