1. **State the problem:** We are given a rectangle with angles labeled as $(9x - 6)^\circ$ and $(2x + 8)^\circ$ at certain vertices, and we need to solve for $x$. 2. **Recall properties of a rectangle:** All interior angles of a rectangle are right angles, i.e., $90^\circ$. 3. **Set up the equations:** Since the angles given are interior angles of the rectangle, each must equal $90^\circ$. \[9x - 6 = 90\] \[2x + 8 = 90\] 4. **Solve the first equation:** $$9x - 6 = 90$$ $$9x = 90 + 6$$ $$9x = 96$$ $$x = \frac{96}{9}$$ $$x = \frac{\cancel{96}}{\cancel{9}} = \frac{32}{3} \approx 10.67$$ 5. **Solve the second equation:** $$2x + 8 = 90$$ $$2x = 90 - 8$$ $$2x = 82$$ $$x = \frac{82}{2}$$ $$x = \frac{\cancel{82}}{\cancel{2}} = 41$$ 6. **Check for consistency:** The two values for $x$ are different, which is not possible if both angles are interior angles of the rectangle. Since the problem states a rectangle, the angles must be $90^\circ$. The only way for both expressions to represent the same angle is if they are equal. 7. **Set the two angle expressions equal:** $$9x - 6 = 2x + 8$$ $$9x - 2x = 8 + 6$$ $$7x = 14$$ $$x = \frac{14}{7}$$ $$x = 2$$ 8. **Verify the solution:** Calculate each angle with $x=2$: $$9(2) - 6 = 18 - 6 = 12^\circ$$ $$2(2) + 8 = 4 + 8 = 12^\circ$$ Since both angles are $12^\circ$, and the rectangle's interior angles are $90^\circ$, these angles must be parts of the rectangle's angles, possibly adjacent angles formed by the diagonal. 9. **Conclusion:** The value of $x$ that satisfies the given angle expressions in the rectangle is $\boxed{2}$.