1. **State the problem:** Solve for $x$ given the angle measures in the circle where one angle is $83^\circ$ and the other is $(9x + 40)^\circ$. 2. **Identify the relationship:** The problem likely involves the fact that the angle at point $K$ and the arc measure $(9x + 40)^\circ$ are related. Since the angle is inscribed and intercepts the arc, the inscribed angle theorem states: $$\text{Inscribed angle} = \frac{1}{2} \times \text{measure of intercepted arc}$$ 3. **Set up the equation:** Using the inscribed angle theorem, $$83 = \frac{1}{2} (9x + 40)$$ 4. **Solve for $x$:** Multiply both sides by 2 to eliminate the fraction: $$2 \times 83 = 9x + 40$$ $$166 = 9x + 40$$ Subtract 40 from both sides: $$166 - 40 = 9x$$ $$126 = 9x$$ Divide both sides by 9: $$x = \frac{126}{9}$$ Show cancellation: $$x = \frac{\cancel{126}}{\cancel{9}} = 14$$ 5. **Final answer:** $$\boxed{14}$$ The correct choice is D. 14.