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I was going through Euclid's elements when I noticed Book I, Proposition 17, which states that:

In any triangle the sum of any two angles is less than two right angles.

And also Euclid's 5th postulate which states that:

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 the two right angles.

So suppose I take two lines AB and CD which would intersect if produced indefinitely and pass a transversal through it as showing in the figure below

enter image description here

Then I would say that AOC is a triangle, where O is the intersection point of the two lines. So can't I just say that by Book I, Proposition 17, 5th Postulate is proven, and therefore technically not a postulate? But it doesn't seem to be the case. Why is it so?

Areen Rath
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  • Postulate 5 says that if the two interior angles on the same side of the transversal add up to less than two right angles, then the two lines will meet. You seem to be proving the converse: that if the two lines meet (forming a triangle), then the two interior angles add up to less than two right angles. – mweiss Dec 28 '23 at 04:49
  • Prop 17 assumes one already has a triangle, and then says something about the sum of two of its angles. Parallel postulate only assumes something about the sum of two angles, and then says there exists the required third point to make a triangle. – coffeemath Dec 28 '23 at 04:49
  • @mweiss Ok, so let's start by proving the converse. Now I make another set of lines that would intersect when produced indefinitely and a transversal. Suppose I found a pair of angles who sum is less than two right angles, so on the other side, sum of angles is more than two right angles. And assume the lines meet on the side where the sum of angles is more than two right angles, forming a triangle. So sum of those two angles must be less than two right angles, which isn't the case, so the lines shall meet on the other side as they would intersect if produced indefinitely. Is there any error? – Areen Rath Dec 28 '23 at 05:21
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    @AreenRath Ok, so let's start by proving the converse. Now I make another set of lines that would intersect when produced indefinitely and a transversal... You are already wrong. You need to conclude that the lines intersect. If you assume that they intersect, you are assuming the thing you are trying to prove. – mweiss Dec 29 '23 at 00:53
  • Oh, I understood now. – Areen Rath Dec 29 '23 at 04:18

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