Answer:
When one illustrates a function and its inverse graphed on the same axes, the symmetry about the line y=x is proved by congruence.
In geometry, proving various sides or angles of triangles identical (in measure) often owes to showing certain triangles are congruent.
In defining the slope of a line, one picks two points and calculates (change in y) divided by (change in x). How do we know that the answer does not depend on which two points you pick? Answer: Draw the picture and obtain similar triangles.
Somebody could probably write a book on this, and perhaps it has been done. (Part of Euclid’s Elements, perhaps?)
