1. **Problem:** Find the missing side $x$ in a right triangle with a $72^\circ$ angle, adjacent side 6, and hypotenuse $x$. 2. **Formula:** Use the cosine function since cosine relates adjacent side and hypotenuse: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 3. **Apply values:** $$\cos(72^\circ) = \frac{6}{x}$$ 4. **Solve for $x$:** $$x = \frac{6}{\cos(72^\circ)}$$ 5. **Calculate $\cos(72^\circ)$:** $$\cos(72^\circ) \approx 0.3090$$ 6. **Substitute and compute:** $$x = \frac{6}{0.3090} \approx 19.4$$ 7. **Answer:** The missing side $x$ (hypotenuse) is approximately **19.4** units.