what does area have to do with the angles to make them add up to more than 180o? If we're just talking about the angles themselves, then I'm afraid ζ = α+β. I checked. If you take your diagram, put it in photoshop, cut α and β and overlay them onto ζ, they fill ζ perfectly. Regardless of what mathematical proofs you think you're using, this is a visual proof that ζ = α+β.
You're joking, right? Angles are sectors of circles, that's why they are denoted with that little arc connecting the two sides, which can be seen as two radii of a circle. This means greater angles correspond to greater circular sector areas and vice versa: the greater the area enclosed by the two sides of an angle, the greater the angle itself. This is self-evident. Therefore, since the area enclosed by the two interior angles α and β is larger than the area enclosed by the exterior angle ζ, it logically follows that the sum of the two interior angles themselves must be larger than the exterior angle.
Your "visual proof" is absolutely ridiculous, and you perfectly well know that. Of course, the difference in the size of the angles in this example is way too small to be measured, it is almost 0, but not exactly 0. That's why Euclidean geometry works well for relatively (relative to the totality, the whole of space) small extensions, as was repeatedly stated in this thread.
With your ludicrous photoshop "proof" you can also deny that a small triangle on the surface of an Euclidean sphere has a spherical excess, i.e. the sum of its interior angles is larger than 180°.
From all this we can conclude that you are either quite slow on the uptake, or else you are dishonest and again one of the disinformation trolls on this forum. From the many experiences gathered here, the latter is more probable.
And since none of the alleged true Flat Earth believers has ever bothered to comment in this thread, it can also be concluded with sufficient certainty that none of them are seriously interested in the true shape of the Earth and the structure of the Cosmos, but only in promoting their silly disk theory, which they themselves do not believe, thus making Flat Earth belief look stupid. However, the most funny thing is that the top dogs here, regardless of the camp they pretend to fight in (RE vs. FE), perfectly well know that what has been presented in this thread is the truth, namely that, as jraffield1 put it on the second page of this thread: "[...] space itself is spherical and the Earth is a plane in that space?"
The Flat Earth Society, which should correctly call itself the Disk Earth Society, is nothing but a diabolical scheme intended to fool and confuse people, but it's not that hard to see through it... So, good luck, anyway.
Lactantius
PS: I fully subscribe to Lord Wilmore's sig:
"I want truth for truth's sake, not for the applaud or approval of men. I would not reject truth because it is unpopular, nor accept error because it is popular. I should rather be right and stand alone than run with the multitude and be wrong." - C.S. DeFord