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This proposition has been concluded without the use of the parallel postulate, because the first time Euclid invokes the parallel postulate is in I.27. Thus, it should apply to all geometries satisfying the other axioms, namely elliptical geometry.

How is not a very large triangle in elliptical geometry not a counterexample to I.16? Take a triangle that nearly stretches over a hemisphere and the interior angles will each be nearly 180 degrees, leaving exterior angles nearly 0. Clearly, 0 is not bigger than 180.

Thanks for reading!

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    $\begingroup$ See Elements, I.16: Elliptic geometry satisfies some of the postulates of Euclidean geometry, but not all of them under all interpretations. Usually, I.Post.1, to draw a straight line from any point to any point, is interpreted to include the uniqueness of that line. But in elliptic geometry a completed “straight line” is topologically a circle so that any pair of points on it divide it into two arcs. Therefore, in elliptic geometry exactly two “straight lines” join any two given “points.” $\endgroup$ Commented 22 hours ago
  • $\begingroup$ Not what you were asking, but that "rebus" style of writing is...not great (personal opinion). In any case, it does seem as if the parallel postulate is needed here. See, e.g., this for a discussion. $\endgroup$ Commented 22 hours ago
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    $\begingroup$ @Lulu: I agree, but these images from Oliver Byrne have eternal life in graphic design. $\endgroup$ Commented 22 hours ago
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    $\begingroup$ @MauroALLEGRANZA: That's a good answer to a good question, so perhaps it should be an actual answer. $\endgroup$ Commented 21 hours ago

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See Elements, I.16:

Elliptic geometry satisfies some of the postulates of Euclidean geometry, but not all of them under all interpretations. Usually, I.Post.1, to draw a straight line from any point to any point, is interpreted to include the uniqueness of that line. But in elliptic geometry a completed “straight line” is topologically a circle so that any pair of points on it divide it into two arcs. Therefore, in elliptic geometry exactly two “straight lines” join any two given “points.

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