# 7.1.2: Properties of Angles (2023)

1. Last updated
2. Save as PDF
• Page ID
62469
• $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

##### Learning Objectives
• Identify parallel and perpendicular lines.
• Find measures of angles.
• Identify complementary and supplementary angles.

## 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 90o (right) angle. And perpendicular line segments also intersect at a 90o (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 90o 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.

$$\ \overleftrightarrow{W X} \| \overleftrightarrow{Y Z}$$

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

##### Exercise

Which statement most accurately represents the image below?

1. $$\ \overleftrightarrow{E F} \| \overleftrightarrow{G H}$$
2. $$\ \overleftrightarrow{A B} \perp \overleftrightarrow{E G}$$
3. $$\ \overleftrightarrow{F H} \| \overleftrightarrow{E G}$$
4. $$\ \overleftrightarrow{A B} \| \overleftrightarrow{F H}$$
1. Incorrect. This image shows the lines $$\ \overleftrightarrow{E G}$$ and $$\ \overleftrightarrow{F H}$$, not $$\ \overleftrightarrow{E F}$$ and $$\ \overleftrightarrow{G H}$$. Both $$\ \overleftrightarrow{E G}$$ and $$\ \overleftrightarrow{F H}$$ are marked with >> on each line, and those markings mean they are parallel. The correct answer is $$\ \overleftrightarrow{F H} \| \overleftrightarrow{E G}$$.
2. Incorrect. $$\ \overleftrightarrow{A B}$$ does intersect $$\ \overleftrightarrow{E G}$$, but the intersection does not form a right angle. This means that they cannot be perpendicular. The correct answer is $$\ \overleftrightarrow{F H} \| \overleftrightarrow{E G}$$.
3. Correct. Both $$\ \overleftrightarrow{E G}$$ and $$\ \overleftrightarrow{F H}$$ are marked with >> on each line, and those markings mean they are parallel.
4. Incorrect. $$\ \overleftrightarrow{A B}$$ and $$\ \overleftrightarrow{F H}$$ intersect, so they cannot be parallel. Both $$\ \overleftrightarrow{E G}$$ and $$\ \overleftrightarrow{F H}$$ are marked with >> on each line, and those markings mean they are parallel. The correct answer is $$\ \overleftrightarrow{F H} \| \overleftrightarrow{E G}$$.

## 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 90o angle, and that a straight angle measures 180o.

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 90o, 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 $$\ \angle D A C$$. $$\ \begin{array}{c} m \angle B A D+m \angle D A C=m \angle B A C \\ 135^{\circ}+m \angle D A C=180^{\circ} \\ m \angle D A C=45^{\circ} \end{array}$$

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

##### Exercise

Find the measurement of $$\ \angle C A D$$.

1. 43o
2. 137o
3. 147o
4. 317o

## Supplementary and Complementary

In the example above, $$\ m \angle B A C$$ and $$\ m \angle D A C$$ add up to 180o. Two angles whose measures add up to 180o are called supplementary angles. There’s also a term for two angles whose measurements add up to 90o; 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 90o angle looks like a corner), while straight and “supplementary” each begin with s (a straight angle measures 180o).

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

Two angles are supplementary. If one of the angles measures 48o, what is the measurement of the other angle?

Solution

 $$\ m \angle A+m \angle B=180^{\circ}$$ Two supplementary angles make up a straight angle, so the measurements of the two angles will be 180o. $$\ \begin{array}{l} 48^{\circ}+m \angle B=180^{\circ} \\ m \angle B=180^{\circ}-48^{\circ} \\ m \angle B=132^{\circ} \end{array}$$ You know the measurement of one angle. To find the measurement of the second angle, subtract 48o from 180o.

The measurement of the other angle is 132o.

##### 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 $$\ \angle C A D$$. $$\ \begin{array}{c} m \angle B A C+m \angle C A D+m \angle D A E=m \angle B A E \\ 25^{\circ}+m \angle C A D+75^{\circ}=180^{\circ} \\ m \angle C A D+100^{\circ}=180^{\circ} \\ m \angle C A D=80^{\circ} \end{array}$$

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

##### Exercise $$\PageIndex{1}$$

Which pair of angles is complementary?

