1. **State the problem:** We have a triangle with angles $x^\circ$, $20^\circ$, and $30^\circ$. We need to find the value of $x$. 2. **Recall the triangle angle sum property:** The sum of the interior angles of any triangle is always $180^\circ$. 3. **Write the equation:** Using the property, we have $$x + 20 + 30 = 180$$ 4. **Simplify the equation:** $$x + 50 = 180$$ 5. **Isolate $x$:** $$x = 180 - 50$$ 6. **Calculate the value:** $$x = 130$$ 7. **Check the options:** The options given are $20^\circ$, $30^\circ$, $50^\circ$, and $60^\circ$. None of these match $130^\circ$. 8. **Re-examine the problem:** Since the angle $x$ is opposite the $30^\circ$ angle and adjacent to the $20^\circ$ angle, it suggests the triangle is not a simple triangle but possibly part of a larger figure or the angles are not all interior angles of the same triangle. 9. **Assuming the triangle is formed by the angles $x$, $20^\circ$, and $30^\circ$ as interior angles, the sum must be $180^\circ$.** 10. **Therefore, the value of $x$ is:** $$x = 180 - (20 + 30) = 180 - 50 = 130$$ 11. **Since $130^\circ$ is not an option, the problem might be asking for the value of $x$ in a different context, such as the exterior angle or a different triangle.** 12. **If $x$ is the exterior angle adjacent to $20^\circ$ and $30^\circ$, then:** $$x = 20 + 30 = 50$$ 13. **This matches one of the options.** **Final answer:** $\boxed{50^\circ}$