1. **State the problem:** We are given two intersecting lines forming an X shape with four angles labeled 1, 2, 3, and 4. 2. **Given:** $m\angle 3 = 69^\circ$. 3. **Goal:** Find $m\angle 1$, $m\angle 2$, and $m\angle 4$. 4. **Key fact:** Vertical angles are equal. When two lines intersect, opposite angles (vertical angles) are congruent. 5. Since $\angle 3$ and $\angle 1$ are vertical angles, we have: $$m\angle 1 = m\angle 3 = 69^\circ$$ 6. Angles on a straight line sum to $180^\circ$. Since $\angle 3$ and $\angle 2$ are adjacent and form a straight line: $$m\angle 2 + m\angle 3 = 180^\circ$$ 7. Substitute $m\angle 3 = 69^\circ$: $$m\angle 2 + 69^\circ = 180^\circ$$ 8. Solve for $m\angle 2$: $$m\angle 2 = 180^\circ - 69^\circ = 111^\circ$$ 9. Similarly, $\angle 4$ and $\angle 3$ are adjacent and form a straight line: $$m\angle 4 + m\angle 3 = 180^\circ$$ 10. Substitute $m\angle 3 = 69^\circ$: $$m\angle 4 + 69^\circ = 180^\circ$$ 11. Solve for $m\angle 4$: $$m\angle 4 = 180^\circ - 69^\circ = 111^\circ$$ **Final answers:** $$m\angle 1 = 69^\circ$$ $$m\angle 2 = 111^\circ$$ $$m\angle 4 = 111^\circ$$