Answer:
- None: B, D
- One: A, C, E
- Many: F
Explanation:
There are several relations that come into play when determining whether a given set of dimensions will form a triangle:
- the angle sum theorem: angles total exactly 180°
- the triangle inequality: a+b>c where a, b are the short side lengths
- the law of sines: a/sin(A) = b/sin(B) = c/sin(C)
For the triangle to be unique, at least one side dimension must be given.
For this question, we only need to consider the first two of these.
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A: the triangle inequality is satisfied. 1 triangle
B: the angles don't add to 180°. 0 triangles
C: a triangle can always be formed when two sides and the enclosed angle are given. 1 triangle
D: the triangle inequality is not satisfied. 0 triangles
E: a triangle can always be formed when two angles and the enclosed side are given. 1 triangle
F: the triangles total 180°, but no sides are given. Many triangles
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Additional comment
It is possible that two triangles can be formed if two sides and an angle are given, such that the angle is opposite the shorter of the given sides. This is where the Law of Sines comes into play.