Corollary

Given two half-lines D, D' with origin 0, there exists one and only one rotation v such that v(D) = D'.

The hypotheses imply in fact that E contains no isotropic lines. Our assertion then follows from cor. 1 of prop. 1 (No. 1).

Henceforth we shall suppose that A is a maximal ordered field, and that the form $\Phi$ is positive. In the set of pairs $(D_1, D_2)$ of lines (resp. half-lines with origin 0) of E, the relation “there exists a direct similitude (resp. a rotation) u such that $u(D_1) = D'_1$ and $u(D_2) = D'_2$” is an equivalence relation between the pairs $(D_1, D_2)$ and $(D'_1, D'_2)$. The equivalence class of the pair $(D_1, D_2)$ is called, by definition, the angle of the lines (resp. half-lines) $D_1, D_2$ (taken in this order); it is denoted by $(\widehat{D_1, D_2})$.