8.1: Angles
Introduction
A Trip to the Art Museum
Mrs. Gilson is taking her math class to the art museum on a field trip. Before leaving for the museum, Mrs. Gilson posed some questions to her class.
“Does anyone know why we are going to an art museum for math class?” She asked.
Matt, who tended to like to make jokes, was the first one to speak up.
“So we can count the paintings?” he joked.
Mrs. Gilson smiled as if she was expecting just such an answer. She looked around and waited for any other responses. Kyle was the next one to speak.
“I think it has to do with geometry. Isn’t most art based on some kind of geometry?” he asked.
“Very nice, and yes you are correct. We can find geometry in many different paintings forms and figures. Some of the first painters, sculptures, actually all kinds of artists used geometry to design their work. Here is a slide that we can look at together,” Mrs. Gilson said, putting an image up on the board from the computer.
“This is to help you practice before we go,” Mrs. Gilson explained. “This is a piece of a stained glass window. You can see the flowers in the painting, but can you see the geometry? Take out your notebooks and make a note of any place that you see angles. We're starting with angles since they are building blocks of geometry.”
The students all took out their notebooks and began to work.
While the students make their notes, you make some notes too. What do you already know about angles?
This lesson is all about angles, the different types and what they look like. Pay attention throughout this lesson and at the end you will be able to find the geometry in the stained glass.
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills:
- Classify angles as acute, obtuse, right or straight.
- Identify angle pairs as complementary, supplementary or neither.
- Given an angle, find the measures of its complement and supplement.
- Use logical reasoning to find angle measures in variable relationships given sufficient classification.
Teaching Time
I. Classify Angles as Acute, Obtuse, Right or Straight
The word “angle” is one of those words that we hear all the time. You might hear someone say “What is the angle of that corner?” or a photographer could use the term “wide angle lens.” Have you ever tried to angle a sofa through a doorway? The way that you turn the sofa makes a big difference in whether the sofa fits through the doorway or not. Notice that the same word “angle” is being used in each of these examples, but each example uses it differently.
In geometry, we use the word angle too. Understanding angles in geometry can help you when you use the angles in real life.
What is an angle?
An angle is a when two lines, line segments or rays connect at a common point. The angle is created by the space between the two lines. We can say that this space “forms the angle.”
We measure an angle in degrees. What we are actually measuring is the distance between the two lines. The space between them near the point where they connect forms the angle. This may sound confusing, but it will make more sense as we continue.
Why are degrees important?
Degrees are important because we classify angles by their size. Knowing the degrees of an angle can help us to be sure that we are using the correct words to classify or identify it. The \begin{align*}^\circ\end{align*} symbol means “degrees.” The number of degrees tells how open or closed the angle is. The smaller the number of degrees, the smaller or more closed the angle is. Angle sizes can range from 0 to \begin{align*}360^\circ\end{align*}, a complete circle.
Here is a diagram that shows some angle measurements.
As you can see, an angle of \begin{align*}360^\circ\end{align*} makes a complete circle. An angle of \begin{align*}270^\circ\end{align*} is three-quarters of a circle, and an angle of \begin{align*}180^\circ\end{align*} is half a circle. A \begin{align*}180^\circ\end{align*} angle is a straight line. Most angles that we deal with are between 0 and \begin{align*}180^\circ\end{align*}.
We classify angles by their size, or number of degrees.
We classify angles as acute, right, obtuse, or straight.
Let’s find out what these names mean.
An acute angle measures less than \begin{align*}90^\circ\end{align*}. If its measure is \begin{align*}1^\circ\end{align*} or \begin{align*}89^\circ\end{align*} or anywhere in between, we call it an acute angle.
Obtuse angles measure more than \begin{align*}90^\circ\end{align*}. Angles greater than \begin{align*}90^\circ\end{align*} and less than \begin{align*}180^\circ\end{align*} are obtuse angles.
Most angles are either acute or obtuse. However, there are two special angles with exact measurements. A right angle measures exactly \begin{align*}90^\circ\end{align*}. Right angles are one of the most important concepts you need to know about geometry. We find them in squares, rectangles, and triangles. They are everywhere in the real world too.
Notice that we use a small box to show when an angle is a right angle.
There are many places in the real world where you can see acute, obtuse and right angles. Here are a few examples. Can you identify the angles?
http://www.flickr.com/photos/ghost_of_kuji/394579484/ (attribution)
http://www.flickr.com/photos/basykes/6558444/ (attribution)
If you look at each of these pictures, you can see the right angles clearly. Also notice that the wires of the bridge stretch to create acute angles on each side of the center beam.
The other special angle is called a straight angle. A straight angle measures exactly \begin{align*}180^\circ\end{align*}. We have already seen that a straight angle forms a line.
Now that we know each kind of angle, let’s try classifying some.
Example
Classify each angle below.
For each angle, it may help to ask yourself: “Is it bigger or smaller than a right angle?”
