the hinge theorem
Open a pair of scissors, or a folding compass, wider and the tips spread farther apart; close it and they draw nearer. The hinge theorem makes that everyday observation exact. Take two triangles that share two pairs of equal sides; then the triangle with the larger included angle (the wider hinge) has the longer third side. The wider you open the hinge, the longer the gap it spans.
Precisely: suppose triangles ABC and DEF have |AB| = |DE| and |AC| = |DF| (two pairs of equal sides), and the included angles satisfy m(angle A) > m(angle D). Then the third sides satisfy |BC| > |EF|. The converse is also true and equally handy: if the two triangles share two pairs of equal sides but |BC| > |EF|, then the included angle at A is larger than at D. The theorem is the inequality cousin of the SAS congruence criterion — SAS says equal hinge gives equal third side; the hinge theorem says a bigger hinge gives a bigger third side.
It is a result of neutral (absolute) geometry, proven without the parallel postulate, and it underlies comparison arguments — for instance reasoning about which of two paths is longer, or why the side opposite the largest angle of a single triangle is the longest. Keep its hypothesis in mind: you need two pairs of equal sides; with unequal sides the simple 'bigger angle, bigger opposite side' picture can break.
Two triangles each have sides 5 and 7 from a common vertex. In the first the angle between them is 80 degrees; in the second it is 50 degrees. By the hinge theorem the first triangle's third side is the longer, because its included angle is wider.
Same two sides, wider included angle, longer third side.
The hinge theorem needs two pairs of equal sides; it compares the included angles against the opposite third sides. It is the inequality version of SAS, valid in neutral geometry without the parallel postulate.