Question

The angles of similar triangles are equal always, sometimes, or never?

Answer 1

Angles of similar triangles are ALWAYS equal

Explanation:

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).

If point `P` is a center of scaling and `f` is a scaling factor, any point `M` on a plane is transformed into a point `N` in such a way that points `P`, `M` and `N` lie on the same line and
`|PM|/|PN| = f`
(positive `f` causes points `M` and `N` to be on the same side of point `P`, negative `f` corresponds to point `N` lying on the opposite side of point `M` from a center point `P`).

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.

These proofs are presented in the course of advanced mathematics for teenagers at Unizor (follow menu items Geometry - Similarity).