1. **State the problem:** We have two intersecting lines forming vertical angles. One angle is labeled $6x^\circ$, and the adjacent angle is labeled $(x - 2)^\circ$ which is also $\angle 2$. We need to find the values of $x$, $\angle 1$, and $\angle 2$ algebraically. 2. **Important rule:** Vertical angles are equal. Also, adjacent angles on a straight line sum to $180^\circ$. 3. Since $6x^\circ$ and $\angle 1$ are vertical angles, $\angle 1 = 6x^\circ$. 4. $\angle 2$ is adjacent to $6x^\circ$, so their sum is $180^\circ$: $$6x + (x - 2) = 180$$ 5. Simplify the equation: $$6x + x - 2 = 180$$ $$7x - 2 = 180$$ 6. Add 2 to both sides: $$7x - \cancel{2} + \cancel{2} = 180 + 2$$ $$7x = 182$$ 7. Divide both sides by 7: $$\frac{7x}{\cancel{7}} = \frac{182}{\cancel{7}}$$ $$x = 26$$ 8. Find $\angle 1$: $$\angle 1 = 6x = 6 \times 26 = 156^\circ$$ 9. Find $\angle 2$: $$\angle 2 = x - 2 = 26 - 2 = 24^\circ$$ **Final answers:** $$x = 26$$ $$\angle 1 = 156^\circ$$ $$\angle 2 = 24^\circ$$