Nathan, Angle-Angle postulate usually refers to similarity of two triangles, not their congruence. There is no way to prove that two triangles are congruent based solely on the fact that the triangles share two equal angles. To prove that triangles are congruent, you must show that at least one pair of corresponding sides are equal (SSS, SAS, ASA, AAS, or Hypotenuse-Leg).
The Angle-Angle postulate (actually you can prove it as a theorem, but the proof is fairly technical and not very useful for teaching or understanding geometry, so it is usually presented as either a postulate or an unproven theorem) states that if two triangles have two corresponding congruent angles, the two triangles are similar, i.e. the triangles three corresponding angles are equal and their three corresponding sides share the same constant of proportionality.
That the third corresponding angle is congruent, follows directly from the fact that the other two angles are equal and the 3 angles of a triangle add up to 180 degrees (in Euclidian geometry).
With the A-A postulate you can then prove SAS-Similarity theorem and SSS-Similarity as theorems. I hope this helps. John