1. **Problem statement:** We are given a geometric figure with two parallel horizontal lines and three angles: 100° between the vertical line and the top horizontal line, 45° between the angled line and the bottom horizontal line, and an unknown angle $x$ between the vertical line and the angled line. We need to find $x$. 2. **Key fact:** The two horizontal lines are parallel. The vertical and angled lines intersect these parallel lines, forming alternate interior angles and supplementary angles. 3. **Step 1:** Note that the angle between the vertical line and the top horizontal line is 100°. Since the top and bottom lines are parallel, the angle between the vertical line and the bottom horizontal line is also 100° (alternate interior angles). 4. **Step 2:** The angle between the angled line and the bottom horizontal line is 45°. 5. **Step 3:** The angle $x$ is between the vertical line and the angled line. At the bottom horizontal line, the vertical and angled lines form a triangle with angles 100°, 45°, and $x$. 6. **Step 4:** The sum of angles in a triangle is 180°, so: $$x + 100 + 45 = 180$$ 7. **Step 5:** Simplify to find $x$: $$x = 180 - 100 - 45 = 35$$ **Final answer:** $$x = 35^\circ$$