Hinge theorem

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In geometry, the hinge 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. This theorem is actually Propositions 24 of Book 1 of Euclid's Elements (sometimes is called the open mouth theorem).

To get a better idea of how to use the theorem, think of an opening gate, and when it gets wider: the largest angle get wider, and the longest segment gets longer. The long segment and wide angle are both associated with each other. How? If the largest angle in the triangle is across from a segment with an unknown length, then the segment across from the largest angle is the longest segment in the triangle. This theorem goes both ways, as in the longest segment being across from the widest angle.

The hinge theorem holds in Euclidean spaces and more generally in simply connected non-positively curved space forms.

It can be also extended from the plane euclidean geometry to higher dimensions euclidean spaces (i.e., for tetrahedra and more general for simplices), as it was done recently by S. Abu-Saymeh, M. Hajja, M. Hayajneh in [1] for orthocentric tetrahedra (i.e., tetrahedra in which altitudes are concurrent)and more generally by M. Hajja and M. Hayajneh in [2] for orthocentric simplices (i.e., simplices in which altitudes are concurrent).

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 SSS Inequality Theorem respectively.



  1. ^ Abu-Saymeh, Sadi (2012). "The open mouth theorem, or the scissors lemma, for orthocentric tetrahedra". Journal of Geometry 103 (1): 1–16. doi:10.1007/s00022-012-0116-4. 
  2. ^ Hajja, Mowaffaq; Mostafa Hayajneh (August 1, 2012). "The open mouth theorem in higher dimensions". Linear Algebra and Its Applications 437 (3): 1057-1069. doi:10.1016/j.laa.2012.03.012.