the ambiguous case

Usually a few measurements pin down exactly one triangle. But there is one stubborn setup where the same data can describe two different triangles, one triangle, or none at all. This is the ambiguous case, and it is the reason SSA is not a congruence criterion the way SAS and ASA are.

It arises when you are given two sides and an angle opposite one of them (side-side-angle, SSA) and you try to solve with the law of sines. Say you know angle A, its opposite side a, and another side b. The law of sines gives sin B = b x sin A / a, but the equation sin B = (some value between 0 and 1) has TWO solutions between 0 and 180 degrees — an acute angle and its obtuse partner (180 minus it). Whether both, only one, or neither actually forms a valid triangle depends on the lengths. Picture swinging side a like a hinged door from the end of side b: if a is too short it never reaches the base and there is no triangle; if it is just long enough it touches once (one right-triangle solution); a bit longer and it crosses the base in two places (two triangles); longer still than b and it can only cross once (one triangle).

The practical rule is to compute sin B, find the acute angle B_1 from your calculator, then test its obtuse partner B_2 = 180 - B_1: if A + B_2 is still less than 180 degrees, that second triangle is also valid and you must report both answers. Forgetting to check for the second solution is one of the most common mistakes in all of trigonometry, because the calculator's inverse-sine button hands you only the acute angle.