7.1.2: Properties of Angles (2023)

Introduction

Imagine two separate and distinct lines on a plane. There are two possibilities for these lines: they will either intersect at one point, or they will never intersect. When two lines intersect, four angles are formed. Understanding how these angles relate to each other can help you figure out how to measure them, even if you only have information about the size of one angle.

Parallel and Perpendicular

Parallel lines are two or more lines that never intersect. Likewise, parallel line segments are two line segments that never intersect even if the line segments were turned into lines that continued forever. Examples of parallel line segments are all around you, in the two sides of this page and in the shelves of a bookcase. When you see lines or structures that seem to run in the same direction, never cross one another, and are always the same distance apart, there’s a good chance that they are parallel.

Perpendicular lines are two lines that intersect at a 90^o (right) angle. And perpendicular line segments also intersect at a 90^o (right) angle. You can see examples of perpendicular lines everywhere as well: on graph paper, in the crossing pattern of roads at an intersection, to the colored lines of a plaid shirt. In our daily lives, you may be happy to call two lines perpendicular if they merely seem to be at right angles to one another. When studying geometry, however, you need to make sure that two lines intersect at a 90^o angle before declaring them to be perpendicular.

The image below shows some parallel and perpendicular lines. The geometric symbol for parallel is ||, so you can show that \(\ \overleftrightarrow{A B} \| \overleftrightarrow{C D}\). Parallel lines are also often indicated by the marking >> on each line (or just a single > on each line). Perpendicular lines are indicated by the symbol \(\ \perp\), so you can write \(\ \overleftrightarrow{W X} \perp \overleftrightarrow{Y Z}\).

If two lines are parallel, then any line that is perpendicular to one line will also be perpendicular to the other line. Similarly, if two lines are both perpendicular to the same line, then those two lines are parallel to each other. Let’s take a look at one example and identify some of these types of lines.

Example

Identify a set of parallel lines and a set of perpendicular lines in the image below.

Solution

	Parallel lines never meet, and perpendicular lines intersect at a right angle. \(\ \overleftrightarrow{A B}\) and \(\ \overleftrightarrow{C D}\) do not intersect in this image, but if you imagine extending both lines, they will intersect soon. So, they are neither parallel nor perpendicular.
	\(\ \overleftrightarrow{A B}\) is perpendicular to both \(\ \overleftrightarrow{W X}\) and \(\ \overleftrightarrow{Y Z}\), as indicated by the right-angle marks at the intersection of those lines.
	Since \(\ \overleftrightarrow{A B}\) is perpendicular to both lines, then \(\ \overleftrightarrow{W X}\) and \(\ \overleftrightarrow{Y Z}\) are parallel.

\(\ \overleftrightarrow{W X} \| \overleftrightarrow{Y Z}\)

\(\ \overleftrightarrow{A B} \perp \overleftrightarrow{W X}, \overleftrightarrow{A B} \perp \overleftrightarrow{Y Z}\)

Finding Angle Measurements

Understanding how parallel and perpendicular lines relate can help you figure out the measurements of some unknown angles. To start, all you need to remember is that perpendicular lines intersect at a 90^o angle, and that a straight angle measures 180^o.

The measure of an angle such as \(\ \angle A\) is written as \(\ m \angle A\). Look at the example below. How can you find the measurements of the unmarked angles?

Example

Find the measurement of \(\ \angle I J F\).

Solution

	Only one angle, \(\ \angle H J M\), is marked in the image. Notice that it is a right angle, so it measures 90o. \(\ \angle H J M\) is formed by the intersection of lines \(\ \overleftrightarrow{I M}\) and \(\ \overleftrightarrow{H F}\). Since \(\ \overleftrightarrow{I M}\) is a line, \(\ \angle I J M\) is a straight angle measuring 180o.
	You can use this information to find the measurement of \(\ \angle H J I\): \(\ \begin{array}{c} m \angle H J M+m \angle H J I=m \angle I J M \\ 90^{\circ}+m \angle H J I=180^{\circ} \\ m \angle H J I=90^{\circ} \end{array}\)
	Now use the same logic to find the measurement of \(\ \angle I J F\). \(\ \angle I J F\) is formed by the intersection of lines \(\ \overleftrightarrow{I M}\) and \(\ \overleftrightarrow{H F}\). Since \(\ \overleftrightarrow{H F}\) is a line, \(\ \angle H J F\) will be a straight angle measuring 180o.
	You know that \(\ \angle H J I\) measures 90o. Use this information to find the measurement of \(\ \angle I J F\): \(\ \begin{array}{c} m \angle H J I+m \angle I J F=m \angle H J F \\ 90^{\circ}+m \angle I J F=180^{\circ} \\ m \angle I J F=90^{\circ} \end{array}\)

\(\ m \angle I J F=90^{\circ}\)

In this example, you may have noticed that angles \(\ \angle H J I, \angle I J F, \text { and } \angle H J M\) are all right angles. (If you were asked to find the measurement of \(\ \angle F J M\), you would find that angle to be 90^o, too.) This is what happens when two lines are perpendicular: the four angles created by the intersection are all right angles.

