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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).
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  for orthocentric tetrahedra (i.e., tetrahedra in which altitudes are concurrent)and more generally by M. Hajja and M. Hayajneh in  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.
- 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.
- 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.
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