Why aren't triangle congruency rules ASA and AAS just combined? AAS says "Any two angles and a non-included side" while
ASA says "Any two angles and the included side".
That's just "any two angles and a side"?  Why do we have both the AAS and ASA rules for triangle congruency?
 A: AAS, unlike ASA, is not unoversal when you consider  non-Euclidean geometries.
Imagine the Earth is a sphere.  Draw the meridian from the North Pole $N$ through London $L$ to the South Pole $S$.  Construct the great circle perpendicular to $NLS$ through $L$, drawing this route westwards until it meets the Equator at pount $M$ west of South America.  Then you construct median $NMS$.
You now have two triangles $LMN$ and $LMS$ that satisfy $AAS$, as each triangle has two right angles and a common side not between the right angles.  But the third angles are different because London is not on the Equator, so the triangles are not congruent.
The Greeks used geometry and trigonometry to model the  Earth and sky as well as an idealized plane, so the failure of $AAS$ on spheres was a big deal. 
A: That's to help you remember that SAS and SSA are different.
A: Surely, because of “the angle sum of triangle”, they can be combined into just only one (ASA, say). But then some one probably will raise the following question:–
From the AAS situation – a not-yet-congruent state, why do I have to write one more step (saying the equality of the third A via “the angle sum of triangle”)  to claim the congruency by the only ASA, when the situation is so obvious?
