The ambiguous SSA case

The ambiguous SSA case: zero, one or two triangles

Learn to determine how many triangles are possible when two sides and a nonincluded angle are given. This precalculus set treats the acute, right and obtuse given-angle cases separately, connecting the sine-law calculation with altitude thresholds and the triangle angle sum.

Single-select and multiselect tasks progress from identifying SSA information to evaluating impossible data, the tangent case, two valid branches, exact side lengths and supplementary angles that must be rejected. Detailed explanations make clear that inverse sine returns one principal value while a triangle may also admit its supplement.

Reliable solution procedure

Check positive lengths and the angle range, compute the candidate sine, then examine every possible angle in the triangle interval. Keep only branches with a positive third angle, and calculate remaining sides separately for each accepted triangle. Use full precision when comparing a side with the altitude; premature rounding can change the apparent number of triangles. Calculator tasks specify degree mode and the required rounding. This set extends the nonambiguous ASA/AAS sine-law practice.