Geometry :: Axiom :: Neutral Protractor

For every angle \(\angle BAC\) there is a real number \(\mu(\angle BAC)\), called the measure of \(\angle BAC\), such that:

1. \(0^\circ \leq \mu(\angle BAC) < 180^\circ\) for every angle \(\angle BAC\);
2. \(\mu(\angle BAC) = 0^\circ\) if and only if \(\overrightarrow{AB} = \overrightarrow{AC}\).
3. For each real number \(r \in (0,180)\), and for each half-plane \(H\) bounded by \(\overleftrightarrow{AB}\) there exists a unique ray \(\overrightarrow{AE}\) such that \(E\) is in \(H\) and \(\mu(\angle BAE) = r^\circ\).
4. If the ray \(\overrightarrow{AD}\) is between rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\), then \(\mu(\angle BAD) + \mu(\angle DAC) = \mu(\angle BAC)\).