There is a unique triangle $ABC$ with $\angle A = (180 - \alpha)$ and $\angle B=\beta$ and $C$ above the line. The problem is to construct or calculate $X$ from $A$, $B$, $\alpha$, $\beta$. For any particular way of giving the angles $\alpha$ and $\beta$ the solution will be relatively easy and does not require trigonometric methods. For physical drawing it is easier to work with a smaller-scale model of the problem, with smaller value of the distance $AB$ but same angles, and scale up to get the answer.
For a Euclidean construction, $\alpha$ and $\beta$ can be given using a protractor (and the problem is reduced to intersecting two known lines), or as angles somewhere in the plane that can be copied to $AB$ using ruler and compass (and the problem is then the same as the protractor case).
If $\tan \alpha$ and $\tan \beta$ are given or measured, there is a simple algebraic solution. If the distances to the base of the tower are $a$ and $b$, and height of the tower is $h$, then we are told the ratios $\frac{a}{h}$, $\frac{b}{h}$, and $(b-a)$ and want to know $h$. Subtracting the ratios provides $(b-a)/h$ and hence $h$. This does not require frightening words like tangent to be used, slopes of lines are enough.