1. **State the problem:** We are given two intersecting lines forming four angles labeled 1, 2, 3, and 4. We know that $m\angle 2 = 126^\circ$. We need to find $m\angle 1$, $m\angle 3$, and $m\angle 4$. 2. **Recall the properties of vertical angles:** Vertical angles are opposite angles formed by two intersecting lines. They are always equal. 3. **Identify vertical angle pairs:** - $\angle 1$ and $\angle 3$ are vertical angles opposite each other. - $\angle 2$ and $\angle 4$ are vertical angles opposite each other. 4. **Use the given measure:** Since $m\angle 2 = 126^\circ$, by vertical angle theorem, $$m\angle 4 = m\angle 2 = 126^\circ$$ 5. **Find adjacent angles:** Adjacent angles formed by intersecting lines are supplementary, meaning their measures add up to $180^\circ$. - $\angle 1$ and $\angle 2$ are adjacent, so $$m\angle 1 + m\angle 2 = 180^\circ$$ 6. **Calculate $m\angle 1$:** $$m\angle 1 = 180^\circ - m\angle 2 = 180^\circ - 126^\circ = 54^\circ$$ 7. **Find $m\angle 3$:** Since $\angle 1$ and $\angle 3$ are vertical angles, $$m\angle 3 = m\angle 1 = 54^\circ$$ **Final answers:** - $m\angle 1 = 54^\circ$ - $m\angle 3 = 54^\circ$ - $m\angle 4 = 126^\circ$