Problem with 2 similar right-angled triangles

Question

There are two similar right-angled triangles, where known information is only one side from one, another side from the other triangle, they are not hypotenuse and right angles.

Is it possible to find all the angles and sides of these triangles?

Answer 1

Here's an example . . .

Let triangle $1$ have sides $12,s,t$, where $12,s$ are the leg lengths, and $t$ is the length of the hypotenuse.

Let triangle $2$ be similar to triangle $1$, with corresponding sides $u,80,w$.

To see that the triangles are not determined by the data, note that you can have \begin{align*} (12,s,t)&=(12,16,20)\\ (u,80,w)&=(60,80,100)\\ \end{align*} but as just one additional example, you could also have \begin{align*} (12,s,t)&=(12,5,13)\\ (u,80,w)&=(192,80,208)\\ \end{align*}