Hinge Theorem

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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.


In the words of Euclid:

If two triangles have two sides equal to two sides, but have one of the angles contained by the equal straight lines greater than the other, they will also have the base greater than the base.

(The Elements: Book $\text{I}$: Proposition $24$)


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

$\blacksquare$


Also known as

This theorem is also known as the side-angle side inequality theorem, or SAS inequality theorem.


Historical Note

This theorem is Proposition $24$ of Book $\text{I}$ of Euclid's The Elements.
It is the converse of Proposition $25$: Converse Hinge Theorem.


Sources