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Consider a triangle with sides, $3,4,5$, let the angle opposite the greatest side $5$ be $\theta$, does $3^2+4^2=5^2$ hold for such a triangle. Now consider a triangle with sides (1,1,$\sqrt{2}$), let $\beta$ be the angle opposite the largest side $\sqrt{2}$. Is $\theta=\beta$.

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$3^2 + 4^2 = 5^2$ based on arithmetic, not geometry. –  Arthur Fischer Aug 14 '12 at 4:52
    
I'm not sure why you're asking this to begin with... unless the Pythagorean theorem isn't something you're familiar with. –  J. M. Aug 14 '12 at 4:53
    
This is true arithmetically but can it be represented geometrically. –  Rajesh K Singh Aug 14 '12 at 5:13
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can we represent the lengths, $3,4,5$ on a plane exactly. i mean does the physical representation of a triangle of sides $3,4,5$ measure $3,4,5$ exactly(with no error). –  Rajesh K Singh Aug 15 '12 at 1:11
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You're trolling, right? –  WacDonald's Aug 15 '12 at 1:44

2 Answers 2

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Answer 1: Since if each side of a triangle has the same length as a corresponding side of the other triangle then the triangles are congruent it holds that your triangle is congruent to a right triangle with sides $3,4,5$ and thus the Pythagorean Theorem applies.

Answer 2: use trigonometry to calculate all angles, given that you know all the lengths

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can we represent the lengths, $3,4,5$ on a plane exactly. i mean does the physical representation of a triangle of sides $3,4,5$ measure $3,4,5$ exactly(with no error). –  Rajesh K Singh Aug 15 '12 at 1:11
    
@RajeshKSingh: That's a question about Physics. The answer, in our current state of knowledge, is no. –  André Nicolas Aug 15 '12 at 4:40

Hint: Converse of Pythagorean Theorem. If $a^2+b^2=c^2$, where $a$, $b$, and $c$ are positive, then there is a triangle with sides $a$, $b$, and $c$, and the angle opposite the side of size $c$ is a right angle.

Or else use the Pythagorean Theorem, and the fact that two triangles with corresponding sides equal are congruent (SSS).

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Link: en.wikipedia.org/wiki/Pythagorean_theorem#Converse –  user2468 Aug 14 '12 at 5:01
    
can we represent the lengths, $3,4,5$ on a plane exactly. i mean does the physical representation of a triangle of sides $3,4,5$ measure $3,4,5$ exactly(with no error). –  Rajesh K Singh Aug 15 '12 at 1:20

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