Constructing triangles

You can construct a triangle from three given side lengths or angle measures using a ruler and protractor, but not every combination works: side lengths of 3, 4, and 5 make exactly one triangle; knowing only three angles, like 60, 60, and 60 degrees, fixes the triangle's shape but not its size, so more than one triangle (infinitely many, in fact) satisfies it; and side lengths of 2, 3, and 10 make no triangle at all, since 2 + 3 is less than 10.

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