1. **State the problem:** We are given two vertical angles, $\angle 1$ and $\angle 2$, with measures $m\angle 1 = 5x + 12$ and $m\angle 2 = 6x - 11$. We need to find the measure of $\angle 1$. 2. **Recall the property of vertical angles:** Vertical angles are congruent, meaning their measures are equal. So, $m\angle 1 = m\angle 2$. 3. **Set up the equation:** $$5x + 12 = 6x - 11$$ 4. **Solve for $x$:** Subtract $5x$ from both sides: $$\cancel{5x} + 12 = \cancel{5x} + 6x - 11 \implies 12 = x - 11$$ Add 11 to both sides: $$12 + 11 = x - 11 + 11 \implies 23 = x$$ 5. **Find $m\angle 1$:** Substitute $x = 23$ into $5x + 12$: $$5(23) + 12 = 115 + 12 = 127$$ **Final answer:** $m\angle 1 = 127$ degrees.