(1) $\triangle ABC$ and $\triangle DBA$ have side $AB$ in common and also $\angle ABC$. What is the other side?
The fact that these two triangles have a side in common is a red herring! The line segment $AB$ just happens to be a side of both triangles, but this is irrelevant to the fact that the triangles are similar.
If you look at $\triangle ABC$ and $\triangle DBA$, you will see that their shorter legs are $AB$ and $DB$ (respectively), their longer legs are $AC$ and $DA$ (respectively), and their hypotenuses are $BC$ and $BA$ (respectively). If you look at each of these pairs of line segments, and calculate the ratio of each pair, you'll find that the three ratios are all equal.
The corresponding angles are $\angle ABC$ with $\angle DBA$ (the wider acute angles); $\angle BCA$ with $\angle BAD$ (the narrower acute angles); and $\angle CAB$ with $\angle ADB$ (the right angles). For each of these three pairs of angles, both of the angles in the pair are congruent.
Does this help explain things?