1. **State the problem:** Given two intersecting lines with angles labeled 1, 2, 3, and 4 around the intersection, and $m\angle 1 = 135^\circ$, find the measures of angles 2, 3, and 4. 2. **Recall the rule for vertical angles:** Vertical angles are opposite angles formed by two intersecting lines and are always equal. 3. **Identify vertical angles:** - $\angle 1$ and $\angle 3$ are vertical angles, so $m\angle 3 = m\angle 1 = 135^\circ$. - $\angle 2$ and $\angle 4$ are vertical angles, so $m\angle 2 = m\angle 4$. 4. **Use the fact that adjacent angles on a straight line sum to $180^\circ$:** - $m\angle 1 + m\angle 2 = 180^\circ$ - Substitute $m\angle 1 = 135^\circ$: $$135^\circ + m\angle 2 = 180^\circ$$ 5. **Solve for $m\angle 2$:** $$m\angle 2 = 180^\circ - 135^\circ = 45^\circ$$ 6. **Find $m\angle 4$ using vertical angles:** $$m\angle 4 = m\angle 2 = 45^\circ$$ **Final answers:** - $m\angle 2 = 45^\circ$ - $m\angle 3 = 135^\circ$ - $m\angle 4 = 45^\circ$