1. **State the problem:** We need to find the height $h$ of a tower given a right triangle where the angle between the ground and the hypotenuse is $45^\circ$, the horizontal distance from the tower is 125 meters, and the observer's height is 1.8 meters. 2. **Formula and rules:** In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ Here, $\theta = 45^\circ$, the opposite side is the height of the tower minus the observer's height $(h - 1.8)$, and the adjacent side is 125 meters. 3. **Set up the equation:** $$\tan(45^\circ) = \frac{h - 1.8}{125}$$ 4. **Evaluate $\tan(45^\circ)$:** $$\tan(45^\circ) = 1$$ 5. **Solve for $h$:** $$1 = \frac{h - 1.8}{125} \implies h - 1.8 = 125 \implies h = 125 + 1.8 = 126.8$$ 6. **Final answer:** The height of the tower is $$h = 126.8 \text{ meters}$$