Converse Of The Hinge Theorem

In geometry, the hinge theorem (sometimes called the open up mouth theorem) states that if two sides of one triangle are congruent to ii sides of another triangle, and the included bending of the outset is larger than the included bending of the second, then the third side of the showtime triangle is longer than the third side of the 2nd triangle.^[1] This theorem is given as Proposition 24 in Book I of Euclid's Elements.

Scope and generalizations [edit]

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

It tin can be too extended from airplane Euclidean geometry to higher dimension Euclidean spaces (e.g., to tetrahedra and more generally to simplices), equally has been done for orthocentric tetrahedra (i.e., tetrahedra in which altitudes are concurrent)^[ii] and more generally for orthocentric simplices (i.e., simplices in which altitudes are concurrent).^[three]

Converse [edit]

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 outset triangle is greater than the third side of the second triangle, and then the included angle of the starting time triangle is larger than the included bending of the 2nd triangle.

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

References [edit]

^ Moise, Edwin; Downs, Jr., Floyd (1991). Geometry . Addison-Wesley Publishing Company. p. 233. ISBN0201253356.
^ Abu-Saymeh, Sadi; Mowaffaq Hajja; Mostafa Hayajneh (2012). "The open mouth theorem, or the scissors lemma, for orthocentric tetrahedra". Journal of Geometry. 103 (ane): ane–16. doi:10.1007/s00022-012-0116-4.
^ Hajja, Mowaffaq; Mostafa Hayajneh (August 1, 2012). "The open up mouth theorem in higher dimensions". Linear Algebra and Its Applications. 437 (three): 1057–1069. doi:ten.1016/j.laa.2012.03.012.