Not all intersections happen at right angles, though. In the example below, notice how you can use the same technique as shown above (using straight angles) to find the measurement of a missing angle.

Example

Find the measurement of \(\ \angle D A C\).

Solution

	This image shows the line \(\ \overleftrightarrow{B C}\) and the ray \(\ \overrightarrow{A D}\) intersecting at point \(\ A\). The measurement of \(\ \angle B A D\) is 135o. You can use straight angles to find the measurement of \(\ \angle D A C\).
	\(\ \angle B A C\) is a straight angle, so it measures 180o.
	Use this information to find the measurement of \(\ \angle D A C\). \(\ \begin{array}{c} m \angle B A D+m \angle D A C=m \angle B A C \\ 135^{\circ}+m \angle D A C=180^{\circ} \\ m \angle D A C=45^{\circ} \end{array}\)

\(\ m \angle D A C=45^{\circ}\)

Supplementary and Complementary

In the example above, \(\ m \angle B A C\) and \(\ m \angle D A C\) add up to 180^o. Two angles whose measures add up to 180^o are called supplementary angles. There’s also a term for two angles whose measurements add up to 90^o; they are called complementary angles.

One way to remember the difference between the two terms is that “corner” and “complementary” each begin with c (a 90^o angle looks like a corner), while straight and “supplementary” each begin with s (a straight angle measures 180^o).

If you can identify supplementary or complementary angles within a problem, finding missing angle measurements is often simply a matter of adding or subtracting.

Example

Find the measurement of \(\ \angle A X Z\).

Solution

	This image shows two intersecting lines, \(\ \overleftrightarrow{A B}\) and \(\ \overleftrightarrow{Y Z}\). They intersect at point \(\ X\), forming four angles. Angles \(\ \angle A X Y\) and \(\ \angle A X Z\) are supplementary because together they make up the straight angle \(\ \angle Y X Z\).
	Use this information to find the measurement of \(\ \angle A X Z\). \(\ \begin{array}{c} m \angle A X Y+m \angle A X Z=m \angle Y X Z \\ 30^{\circ}+m \angle A X Z=180^{\circ} \\ m \angle A X Z=150^{\circ} \end{array}\)

\(\ m \angle A X Z=150^{\circ}\)

Example

Find the measurement of \(\ \angle B A C\).

Solution

	This image shows the line \(\ \overleftrightarrow{C F}\) and the rays \(\ \overrightarrow{A B}\) and \(\ \overrightarrow{A D}\), all intersecting at point \(\ A\). Angle \(\ \angle B A D\) is a right angle. Angles \(\ \angle B A C\) and \(\ \angle C A D\) are complementary, because together they create \(\ \angle B A D\).
	Use this information to find the measurement of \(\ \angle B A C\). \(\ \begin{array}{c} m \angle B A C+m \angle C A D=m \angle B A D \\ m \angle B A C+50^{\circ}=90^{\circ} \\ m \angle B A C=40^{\circ} \end{array}\)

\(\ m \angle B A C=40^{\circ}\)

Example

Find the measurement of \(\ \angle C A D\).

Solution

	You know the measurements of two angles here: \(\ \angle C A B\) and \(\ \angle D A E\). You also know that \(\ m \angle B A E=180^{\circ}\).
	Use this information to find the measurement of \(\ \angle C A D\). \(\ \begin{array}{c} m \angle B A C+m \angle C A D+m \angle D A E=m \angle B A E \\ 25^{\circ}+m \angle C A D+75^{\circ}=180^{\circ} \\ m \angle C A D+100^{\circ}=180^{\circ} \\ m \angle C A D=80^{\circ} \end{array}\)

\(\ m \angle C A D=80^{\circ}\)

Summary

Parallel lines do not intersect, while perpendicular lines cross at a 90^o. angle. Two angles whose measurements add up to 180^o are said to be supplementary, and two angles whose measurements add up to 90^o are said to be complementary. For most pairs of intersecting lines, all you need is the measurement of one angle to find the measurements of all other angles formed by the intersection.