1. **State the problem:** We need to find the size of angle $x$ in the given triangle. 2. **Recall the rule:** The sum of the interior angles in any triangle is always $180^\circ$. 3. **Identify the known angles:** The triangle has angles $77^\circ$, $41^\circ$, and $x$. 4. **Set up the equation:** $$77 + 41 + x = 180$$ 5. **Calculate the sum of known angles:** $$77 + 41 = 118$$ 6. **Solve for $x$:** $$x = 180 - 118$$ $$x = 62$$ 7. **Answer:** The size of angle $x$ is $62^\circ$. Note: The $36^\circ$ angle mentioned is outside the triangle or part of a different figure and does not affect the calculation of $x$ inside the triangle.