#### 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 $\measuredangle \left(g,h\right)$.

Angle enclosed by the rays $g$ and $h$.

For the notation of the angle $\measuredangle \left(g,h\right)$, the order of $g$ and $h$ is relevant. $\measuredangle \left(g,h\right)$ 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, $\measuredangle \left(h,g\right)$ denotes the angle from $h$ to $g$ as illustrated by the figure below.

Angle enclosed by the rays $h$ and $g$

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 $\measuredangle \left(g,h\right)$ can also be denoted by $\measuredangle \left(ASB\right)$. In this way, angles between line segments $\stackrel{‾}{SA}$ and $\stackrel{‾}{SB}$ 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.
##### Vertical Angles and Supplementary Angles 5.2.1

Let $g$ and $h$ be two lines intersecting in a point $S$.

• The angles $\phi$ and $\phi \text{'}$ are called vertical angles.
• The angles $\phi$ and $\psi$ are called supplementary angles with respect to $g$.

The figure above contains further vertical and supplementary angles.
##### Exercise 5.2.2
Find all vertical and supplementary angles occurring in the figure above.

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 $h$.
##### Names of Special Angles 5.2.3
Let $g$ and $h$ be half-lines with the intersection point $S$.
• The angle covering the entire plane is called the complete angle.
• If the rays $g$ and $h$ form a line, the angle between $g$ and $h$ is called a straight angle.
• The angle between two half-lines bisecting a straight angle is called the right angle. One also says that $g$ and $h$ are perpendicular (or orthogonal) to each other.

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.
##### Angles at Parallel Lines 5.2.4
Let two parallel lines $g$ and $h$ be given cut by another transversal line $j$.

• Then the angle $\alpha \text{'}$ is called a corresponding angle of $\alpha$ and
• the angle $\beta \text{'}$ is called an alternate angle of $\beta$.
Since the lines $g$ and $h$ are parallel, the angles $\alpha$ and $\alpha \text{'}$ are equal. Likewise, the angles $\beta$ and $\beta \text{'}$ are equal.

##### Exercise 5.2.5
The figure shows two parallel lines $g$ and $h$ cut by another line $j$. Explain which angles are equal and which angles are corresponding angles or alternate angles to each other, respectively.