The angles of similar triangles are equal always, sometimes, or never?
Angles of similar triangles are ALWAYS equal
We have to start from a definition of similarity.
There are different approaches to this. The most logical one I consider to be the definition based on a concept of scaling.
Scaling is a transformation of all points on a plane based on a choice of a scaling center (a fixed point) and a scaling factor (a real number not equal to zero).
Then the definition of similarity is:
"two objects are called 'similar' if there exists such a center of scaling and scaling factor that transform one object into an object congruent to another."
Next, we have to prove that a straight line is transformed into a straight line parallel to an original.
That causes angles to be transformed into equal angles, which is a subject of this question.