The Ambiguous Case (SSA)
SSA is called “ambiguous” because the given information can produce no triangle, exactly one, or two valid triangles. Learning to test for each outcome is the key skill.
Introduction
When you're given two sides and an angle opposite one of them (SSA), the swing of the unknown side can form zero, one, or two triangles. You must test each possibility.
Past Knowledge
Law of Sines (7.2), inverse sine range, angle-sum property.
Today's Goal
Determine how many triangles exist in SSA and solve each one.
Future Success
Mastering ambiguity here prevents errors in applied problems throughout the rest of this chapter.
Key Concepts
The 3 Possible Outcomes
Given sides and angle (opposite side ), compute :
| If… | # Triangles |
|---|---|
| 0 (no triangle) | |
| 1 (right triangle) | |
| 1 or 2 — test both |
Testing for Two Triangles
When , there are two candidate angles:
- (acute)
- (obtuse)
Check: if , both triangles are valid. Otherwise only works.
The Supplement Test
Always check . If , reject and you have one triangle. If it fits, solve both triangles completely.
Worked Examples
No Triangle
Given: .
Since , no triangle exists.
Answer: No solution.
One Triangle
Given: .
,
Check: → reject
Answer: One triangle with
Two Triangles
Given: .
,
Check: ✓ → both work!
Triangle 1: → solve for
Triangle 2: → solve for
Answer: Two valid triangles.
Common Pitfalls
Forgetting to Check the Supplement
When , you must test . Skipping this step misses valid triangles (or wrongly includes invalid ones).
Real-Life Applications
Air Traffic Control
Radar returns give a distance and bearing to an aircraft — sometimes matching two possible flight paths. Controllers must resolve this ambiguity (just like SSA) using additional data points or radar sweeps.
Practice Quiz
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