How many different triangles can be formed from three line segments if two of the segments have the same length?

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Side-side-side (SSS) is one of those congruence theorems in geometry class that proves if two triangles have the same side lengths, they are congruent triangles. Therefore, there can only be one unique triangle formed by a set of three line segments. In addition, using Law of Cosines to solve for any angle yields only one possible solution when given SSS which provides more evidence that only one unique triangle is possible. 

There are situations where more than one unique triangle is possible given three pieces of information about the triangle. SSA is a situation that can result in more than one unique triangle. 

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Given any three line segments, either you can form a triangle (exactly one) or you cannot form a triangle.

So if it is possible to form the triangle, there can be only one. Regardless of the lengths.

(If you are just given the two segments of equal length, say a, then there are an infinite number of triangles that can be formed. The third side can be any 0<b<2a )

(( If you are asked how many different ways can you name the triangle, you would need to know how the segments are labelled. The triangles will be congruent -- the names will be permutations of the names on the vertices (i.e. ABC,ACB,BAC,BCA,CAB,CBA) ))

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