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© Copyright 2003, Jim Loy
Normally, if you have defined one side of a triangle and two angles, or two sides and one angle, or all three sides, then you have defined a unique triangle (you can then deduce the other unknown elements of the triangle). The exception is the ambiguous angle-side-side (ASS) "theorem," where (given an angle, next to a side, next to a side) there are usually two triangles which fit those conditions. Here you can distort the triangle on the left by moving two of its vertices. The green angle and the two green sides are the known elements which I then duplicate for the triangle on the right, which shows the one or two triangles which are possible with those elements. The only two times when the triangle is unique are (1) if the upper angle (in this diagram) is a right angle, or (2) if the second known side is longer than the first known side. The right angle version is a useful theorem for congruence:
The above Java interactive demonstration was created with Cinderella (a geometry program).
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