Hinge theorem
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.[1] This theorem is given as Proposition 24 in Book I of Euclid's Elements.
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
Converse
The
In some textbooks, the theorem and its converse are written as the SAS Inequality Theorem and the AAS Inequality Theorem respectively.
References
- ISBN 0201253356.
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