>>14197216At the place where the arc of the circle meets the line we have a horn angle. If we draw tangents there we would have a 90 degree or right angle. We can say that the measure is infinitesmally less than 90 degreees.
1) Horn angles and variations thereof appear to be well studied in conformal geometry, see e.g. the papers by Ladue, Mary Elizabeth "Conformal geometry of horn angles of higher order", Amer. J. Math. 65, (1943), 455–476, and "Trihornometry: A New Chapter of Conformal Geometry" by Edward Kasner, Proceedings of the National Academy of Sciences of the United States of America, Vol. 23, No. 6 (Jun. 15, 1937), pp. 337-341
2) You can find more about the theory of horn angles in the paper "The recent theory of the horn angle" by E. Kasner, Scripta Math. 11, (1945). 263–267, 53.0X. The paper is from 1945, but it should give you some idea about the directions of study in that area.
3) Your intuition is right as horn angles can be measured within the model of superreal numbers (in the sense of D. O. Tall), see the paper by J. Bair and V. Henry "Angles corniculaires et nombres superréels" (French) ("Horn angles and superreal numbers), Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 1, 77–86.
