Hinge Theorem

Theorem
If two triangles have two pairs of sides which are the same length, the triangle with the larger included angle also has the larger third side.

Proof

 * Hinge Theorem.png

Let $\triangle ABC$ and $DEF$ be two triangles in which $AB = DE$, $AC = DF$, and $\angle CAB > \angle FDE$.

Construct $\angle EDG$ on $DE$ at point $D$.

Place $G$ so that $DG = AC$.

Join $EG$ and $FG$.

Since $AB = DE$, $\angle BAC = \angle EDG$, and $AC = DG$, by Triangle Side-Angle-Side Equality:
 * $BC = GE$

By Euclid's first common notion:
 * $DG = AC = DF$

Thus by Isosceles Triangle has Two Equal Angles:
 * $\angle DGF = \angle DFG$

So by Euclid's fifth common notion:
 * $\angle EFG \, > \, \angle DFG = \angle DGF \, > \, \angle EGF$

Since $\angle EFG > \angle EGF$, by Greater Angle of Triangle Subtended by Greater Side:
 * $EG > EF$

Therefore, because $EG = BC$, $BC > EF$.