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# 1.7: Angles

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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 $$\measuredangle AOB = \alpha$$.