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

Question

Sophia Alfred · Accepted Answer

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). 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).