1. **State the problem:** We are given two parallel lines connected by two slanted segments forming a triangle-like shape with angles 100°, $x^\circ$, and 45°. We need to find the value of $x$. 2. **Identify the relationships:** Since the two horizontal lines are parallel, the exterior angle of 100° and the interior angle adjacent to it form a linear pair, summing to 180°. 3. **Calculate the adjacent interior angle:** $$180^\circ - 100^\circ = 80^\circ$$ 4. **Use the triangle angle sum rule:** The sum of angles inside a triangle is always 180°. 5. **Set up the equation:** $$x + 45^\circ + 80^\circ = 180^\circ$$ 6. **Simplify the equation:** $$x + 125^\circ = 180^\circ$$ 7. **Solve for $x$:** $$x = 180^\circ - 125^\circ$$ $$x = 55^\circ$$ **Final answer:** $$\boxed{55^\circ}$$