Angle Relationships

In this lesson, you will draw on prior knowledge and recognize angles as geometric shapes. You will learn how to measure angles in whole nonstandard units as well as with a protractor. Then, you will extend your knowledge of different types of angles to learning about relationships among pairs of angles. You will also learn to write and solve equations that will allow you to find the measure of unknown angles in different figures composed of angles.

We will start out with a quick review of basic angles. Remember that angles are formed by two rays with a common endpoint (vertex). Congruent angles are angles that have the same measure. It is important to remember that when naming angles, the vertex always goes in the middle. If you need a bit more review for angle classifications, please watch the following video. Once you have seen how to name and classify angles, we will move forward to angle pairs.

Classifying Angles

Now that you’ve refreshed your memory, let’s talk about the measures of angles. An angle measure is the degrees of an angle. It tells us how big an angle is in the unit we call degree. One method you can use to find the angle degree is to use a circle. Remember, a circle has a total of 360 degrees. We can use this information to help us divide the circle and find the measurement of even the smallest angle. For instance, the unit angle of a circle is an angle that cuts 1/360 of a circle and measures 1 degree.

Showing how to divide a circle in half (360/2 = 180), fourths (360/4 = 90) and eighths (360/8 = 45)

What if you need more than one section or wedge of a circle?

For example, what is the measurement in degrees of the part of the circle that is colored in? First, we need to find out the measurement of one of the wedges. Do you remember how to do that? 360/6 = 60

Each wedge is 60º.

If each wedge is 60º, and we have two wedges, how could we find the total for both? We could add! (Or multiply)

60 +60 =120° or 2(60)=120°

Another way to find the measurement of angles is to use a tool called a protractor. You can see a picture of a protractor below.

protractor image

You can print your very own protractor by visiting the link below or finding any free online printable protractor.

Printable Protractor

Let’s look at a video showing how to use the protractor by clicking here.

Investigate

Find the measurement of each given angle using the protractors.

protractor with a certain angle on it

Did you get 55°? Great job!

protractor with a certain angle on it

Did you get 125°? Great job!

Types of Angles

In earlier years, you learned how to recognize right angles, obtuse angles, acute angles, and straight angles. We know that a right angle measures 90° and a straight angle always measures 180°.

An acute angle is an angle that measures less than 90°. How can you use the idea of a right angle to help you identify an acute angle? If the opening of an angle is smaller than the opening of a right angle, it is acute.

The same goes for an obtuse angle. This is an angle that measures more than 90 but less than 180. If the opening of an angle is larger than the opening of a right angle, but smaller than the opening of a straight angle, the angle is obtuse.

Let us look at some of the relationships that are common among angles.

Adjacent angles are pairs of angles that share a vertex and one side but do not overlap.

adjacent angles AB adjacent angles AB that splits a 90º angle

Supplementary angles, (also known as linear pairs), are two angles whose measure have a sum of 180°. Remember, the key word is "pair", which means two angles. Like complementary angles, these angle pairs do not have to be adjacent.

supplementary adjacent angles ab Supplementary angles of 139º and 41º

Here are some basic word problems that we can solve without diagrams.

Angles A and B are supplementary. If m ∠ A is 32 °, what is the m ∠ B ?

Given the fact that they are supplementary, we know that the two add up to 180°. You can set up an equation to read:

B + 32 = 180 and solve for B.

-32 = -32

B = 148

The measure of angle B equation image indicator $LaTeX: \left(m\angle B=148^\circ\right)$

Angles C and D are complementary. If m ∠C is 22 °, what is the m ∠ D ?

Given the fact that they are complementary, we know that the two add up to 90°. You can set up an equation to read:

D + 22 = 90 and solve for D

-22 -22

D = 68

The measure of angle D equation image indicator $LaTeX: \left(m\angle D=68^\circ\right)$

You can also use diagrams to find unknown angle measures in complementary and supplementary angles.

find x where MGH is 64.9º

Angles LGM and MGH are complementary.

