1. **State the problem:** We need to find the size of angle $x$ in a quadrilateral where the given angles are $105^\circ$, $72^\circ$, $218^\circ$, and $51^\circ$. The $218^\circ$ angle is likely an exterior angle. 2. **Recall the rule:** The sum of interior angles of any quadrilateral is $360^\circ$. 3. **Identify interior angles:** Given $218^\circ$ is an exterior angle, its adjacent interior angle is $180^\circ - 218^\circ = -38^\circ$, which is not possible. Instead, the $218^\circ$ angle likely represents the reflex angle around a vertex, so the interior angle at that vertex is $360^\circ - 218^\circ = 142^\circ$. 4. **Sum interior angles:** The four interior angles are $105^\circ$, $72^\circ$, $142^\circ$, and $x$. 5. **Set up the equation:** $$105 + 72 + 142 + x = 360$$ 6. **Calculate:** $$319 + x = 360$$ $$x = 360 - 319 = 41$$ 7. **Conclusion:** The size of angle $x$ is $41^\circ$.