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It does not.
If you consider a right angled triangle with minor sides of length 1 unit each, then the Pythagorean theorem shows the third side (the hypotenuse) is sqrt(2) units in length. So the theorem proves that a side of such a length does exist. However, it does not prove that the answer is irrational. The same applies for some other Irrational Numbers.
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How do you prove the hypotenuse leg theorem without using Pythagorean theorem?
I have to prove http://s5.tinypic.com/19ldma.jpg http://img22.imageshack.us/img22/9263/mathhlproofou4.jpg without using pythagorean theorem
Does pythagorean theorem prove a triangle to be a right triangle?
Yes
What type of triangle proves the pythagorean theorerm?
The Pythagorean Theorem applies only to right triangles. (But they don't prove it.)
When did James Garfield prove the pythagorean theorem?
Somewhere around 1875 and 1876
Why was Pythagorean Theorem become necessary to incorporate in the number system?
Your question is so confusing that I almost trashed it and am not sure yet what you want to know but I have a possible idea : consider a right triangle each of whose legs have length 1. By the Pythagorean theorem, the hypotenuse has length equal to the square root of 2. The square root of 2 is irrational- one can prove it is not equal to any fraction of integers, yet it is obviously is a number of some kind. Thus the number system had to be extended to include numbers of this kind.