Create an equation to solve for x.

x + 64.9 = 90

-64.9 -64.9

x = 25.1

The measure of equation image indicator $LaTeX: \angle LGM\:is\:25.1^\circ$ .

Take a look at the video lesson to see examples of complementary and supplementary angles.

Adjacent, Complementary, and Supplementary Angles

Now, practice what you have learned about adjacent, complementary, and supplementary angles.

A, B, C, D, E lines with 40º, 50º, and 90º angles

1. Name the right angles.

Solution: $LaTeX: \angle CFD\:or\:\angle CFA$

2. Name 5 pairs of adjacent angles.

Solution: $LaTeX: \angle BFA\:and\:BFC,\:\angle BFC\:and\:\angle CFD,\:\angle CFD\:and\:\angle DFE,\:\angle DFE\:and\:\angle EFA,\:\angle EFA\:and\:\angle AFB,\:\angle CFE\:and\:CFB$

3. Name a pair of complementary angles.

Solution: $LaTeX: \angle BFA\:and\:\angle BFC$

4. Name an angle that is supplementary to equation image indicator $LaTeX: \angle CFE$ .

Solution: $LaTeX: \angle CFB$

5. Name an angle that is supplementary to equation image indicator $LaTeX: \angle BFD$ .

Solution: $LaTeX: \angle BFA$

6. Name an angle that is supplementary to equation image indicator $LaTeX: \angle CFD$ .

Solution: $LaTeX: \angle CFA$

7. Are equation image indicator $LaTeX: \angle CFB\:and\:\angle DFE$ adjacent angles?

Solution: No, they share a vertex but have no common side

8. Are equation image indicator $LaTeX: \angle BFD\:and\:\angle AFE$ adjacent angles?

Solution: No, they share a vertex but have no common side

We have learned about angle pairs. Now, let us go a bit deeper and learn to use line and angle relationships to solve problems with figures. First of all, we know that adjacent angles have a common vertex and a common ray.

Vertical angles, also known as opposite angles, are opposite angles formed by two intersecting lines. Vertical angles are ALWAYS congruent and tend to resemble bow ties.

Check it out! Look at the figure below where the lines intersect and the angles formed. You can see that the opposite angles are congruent.

We can use what we have learned about supplementary angles and vertical angles to write and solve equations to find unknown angle measures. In the following figure, we are going to use the vertical angle relationship and the supplementary angle pair to find the measure of ∠ 1.

The given angle is vertical to ∠ 2 and is 54°. Remember that vertical angles are congruent. Therefore, the measure of ∠ 2 is 54°.

equation image indicator $LaTeX: \angle1\:and\:\angle2$ are supplementary. That means the sum of their measures is 180°.

We can write an equation now to find the measure of the unknown angle:

∠ 1 + 54 =180

-54 -54

∠ 1 = 126°

Watch this video to see other ways to use vertical angles to find unknown angle measures.

There are other lines that can be used to help us find angles. We know that parallel lines are lines that lie in the same plane and never intersect. If you draw a line that intersects both of these parallel lines, you have created a transversal. The transversal creates eight angles through its intersection with the lines and if you know the measure of one of those angles, you can use what you have learned about adjacent and vertical angles to find the measures of the other seven angles.

Notice the figures below to get an idea of how this works.

transversals in parallel lines, one figure indicating adjacent angles, and four sets of vertical angles

Watch this short video to see how transversals work with parallel lines to help you find angle measures of vertical and adjacent angles that form complementary and supplementary angles.

Angle Relationships Practice

Now that you have spent some time learning strategies for solving angle and line relationships problems, you are ready to complete your Geometrical Figures: Angle Relationships Practice. Download your practice by CLICKING HERE.

Once you have completed your practice, AND MAKE SURE YOU ATTEMPTED AND WORKED THE PROBLEMS OUT ON YOUR OWN, click here to download your practice key.

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