ACT® · Math: Geometry — Similarity, Right Triangles, and Trigonometry
The Ambiguous Case of the Law of Sines
When you know two sides and an angle that is not between them (SSA), the Law of Sines can produce two different triangles that both fit the facts. You have to check whether the second, obtuse angle is actually possible.
Give someone an angle, the side across from it, and one more side, and they can sometimes build two different triangles, not one. Picture angle A pinned down with side b laid along one ray. Side a has a fixed length, so swing it like a compass from the far end of b. If a is long enough it only grazes the base line once, but if it's shorter it can cross the base line twice, landing two different spots for point C and creating two different triangles, each with a different angle B.
The Law of Sines hands you sin B = b sin A / a, and inverse sine always returns the acute angle whose sine matches that number. But sine is positive for both an angle and its supplement, so if that acute angle works, 180 minus that angle might work too. You only keep the obtuse version if adding it to angle A still leaves room under 180 degrees for angle C. If it doesn't, only the acute triangle survives.
This is exactly why side-side-angle never makes the list of congruence shortcuts next to side-angle-side or angle-side-angle. SSA data can lock in two genuinely different shapes, so knowing those three measurements never pins down one triangle for certain. Side-angle-side and angle-side-angle work as congruence rules precisely because the angle sits between the two sides, which removes that swinging freedom and leaves only one triangle possible.
Key terms
- Law of Sines
- The relationship a/sin A = b/sin B = c/sin C linking each side of a triangle to the sine of its opposite angle.
- SSA
- Side-side-angle data: two side lengths and an angle not between them, the setup that can produce two triangles.
- Ambiguous case
- The situation where given SSA measurements fit two distinct, non-congruent triangles instead of just one.
- Supplement
- The angle that, added to a given angle, totals 180 degrees.
- Congruence criterion
- A set of matching measurements (like SAS or ASA) guaranteed to force exactly one triangle shape.
Watch it explained
- How it worksHow does swinging a fixed side from one endpoint sometimes cross the base line in two places?
- Worked exampleFind both possible measures of angle B when angle A is 35 degrees, side a is 8 and side b is 12.
- Common mistakeWhy isn't the angle your calculator returns from inverse sine always the angle sitting in the triangle?
- Ties togetherHow does this two-triangle ambiguity explain why side-side-angle fails as a congruence criterion?
Check yourself
End of topic.