ASA congruence
ASA says: if two angles of one triangle and the side between them are equal to two angles and the included side of another, the triangles are congruent. Picture a side of fixed length with two roads leaving its endpoints at fixed angles; the two roads meet at exactly one point, so the whole triangle is determined. Fix a base and the two base angles, and the apex has nowhere else to go.
Precisely: if in triangles ABC and DEF you have angle B = angle E, |BC| = |EF|, and angle C = angle F (the side BC lies between the two named angles), then triangle ABC is congruent to triangle DEF. The included side is the one whose two endpoints carry the two known angles. Because the angle sum forces the third angle as well, knowing two angles is as good as knowing all three for the purpose of fixing shape; the single known side then fixes the size.
ASA is close cousin to AAS, where the known side is not between the two angles but opposite one of them; both are valid because the third angle is determined by the angle sum, so AAS quietly reduces to ASA. Do not confuse either with the invalid 'AAA': three equal angles fix the shape but not the size, giving only similarity, not congruence.
In triangles ABC and DEF, angle B = angle E = 55 degrees, |BC| = |EF| = 7, and angle C = angle F = 65 degrees. The equal side BC is between the two equal angles, so ASA gives the triangles congruent.
A side with its two adjacent angles fixed determines the triangle.
In ASA the side is between the two angles; in AAS it is opposite one of them. Both are valid because the angle sum determines the third angle. 'AAA' is not valid — it gives only similarity.