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

Dec 1, 2015

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
$| P M \frac{|}{|} P N | = f$
(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).