The arbitrariness lies somewhere else. With history or linguistics or biology, you could say that the arbitrariness appears when you look at the world. The laws of the universe are fixed, but what happened in 1066 or which word we use to refer to a rose or which proteins take apart DNA are arbitrary in that the world might have happened differently. In math, the arbitrariness comes in setting up the rules of the game (contrast this with the fixed laws of nature), but once you have done that, everything that comes after necessarily follows from the rules you started with.
Alorael, it's interesting that you cite angles of a triangle adding up to 180 degrees as an example of the non-arbitrariness of math. There are models of geometry — spherical geometry, hyperbolic geometry — where the angles of a triangle add up to greater than or less than 180 degrees, respectively. Of course, once you define what your model is, every truth that follows is a necessary truth.