For the acute-angle SSA case shown, \(A~=~35°\), \(a~=~12\), and \(b~=~8\). Compute altitude \(h\), compare \(h\), \(a\), and \(b\), then list the valid \(B\) and \(C\) values and the number of triangles.

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Keep only supplementary B≈157.5°, which gives C≈-12.5°, but report 1 triangle.

Reverse the side ratio to get sin(B)=12sin(35°)/8≈0.860; use B≈59.4° or 120.6° and report 2 triangles.

Because 12>4.59, use B≈22.5° and 157.5°; even though the second gives C≈-12.5°, report 2 triangles.

h=8sin(35°)≈4.59. Since a=12≥b=8, B≈22.5° and C≈122.5° give the only valid configuration, so 1 triangle is possible.

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Curriculum context

Course: Math III
Standard: G-SRT.10
Category: Geometry
Domain: Similarity, Right Triangles, and Trigonometry
Objective: Prove the Laws of Sines and Cosines and use them to solve problems.
Problem type: Classify acute SSA with altitude and list all angles