1. **Stating the problem:** We need to find the size of angle $x$ in a triangle formed by two parallel lines and a transversal, with given angles $87^\circ$, $45^\circ$, and $36^\circ$. 2. **Identify relationships:** Since the lines are parallel, alternate interior angles and corresponding angles are equal. The $45^\circ$ angle is exterior to the triangle but related to the interior angles. 3. **Use the triangle angle sum rule:** The sum of interior angles in any triangle is $180^\circ$. 4. **Calculate the missing angle:** The triangle has angles $87^\circ$, $x$, and $36^\circ$. So, $$87^\circ + x + 36^\circ = 180^\circ$$ 5. **Simplify the equation:** $$x + 123^\circ = 180^\circ$$ 6. **Solve for $x$:** $$x = 180^\circ - 123^\circ$$ $$x = 57^\circ$$ 7. **Conclusion:** The size of angle $x$ is $57^\circ$.