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7.G.A.2 says:
… Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
I am parsing this to mean that students should be able to identify whether there is a unique, multiple, or no solutions for the given constraints.
The uniqueness question can be tricky. Are we considering reflections of triangles in the plane to be different triangles?
Is the black half of this triangle the “same” as the white half in the following unicode character ◭ ?Example: If I specify three side lengths say 3,4,5, then you can build a left-hand and a right-hand version of this triangle in the plane.
Should these be considered as multiple solutions or a unique solution?Workaround: We could interpret the standard as saying “unique up to symmetries” and interpret the multiple solutions as infinitely-multiple solutions, e.g. when we do not specify the third side-length or the third angle, as in example above.
After looking at the high school geometry standards, I understand that the notion of uniqueness of a triangle is up to congruence relations. Thus the triangle with side lengths 3,4,5 and the triangle with side lengths 3,5,4 are the “same” since one can be obtained from the other by a reflection.
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