Two Triangles
Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 11 of 44.
Sometimes both angles fit. Take A = 30°, b = AC = 10 and a = BC = 6. Swing the side of length 6 from C: it reaches the base line at two points, B₁ and B₂, and both make a triangle.
The law of sines gives sin(B) = 5/6, so B is about 56.44° or about 123.56°. Each leaves room: 30° + 123.56° is still under 180°. The two triangles share A, b and a but have different third sides and different angles at C.
This is why SSA is called the ambiguous case. Do not throw the obtuse angle away out of habit: test it against the given angle. On the board both cases stay when both fit.
Challenge
In triangle ABC, A = 35°, a = 9 cm and b = 14 cm. Find every possible angle θ at B, to two decimal places, then press Check.
Try it yourself
Use the figure to find theta
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Triangles Without Right Angles
2Law of Sines: AAS and ASA
Why the Law of Sines HoldsPair Side With AngleThe Third Angle FirstTwo Coast StationsRecap: Law of Sines8Bearings and Navigation
Reading BearingsTwo Ships From PortTurning Off CourseFinding a Course AngleRecap: Ships at Sea3The Ambiguous Case
Finding an AngleTwo TrianglesThe Height TestAn Obtuse Given AngleRadio CoverageRecap: How Many Triangles?