#### 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 $\stackrel{\u203e}{SA}$ and $\stackrel{\u203e}{SB}$ 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.

##### **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 $.