Solving the Ambiguous Case

Pre-Calculus · Axiom Academy

EXAMPLE Solving the Ambiguous Case Working a two-triangle SSA problem with the Law of Sines We are given angle A = 35° with its opposite side a = 8 , and a second side b = 10 . This is the side-side-angle (SSA) configuration, the "ambiguous case" — it may give zero, one, or two triangles. Find every triangle that fits, solving each one completely. Side b = 10 (blue) leaves A at the 35° angle and ends at C . From C , the swing side a = 8 (gold) is longer than the altitude but shorter than b , so it reaches the base in two places — giving an acute triangle (green, B_1 ) and an obtuse one (pink, B_2 ). Nice work — you worked a full ambiguous (SSA) case and found that it produces two distinct triangles. Count first: compare side a with the altitude . Here , the signature of the two-triangle case. The supplement is the second triangle: has two angle solutions, B_1 and B_2 = 180° - B_1 . Keep B_2 only if . Both checked out: Triangle 1 is ; Triangle 2 is . Each set of angles sums to 180° . Whenever you are handed SSA, always test for the supplement — a second valid triangle is easy to miss but just as real as the first.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.