1.7: Angles

Our next goal is to introduce angles and angle measures; after that, the statement “we can measure angles” will become rigorous; see (iii) on Section 1.1.

An ordered pair of half-lines that start at the same point is called an angle. The angle $AOB$ (also denoted by $\angle AOB$) is the pair of half-lines $[OA)$ and $[OB)$. In this case the point $O$ is called the vertex of the angle.

Intuitively, the angle measure tells how much one has to rotate the first half-line counterclockwise, so it gets the position of the second half-line of the angle. The full turn is assumed to be $2 \cdot \pi$; it corresponds to the angle measure in radians. (For a while you may think that $\pi$ is a positive real number that measures the size of a half turn in certain units. Its concrete value $\pi \approx 3.14$ will not be important for a long time.

The angle measure of $\angle AOB$ is denoted by $\measuredangle AOB$; it is a real number in the interval $(-\pi, \pi]$.

The notations $\angle AOB$ and $\measuredangle AOB$ look similar; they also have close but different meanings which better not be confused. For example, the equality $\angle AOB = \angle A'O'B'$ means that $[OA) = [O'A')$ and $[OB) = [O'B')$; in particular, $O = O'$. On the other hand the equality $\measuredangle AOB = \measuredangle A'O'B'$ means only equality of two real numbers; in this case $O$ may be distinct from $O'$.

Here is the first property of angle measure which will become a part of the axiom.

Given a half-line $[OA)$ and $\alpha \in (-\pi, \pi]$ there is a unique half-line $[OB)$ such that $\measuredangle AOB = \alpha$.