Hinge theorem
Encyclopedia
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.
The hinge theorem holds in Euclidean space
s and more generally in simply connected non-positively curved space form
s.
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.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the hinge theorem states that if two sides of one triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
are congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
to two sides of another triangle, and the included angle
Included angle
The concept of included angle is discussed at:* Congruence of triangles* Solution of triangles...
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.
The hinge theorem holds in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
s and more generally in simply connected non-positively curved space form
Space form
In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three obvious examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.-Reduction to generalized crystallography:It is a...
s.
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
Solution of triangles
In trigonometry, to solve a triangle is to find the three angles and the lengths of the three sides of the triangle when given some, but not all of that information...
Inequality Theorem and the SSS
Solution of triangles
In trigonometry, to solve a triangle is to find the three angles and the lengths of the three sides of the triangle when given some, but not all of that information...
Inequality Theorem respectively.