The Triangle Inequality theorem, The Hinge Theorem, and Its Converse
Architectures may need to build windows or doors, but first, they need to know what side lengths will create a triangle. For example, if an architect needs to create a triangular window in which no side is greater, then 5, they need to know that a triangle with side lengths of 1, 2, and 4 will NOT work because of the triangle inequality theorem ( 1+2 is less than 4). They should also know, though, that side lengths such as 2, 3, and 4 will work. This is just one example of a triangle theorem used in architecture. The hinge theorem can also be used. The hinge theorem states that if two sides of a triangle are congruent to 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. When architects are trying to create two different types of triangles that have 2 congruent sides, but different angle measures, but they don't know which triangle's third side should be bigger in order to ensure that the sides form a triangle, they can use the hinge theorem to find out. Similarly, the converse of the hinge theorem can also be used in architecture. The converse of the hinge theorem states that when two sides of triangle are congruent to two sides of another triangle,then the included angle of the first triangle is larger than the included angle of the second triangle. When an architect knows they want two different windows and know that two sides of both the windows are congruent to each other, and they know what they want to size the third lengths but do not know which included angle to make bigger, they can use the converse of the hinge theorem.
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