Problem

For the triangle shown, \(A~=~42°\), \(B~=~68°\), and \(a~=~12\). Classify the given data, find \(C\), set up the appropriate law for \(b\) and \(c\), and state the number of triangles.

Prompt AAS triangle with A=42°, B=68°, and opposite side a=12; classification, law, C, b, c, and ambiguity decision are withheld.

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What this problem is really about

Classify only the measurements actually given before deciding on a triangle law. Two known angles fix the third angle and eliminate SSA ambiguity; once a known side is matched with its opposite angle, the Law of Sines can anchor every remaining side. Reserve the Law of Cosines for SAS or SSS data patterns.

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Treat B=68° as an SSA inverse-sine result, replace it by 112°, get C=26°, and claim a second triangle.

Use C=42°+68°=110° and 12/sin(42°)=b/sin(68°)=c/sin(110°), so the listed angles total 220°.

Treat the data as SAS with two invented 12-unit sides: b²=12²+12²-2·12·12cos(70°), giving b≈13.8.

The data are AAS. C=180°-42°-68°=70°, and 12/sin(42°)=b/sin(68°)=c/sin(70°); there is one unambiguous triangle.

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For the triangle shown, \(A~=~42°\), \(B~=~68°\), and \(a~=~12\). Classify the given data, find \(C\), set up the appropriate law for \(b\) and \(c\), and state the number of triangles.

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