Hinge theorem
{{Short description|Geometry theorem relating to triangles}}
In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included 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.{{cite book|last1=Moise|first1=Edwin|last2=Downs, Jr.|first2=Floyd|title=Geometry|date=1991|publisher=Addison-Wesley Publishing Company|isbn=0201253356|page=[https://archive.org/details/geometry00mois/page/233 233]|url-access=registration|url=https://archive.org/details/geometry00mois/page/233}} This theorem is given as Proposition 24 in Book I of Euclid's Elements.
Proof from the law of cosines
The theorem is an immediate corollary of the law of cosines.{{cite journal |journal=Journal of Geometry |title=The open mouth theorem, or the scissors lemma, for orthocentric tetrahedra |year=2012 |volume=103 |pages=1–16 |last1=Abu-Saymeh |first1=Sadi |last2=Hajja |first2=Mowaffaq |last3=Hayajneh |first3=Mostafa |doi=10.1007/s00022-012-0116-4 }} For two triangles with sides and with angles and opposite the respective sides and , the law of cosines states:
c^2 &= a^2 + b^2 - 2ab\cos \gamma, \\
\hat{c}^2 &= a^2 + b^2 - 2ab\cos \hat \gamma.
\end{align}
The cosine function is monotonically decreasing for angles between and radians, so implies (and the converse as well).
Scope and generalizations
The hinge theorem holds in Euclidean spaces and more generally in simply connected non-positively curved space forms.
It can be also extended from plane Euclidean geometry to higher dimension Euclidean spaces (e.g., to tetrahedra and more generally to simplices), as has been done for orthocentric tetrahedra (i.e., tetrahedra in which altitudes are concurrent){{r|ahh}} and more generally for orthocentric simplices (i.e., simplices in which altitudes are concurrent).{{cite journal|last=Hajja|first=Mowaffaq|author2=Mostafa Hayajneh|title=The open mouth theorem in higher dimensions|journal=Linear Algebra and Its Applications|date=August 1, 2012|volume=437|issue=3|pages=1057–1069|doi=10.1016/j.laa.2012.03.012|doi-access=free}}
Converse
The converse 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 the SAS Inequality Theorem and the AAS Inequality Theorem respectively.
References
{{reflist}}
{{Ancient Greek mathematics}}