1. **Problem:** \nGiven that $\angle 1$ and $\angle 2$ are vertical angles, and $m\angle 1 = (5x + 12)^\circ$, $m\angle 2 = (6x - 11)^\circ$, find $m\angle 1$.\n\n2. **Rule:** Vertical angles are congruent, so $m\angle 1 = m\angle 2$.\n\n3. **Set up the equation:**\n$$5x + 12 = 6x - 11$$\n\n4. **Solve for $x$:**\n$$5x + 12 = 6x - 11$$\n$$5x + 12 - 6x = 6x - 11 - 6x$$\n$$\cancel{5x} + 12 - \cancel{6x} = -11$$\n$$-x + 12 = -11$$\n$$-x = -11 - 12$$\n$$-x = -23$$\n$$x = 23$$\n\n5. **Find $m\angle 1$:**\n$$m\angle 1 = 5x + 12 = 5(23) + 12 = 115 + 12 = 127^\circ$$\n\n**Final answer:** \n$$m\angle 1 = 127^\circ$$