1. **State the problem:** We need to find the size of the exterior angle $x$ adjacent to the interior angle of $65^\circ$ in a quadrilateral. 2. **Recall the rule for exterior angles:** The exterior angle and its adjacent interior angle are supplementary, meaning their sum is $180^\circ$. 3. **Write the equation:** $$x + 65^\circ = 180^\circ$$ 4. **Solve for $x$:** $$x = 180^\circ - 65^\circ$$ 5. **Calculate the value:** $$x = 115^\circ$$ 6. **Conclusion:** The exterior angle $x$ is $115^\circ$.