Triangle Sum 180 from Euclid's 5th axiom

Aug 2008
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I am working on a proof of the Triangle Sum Theorem by using Euclid's 5th axiom. Attached the picture so that you can see.Screen Shot 2016-01-22 at 9.03.13 PM.png

So I started going in the other direction and saying well if \(\displaystyle \alpha+\gamma+\beta\prime \ge 180\) then also \(\displaystyle \beta+\gamma+\alpha\prime \ge 180\) and started working out that
\(\displaystyle \alpha+\beta\prime \ge 180-\gamma\) and \(\displaystyle \beta+\alpha\prime \ge180-\gamma\)

so \(\displaystyle \alpha+\beta\prime +\beta +\alpha\prime \ge 2(180-\gamma)\)

\(\displaystyle \alpha +\beta +\alpha\prime +\beta\prime + 2\gamma \ge 360\)

\(\displaystyle \alpha +\beta +\gamma \ge 360-(\alpha\prime +\beta\prime + \gamma)\)


But this is as far as I've gotten and I'm not sure if I'm going in the right direction.
I know I was given that it is less than 180 so I need to show greater than 180 so then I can conclude it must equal 180.
 
Feb 2014
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I am clearly missing something here.

$\alpha ^ \prime + \gamma + \beta ^ \prime = sum\ of\ two\ right\ angles.$

$\alpha = \alpha ^ \prime\ and\ \beta = \beta^ \prime \implies$

$\alpha + \gamma + \beta = \alpha ^ \prime + \gamma + \beta ^ \prime = sum\ of\ two\ right\ angles.$

$\alpha + \gamma + \beta = sum\ of\ triangle's\ angles.$

$sum\ of\ triangle's\ angles = sum\ of\ two\ right\ angles.$
 
Last edited:
Aug 2008
45
2
I would totally agree but the definition of Euclid's 5th axiom we have to work with states, "If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles."

So while this proof gave the part of if the interior angles are less than 180 then the exterior (\(\displaystyle \alpha\prime\) and \(\displaystyle 180-\alpha\)) will be greater than 180.

That lead to \(\displaystyle \alpha +\beta +\gamma \le 180\).

So I must prove that \(\displaystyle \alpha +\beta +\gamma \ge 180\) for me to say it must be true that they are equal.
 
Feb 2014
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We are, I fear, still talking past each other. What I was saying was that my little proof, one of the steps can be derived only using Postulate 5.

Euclid of course does not talk about degrees or radians. He reasons in terms of right angles.

His definition of right angles is that they are the equal angles formed by a straight line perpendicular to another straight line. Definition 10.

His definition of parallel lines is that two straight lines in a plane that never intersect no matter how far extended are parallel. Definition 23.

His Postulate 4 says that all right angles are equal to each other.

Proposition 13 proves that the sum of the angles sum formed by a straight line standing on another straight line equals the sum of two right angles. From that proposition, you can easily derive the first line in my proof.

$\alpha ^ \prime + \gamma + \beta ^ \prime = sum\ of\ two\ right\ angles.$

Proposition 15 shows the equality of vertical angles formed by the intersection of two straight lines.

So far Postulate 5 has not come into play.

$\alpha = \alpha ^ \prime\ and\ \beta = \beta^ \prime$ depends on Propositions 13, 15, and 29, the last of which depends on Postulate 5.

See whether that does not work.
 

Plato

MHF Helper
Aug 2006
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I am working on a proof of the Triangle Sum Theorem by using Euclid's 5th axiom. Attached the picture so that you can see.View attachment 35182
Do you realize that this question is the most written about in the entire history of mathematics?
If you are near a good mathematics library you can many books on this topic.
A most readable book is Deductive Systems by Runion and Lockwood.

A more rigorous textbook is An Introduction to NON-EUCLIDEAN GEOMETRY by David Gans.
 
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Jul 2005
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My house.
Do you realize that this question is the most written about in the entire history of mathematics?
To be perfectly honest, the second most written about question in the entire history of mathematics, involves whether Gosta Mittag-Leffler was involved with Alfred Nobel's wife. :p
 
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Apr 2008
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So that's why Nobel hated Mittag-Leffler's guts. Fascinating.