The Ambiguous Case (SSA)

Pre-Calculus · Axiom Academy

LESSON The Ambiguous Case (SSA) Why knowing two sides and a non-included angle can give zero, one, or two triangles — and how one swinging side decides. 1. The Setup: One Side Is Free to Swing In SSA you are given angle A , the side b next to it, and the side a opposite A . Pin angle A at the origin and lay b along it — that fixes the far vertex C . The known side a hangs from C and can swing , tracing an arc of radius a . Wherever that arc lands on the base is a possible third vertex B . The altitude — the shortest reach from C to the base 2. Case 1 — When : No Triangle If side a is shorter than the altitude h , the swinging arc never reaches the base. The side simply cannot bridge the gap, so no triangle exists. Watch the arc fall short of the line entirely. 3. Case 2 — When a = h : Exactly One (Right) Triangle When a equals the altitude exactly, the arc just grazes the base at a single point — the foot of the altitude. That one touch gives a single triangle, and because a meets the base perpendicularly, the angle at B is a right angle. 4. Case 3 — When : Two Triangles This is the genuinely ambiguous case. When a is longer than the altitude but still shorter than b , the arc crosses the base twice . Each crossing is a valid vertex B , so the same data describes two different triangles — one acute, one obtuse. 5. Case 4 — When : Exactly One Triangle

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