What special cases of similar triangles are there?

Similar triangles are a special case of similar polygons, so one should prove triangles similar in exactly the same way as for polygons in general.

However, the interesting thing is that in the special case of triangles (and this applies only to triangles), it is not necessary to prove both conditions, because it can be proved that the one implies the other.

Two triangles are similar iff:

1. all the corresponding angles are equal, or

2. all the corresponding sides are proportional.