Definition: A triangle the shape formed from the intersection of three noncolinear geodesics at three points.
Side-Angle-Side (SAS): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
Three different definitions of “small triangle.”
Def 1: A triangle on the sphere is small if all angles are less than 180 degrees
Def 2: A triangle on the sphere is small if all sides are less than ½ of a great circle
Def 3: A triangle on the sphere is small if it is contained in a hemisphere
- Given two small triangles (small per definition 1) that have side-angle-side congruent, prove that the two triangles are congruent.
- Given two small triangles (small per definition 2) that have side-angle-side congruent, prove that the two triangles are congruent.
- Given two small triangles (small per definition 3) that have side-angle-side congruent, prove that the two triangles are congruent.
- Are all three definitions of small triangles the “same”? For example, is Def 1 the same as Def 2? The answer is yes if every triangle that meets the criteria for Def 1 also meets the criteria for Def 2 and conversely, every triangle that meets the criteria for Def 2 also meets the criteria for Def 1. If both of these things are true, then we say the two definitions are the same (or in more mathy terms, they are equivalent).
- Is Def 1 the same as Def 2? Prove or disprove.
- Is Def 1 the same as Def 3? Prove or disprove.
- Is Def 2 the same as Def 3? Prove or disprove.
5. Which definition of the small triangle do you prefer and why? [full credit for a conscientious answer]
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