Why AAS feels unique and SSA does not
If you know two angles, the third angle is fixed because the interior angles add to 180°. One side then scales the triangle, and there is only one shape of that size. SSA does not hand you two angles. It hands you one angle and two sides. The second angle is recovered from the sine ratio, and that ratio is not one-to-one on (0°, 180°). The equation sin(B) = k can have two candidates: an acute angle B₁ and its supplement 180° − B₁.
Both candidates are mathematically legal until you form the third angle. If 180° − A − B₂ is still positive, a second triangle exists. That second triangle is the ambiguous case. If the third angle would be zero or negative, the supplement is discarded and only one triangle remains.
The height test
Draw angle A at one end of side b. The opposite side a must reach the other ray. The shortest path to that ray is the perpendicular of length h = b · sin(A). Compare a with h and with b.
When A is acute and the opposite side is shorter than h, that side cannot reach the ray. No triangle exists. Equality with the height means the side meets the ray at a right angle, and there is one right triangle. A length between h and b allows both the acute and the obtuse placements of B, so two triangles exist. If the opposite side is at least as long as b, the obtuse placement of B would force the original angle A to sit opposite a side that is no longer the longer of the two, and only one triangle survives.
If A is a right angle or an obtuse angle, the picture is stricter. Side a must be longer than side b, or the pieces cannot close. There is never a second triangle in that obtuse setting on this page.
A worked pair of numbers
Take A = 40°, opposite side 6, and adjacent side 8. The height is h = 8 · sin(40°), about 5.14. The opposite side is longer than that height and shorter than the adjacent side, so two triangles fit. The law of sines gives sin(B) = 8 · sin(40°) / 6, about 0.857. One solution is B about 59.0°. The supplement is B about 121.0°. The third angles are then about 81.0° and 19.0°. Both are positive, so both triangles are kept. The calculator prints both and labels the situation as the ambiguous case.
Change the opposite side to 5. That value is under the height 5.14, so the same angle and the same adjacent side produce no triangle. The calculator reports that failure instead of inventing an angle.
What this guide does not do
The guide does not treat spherical triangles, and it does not discuss SSA on a coordinate plane with directed sides. It also does not replace the law of cosines. If you know two sides and the included angle, that is SAS, and you should start on the law of cosines calculator.
When you want to see the arithmetic with your own numbers, enter SSA on the law of sines page. The steps there use the same height test described in these paragraphs.