Chapter 5 Geometry

Section 5.2 Angles and Angle Measurement

5.2.2 Angles

Two rays (half-lines) $g$ and $h$ in the plane starting from the same initial point $S$ enclose an angle $\measuredangle (g,h)$.

For the notation of the angle $\measuredangle (g,h)$, the order of $g$ and $h$ is relevant. $\measuredangle (g,h)$ denotes the angle shown in the figure above. It is defined by rotating the half-line $g$ counter-clockwise to the half-line $h$.
In contrast, $\measuredangle (h,g)$ denotes the angle from $h$ to $g$ as illustrated by the figure below.

The point $S$ is called a vertex of the angle, and the two half-lines enclosing the angle are called the arms of the angle. If $A$ is a point on the line $g$ and $B$ is a point on the line $h$, then the angle $\measuredangle (g,h)$ can also be denoted by $\measuredangle \left(ASB\right)$. In this way, angles between line segments $\overline{SA}$ and $\overline{SB}$ are described.
Angles are often denoted by lower-case Greek letters to distinguish them from variables, which are generally denoted by lower-case Latin letters (see Table 1.1.8 in module 1). Further angles can be found by considering angles formed by intersecting lines.

The figure above contains further vertical and supplementary angles.

Some special angles have their own dedicated name. For example, the angle bisector $w$ is the half-line whose points have the same distance from the two given half-lines $g$ and $h$. Then, it can be said that $w$ bisects the angle between $g$ and $h$.

Next, three lines are considered. Two of the three lines are parallel, while the third line is not parallel to the others. It is called a transversal. These lines form eight cutting angles. Four of the eight angles are equal.