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.

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Triangle Inequality and Its Proof

Triangle Inequality states that for any real numbers a and b,

|a| + |b| ≥ |a + b|.

In proving this theorem, we use the definition of absolute value of a number. The absolute value of a number is x if x > 0 (*), –x  if x < 0 (**) and 0 if x = 0 (**). For instance, the absolute value of 5 is 5 since 5 > 0. On the other hand, the absolute value of -3 is -(-3) = 3. Clearly, the absolute value of 0 is 0. Triangle Inequality The geometric representation of triangle inequality is shown above (see Demystifying Triangle Inequality). It states that a triangle can only be formed if the combined length of any two sides is greater than that of the third sides. Using the definition of absolute value, we now prove the theorem. Continue reading