AA similarity
AA similarity is the easiest way to prove two triangles have the same shape: just show that two angles of one triangle equal two angles of the other. That is all. You do not need to measure a single side. Because two matching angles is such a light requirement, AA is the workhorse criterion used in almost every similar-triangle problem.
Why two angles is enough: the three angles of any triangle add to 180 degrees, so if two angles of triangle ABC equal two angles of triangle DEF, the third angles must be equal too (each is 180 minus the same sum). All three angles match, and a theorem then guarantees the sides are automatically in proportion. In practice you hunt for equal angles — shared angles, vertical angles, or the equal angles that a parallel line creates with a transversal — and once you find two pairs, the triangles are similar and you may write a proportion among the sides.
Here is the honest subtlety that trips many students. For congruence, knowing all three angles match (AAA) is NOT enough — equal angles fix the shape but not the size, so two triangles can be equiangular yet wildly different in scale. That very failure is exactly what makes AA a similarity criterion rather than a congruence criterion: it pins down the shape and lets the size float free. So AA proves similar, never congruent.
In a right triangle, drop the altitude to the hypotenuse. Each smaller triangle shares one acute angle with the big triangle and has its own right angle, so by AA all three triangles are similar — the key fact behind the geometric-mean relations.
Two shared/equal angles are enough; AA quietly powers the altitude-on-hypotenuse picture.
AAA and AA say the same thing for triangles (the third angle comes free), so both prove similarity — but neither proves congruence. Equal angles never determine size; that is the whole point of similarity.