I was given a simple geometry problem. You need to find the angle ADB. Line DB touches the circle. The angle can be easily found by making use of properties of circle and interior angles taught in schools. I solved it in a different way:
I made the following argument: since we need to find the angle ADB, that means that it is invariant relative to change in the position of the point B on the circle (the upper drawing in the photo I attached has two degrees of freedom - coordinate of X and B, for example). Hence, the problem is equivalent to the situation where we put the point B in the same position with the point X (drawn below in the picture), reducing it to having only one degree of freedom, and the angle is found trivially, without using properties of circle. And is equal to 70 degrees, which is right. Is it a correct solution?
Several questions arise under your proposed method. As you move $B$, what else changes? The location of $D$? The location of $A$? The location of $X$?
You have imagined what might happen if we moved things so $B$ and $X$ are in the same place. Where now is the angle $\angle AXB$? How does this relate to $\angle ABD$?
Fundamentally, the claim of equivalence of your two figures is just a claim of the Alternate Segment Theorem (aka Tangent-Chord Theorem). I do not think your "proof" of it is valid, however. I think you need something more if you want a valid proof of the theorem.
In short, the fact that $\angle AXB = \angle ABD$ is a non-trivial property of circles.
Note that even before you move $B$ onto $X,$ the notion that $\angle ABD$ stays the same as you move stuff around is an unproven assertion. The fact that the problem is stated the way it is is a clue that this assertion might be true, but that's a matter of guessing the intention of the person posing the question, not mathematics. It could be that this exercise is from a book in which the answers to several exercises are "cannot be determined."