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, $\angle DGF = \angle DFG$.

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

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

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