Ingeometry, thehinge theorem (sometimes called theopen mouth theorem) states that if two sides of onetriangle arecongruent to two sides of another triangle, and theincluded angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.^[1] This theorem is given as Proposition 24 in Book I ofEuclid'sElements.

The theorem is an immediate corollary of thelaw of cosines.^[2] For two triangles with sides${\displaystyle \{a,b,c\}}$ and${\displaystyle \{a,b,\hat{c}\}}$ with angles${\displaystyle \gamma }$ and${\displaystyle \hat{\gamma }}$ opposite the respective sides${\displaystyle c}$ and${\displaystyle \hat{c}}$, the law of cosines states:

Thecosine function ismonotonically decreasing for angles between${\displaystyle 0}$ and${\displaystyle \pi }$radians, so${\displaystyle \hat{\gamma }>\gamma }$ implies${\displaystyle \hat{c}>c}$ (and the converse as well).

The hinge theorem holds inEuclidean spaces and more generally in simply connected non-positively curvedspace forms.

It can be also extended from plane Euclidean geometry to higher dimension Euclidean spaces (e.g., totetrahedra and more generally tosimplices), as has been done fororthocentric tetrahedra (i.e., tetrahedra in which altitudes are concurrent)^[2] and more generally for orthocentric simplices (i.e., simplices in which altitudes are concurrent).^[3]

Theconverse of the hinge theorem is also true: If the two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is greater than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

In some textbooks, the theorem and its converse are written as theSAS Inequality Theorem and theSSS Inequality Theorem respectively.

• Elementary geometry
• Theorems about triangles

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