[2] The only primitive notions in ordered geometry are points A, B, C, ... and the ternary relation of intermediacy [ABC] which can be read as "B is between A and C".
An angle consists of a point O (the vertex) and two non-collinear rays out from O (the sides).
A triangle is given by three non-collinear points (called vertices) and their three segments AB, BC, and CA.
[4][1]: 181, 2 Gauss, Bolyai, and Lobachevsky developed a notion of parallelism which can be expressed in ordered geometry.
[5] Therefore, the "ordered" concept of parallelism does not form an equivalence relation on lines.