The Proof of the Hinge Theorem

The Hinge Theorem can be understood by exploring real hinges. If the two hinges are of the same size and the angle of the first hinge is opened wider than the second, then the distance between the edges of the first hinge, is farther than that of the second.

 Hinge Theorem

If a string is placed connecting the hinges, then a triangle is formed. As we shall see, hinges are connected to theorems about triangles.

Theorem

The Hinge Theorem states if two sides of one triangle is congruent, respectively, to two sides of another triangle, and the included angle of the first angle is larger than the included  angle of the second, then the third side o f the first triangle is longer than the third side of the second.

hinge theorem 1

The Hinge Theorem is illustrated in the first figure. Given triangle ABC and triangle DEF, with AB = DE, and AC = DF. If angle A > angle D, then BC > EF.

Proof

First we construct AGC, with G in the interior of angle BAC such that triangle AGC is congruent to triangle DEF. This can be done using compass and straightedge construction. First copy angle EDC to angle BAC, then locate AG = DE

hinge theorem 2

Now, bisect angle BAG and let M be the intersection of the bisector and BC. By SAS Congruence, triangle AMB is congruent to triangle AMG. Therefore, MB = MG.

hinge theorem 3

Now, by the Triangle Inequality Theorem,

CG < CM + MG

Therefore,

CG < CM + MB

because MB = MG.

Since CG = EF, and CM + MB = BC,

We have EF < BC which is what we want to show.

 

3 thoughts on “The Proof of the Hinge Theorem

  1. This seems like such an obvious concept, yet the proof is surprisingly more complex! I think there is an error though. In the second paragraph of your proof, you say MD = MK, but I think MK should be MG?

  2. The proofs are more interesting than all the problems I had to solve in high school geometry ca.1965. I don’t recall any proofs, so it is wonderful to encounter them again. Much appreciate their beauty.

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