1. $$\ \angle P K O \text { and } \angle M K N$$
2. $$\ \angle P K O \text { and } \angle P K M$$
3. $$\ \angle L K P \text { and } \angle L K N$$
4. $$\ \angle L K M \text { and } \angle M K N$$
1. Incorrect. The measures of complementary angles add up to 90o. It looks like the measures of these angles may add up to 90o, but there is no way to be sure, so you cannot say that they are complementary. The correct answer is $$\ \angle L K M \text { and } \angle M K N$$.
2. Incorrect. $$\ \angle P K O \text { and } \angle P K M$$ are supplementary angles (not complementary angles) because together they comprise the straight angle $$\ \angle O K M$$. The correct answer is $$\ \angle L K M \text { and } \angle M K N$$.
3. Incorrect. $$\ \angle L K P \text { and } \angle L K N$$ are supplementary angles (not complementary angles) because together they comprise the straight angle $$\ \angle P K N$$. The correct answer is $$\ \angle L K M \text { and } \angle M K N$$.
4. Correct. The measurements of two complementary angles will add up to 90o. $$\ \angle L K P$$ is a right angle, so $$\ \angle L K N$$ must be a right angle as well. $$\ \angle L K M+\angle M K N=\angle L K N$$, so $$\ \angle L K M \text { and } \angle M K N$$ are complementary.

## Summary

Parallel lines do not intersect, while perpendicular lines cross at a 90o. angle. Two angles whose measurements add up to 180o are said to be supplementary, and two angles whose measurements add up to 90o 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.

## FAQs

### What are two properties of angles? ›

Properties of Angles

Important properties of the angle are: For one side of a straight line, the sum of all the angles always measures 180 degrees. The sum of all angles always measures 360 degrees around a point. An angle is a figure where, from a common position, two rays appear.

What are the properties of angles on parallel lines? ›

Properties of Parallel Lines

Corresponding angles are equal. Vertical angles/ Vertically opposite angles are equal. Alternate interior angles are equal. Alternate exterior angles are equal.

Which is a property of an angle answer? ›

The following are the important properties of angles: The sum of all the angles on one side of a straight line is always equal to 180 degrees, The sum of all the angles around the point is always equal to 360 degrees.

What is the angle formula? ›

Angles Formulas at the center of a circle can be expressed as, Central angle, θ = (Arc length × 360º)/(2πr) degrees or Central angle, θ = Arc length/r radians, where r is the radius of the circle.

What are 2 examples of angles? ›

There are many daily life examples of an angle, such as cloth-hangers, arrowheads, scissors, partly opened doors, pyramids, edge of a table, the edge of a ruler, etc.

What is right angle properties? ›

Right Angle Triangle Properties

One angle is always 90° or right angle. The side opposite angle of 90° is the hypotenuse. The hypotenuse is always the longest side. The sum of the other two interior angles is equal to 90°. The other two sides adjacent to the right angle are called base and perpendicular.

What are the properties of the angles of a triangle? ›

The sum of the angles of a triangle is always 180 degrees. The exterior angles of a triangle always add up to 360 degrees. The sum of consecutive interior and exterior angle is supplementary. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

What are the 4 types of angles in parallel lines? ›

To do this, we use three facts about angles in parallel lines: Alternate angles, co-Interior angles, and corresponding angles. Sometimes called 'Z angles'. Sometimes called 'C angles'.

What are the properties of parallel and perpendicular lines? ›

Parallel lines are lines that never intersect, and they form the same angle when they cross another line. Perpendicular lines intersect at a 90-degree angle, forming a square corner. We can identify these lines using angles and symbols in diagrams.

What are all the angles of 2 parallel lines? ›

If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary, i.e. they add up to 180°.

### What are the 3 ways to identify an angle? ›

There are three ways to name an angle--by its vertex, by the three points of the angle (the middle point must be the vertex), or by a letter or number written within the opening of the angle. This video demonstrates each method for naming an angle. These resources are part of KET's Measurement and Geometry collection.

## References

Top Articles
Latest Posts
Article information

Author: Clemencia Bogisich Ret

Last Updated: 15/11/2023

Views: 5716

Rating: 5 / 5 (60 voted)

Author information

Name: Clemencia Bogisich Ret

Birthday: 2001-07-17

Address: Suite 794 53887 Geri Spring, West Cristentown, KY 54855

Phone: +5934435460663

Job: Central Hospitality Director

Hobby: Yoga, Electronics, Rafting, Lockpicking, Inline skating, Puzzles, scrapbook

Introduction: My name is Clemencia Bogisich Ret, I am a super, outstanding, graceful, friendly, vast, comfortable, agreeable person who loves writing and wants to share my knowledge and understanding with you.