1. **State the problem:** We need to find the height $h$ of a right triangle where the base is 125 m, the angle adjacent to the base is $45^\circ$, and a man of height 1.8 m stands at the left vertical segment. 2. **Identify the triangle and known values:** The triangle has a base $b = 125$ m and an angle $\theta = 45^\circ$ at the base. The height $h$ is the vertical side opposite the angle $45^\circ$. 3. **Use the tangent function:** In a right triangle, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. 4. **Apply the formula:** Here, $\tan(45^\circ) = \frac{h}{125}$. 5. **Calculate $\tan(45^\circ)$:** $\tan(45^\circ) = 1$. 6. **Set up the equation:** $1 = \frac{h}{125}$. 7. **Solve for $h$:** Multiply both sides by 125: $$ 1 \times 125 = \cancel{125} \times \frac{h}{\cancel{125}} \implies 125 = h $$ 8. **Interpret the result:** The height $h$ of the structure is 125 m. 9. **Note about the man’s height:** The man’s height (1.8 m) is given but does not affect the calculation of $h$ in this problem. **Final answer:** $$h = 125$$