Cauchy's Arm Lemma is used in the proof of Cauchy's Rigidity Theorem for convex polyhedra. The Lemma states that in the plane or on the sphere that if all but one of the side lengths of two convex polygons $P$ and $P'$ are the same, and the angles formed by the remaining sides of P are less than or equal to those of the remaining sides of $P'$, then the ommitted side length from $P$ is less than or equal to the omitted side length for $P'$, with equality occurring iff the angles are all the same. I know of a couple of extensions of this theorem (a nice presentation of this sort of thing is available in O'Rourke's paper [O'R01]). The Lemma can also be used to show that convex linkages may be (unsurprisingly) straightened.
I'm seeking other applications of this Lemma, particularly those that might be suitable for use as exercises in an advanced undergraduate course, or other applications in the theory of polyhedra.
[O’R01] Joseph O’Rourke, An extension of Cauchy’s arm lemma with application to curve development, Discrete and computational geometry (Tokyo, 2000), Lecture Notes in Comput. Sci., vol. 2098, Springer, Berlin, 2001, pp. 280–291. MR MR2043660 (2004m:52052)