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 $AC = DG$, $BC = GE$.

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 $\angle EFG > \angle EGF$, $EG > EF$.

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