Remember, right angles always measure \begin{align*}90^\circ\end{align*}, and we use \begin{align*}90^\circ\end{align*} to tell whether an angle is acute or obtuse.
Is Figure 1 larger or smaller than a right angle? A right angle looks like a perfect corner, often with one arm pointing straight up. This angle is wider than that, so it is an obtuse angle.
The angle in Figure 2 looks like a straight line... you know what that means! It must be a straight angle.
Is Figure 3 larger or smaller than a right angle? It is smaller than \begin{align*}90^\circ\end{align*}, so it is an acute angle.
The angle in Figure 4 does resemble a perfect corner, so it could be a right angle. Now take a closer look. The small box tells you that it definitely is a right angle.
We can also identify an angle by using a symbol. Here is the symbol for angle.
\begin{align*}\angle A\end{align*}
This means “Angle \begin{align*}A\end{align*}”. You will see this symbol used when we work with angles.
8A. Lesson Exercises
Identify each type of angle described.
- An angle greater than \begin{align*}90^\circ\end{align*}
- An angle that measures \begin{align*}15^\circ\end{align*}
- An angle that measures exactly \begin{align*}90^\circ\end{align*}
Take a few minutes to check your answers with a friend.
Write down a definition for each type of angle in your notebook. You will need these throughout this lesson.
Once you memorize the different definitions for each angle, identifying them will become easier and easier. Now you know the four kinds of angles: acute, obtuse, right, and straight. We can classify any angle into one of these four categories. Next, let’s look at combining angles to make special pairs
II. Identify Angle Pairs as Complementary, Supplementary or Neither
When we have two angles together, we can say that we have angle pairs. Sometimes, the measures of these angles add up to form a special relationship. Sometimes they don’t. There are two special angle pair relationships for you to learn about. The first one is called complementary angles and the second one is called supplementary angles.
Complementary angles are two angles whose measurements add up to exactly \begin{align*}90^\circ\end{align*}. In other words, when we put them together they make a right angle. Below are some pairs of complementary angles.
Supplementary angles are two angles whose measurements add up to exactly \begin{align*}180^\circ\end{align*}. When we put them together, they form a straight angle. Take a look at the pairs of supplementary angles below.
Let’s practice classifying some pairs of angles.
Example
Classify the following pairs of angles as either complementary or supplementary.
The sum of the angles in Figure 1 is \begin{align*}180^\circ\end{align*}. Therefore these angles are supplementary angles.
The sum of the angles in Figure 2 is \begin{align*}90^\circ\end{align*}. Therefore these angles are complementary angles.
Remember, complementary angles add up to \begin{align*}90^\circ\end{align*} and supplementary angles add up to \begin{align*}180^\circ\end{align*}. In order to classify the pairs as complementary or supplementary, we need to add the measures of the angles in each pair together to find out the total.
Example
Are angles \begin{align*}X\end{align*} and \begin{align*}Y\end{align*} complementary or supplementary?
The question asks us to classify angles \begin{align*}X\end{align*} and \begin{align*}Y\end{align*} as either complementary or supplementary. Look at the figure. This time we do not know the measures of any of the angles. Can we still answer the question?
We can. We know that complementary angles add up to \begin{align*}90^\circ\end{align*} and supplementary angles add up to \begin{align*}180^\circ\end{align*}. We also know that \begin{align*}90^\circ\end{align*} is a right angle and that \begin{align*}180^\circ\end{align*} is a straight angle. Now take a good look at angles \begin{align*}X\end{align*} and \begin{align*}Y\end{align*}. If we put them together as a whole, do they form a right angle or a straight angle? They form a straight angle, so they must be supplementary.
8B. Lesson Exercises
Identify the following angle pairs as complementary, supplementary or neither.
- Angle \begin{align*}A = 23^\circ\end{align*}, Angle \begin{align*}B = 45^\circ\end{align*}
- Angle \begin{align*}A = 45^\circ\end{align*}, Angle \begin{align*}B = 45^\circ\end{align*}
- Angle \begin{align*}A = 103^\circ\end{align*}, Angle \begin{align*}B = 77^\circ\end{align*}
Take a few minutes to check your answers with a friend. Did you find the sum of each pair first?
Write down the definitions of complementary and supplementary angles in your notebook.
III. Given an Angle, Find the Measure of its Complement and Supplement
As we have seen, we identify complementary and supplementary angles by their sum. This means that we can also find the measure of one angle in a pair if we know the measure of the other angle. For instance, because we know that complementary angles always add up to \begin{align*}90^\circ\end{align*}, we can calculate the measurement of one angle in a pair of complementary angles. Let’s see how this works.
We can see that together, \begin{align*}C\end{align*} and \begin{align*}D\end{align*} form a right angle. Therefore they are complementary, and they add up to \begin{align*}90^\circ\end{align*}. We know that \begin{align*}C\end{align*} has a measure of \begin{align*}44^\circ\end{align*}. How can we find the measure of angle \begin{align*}D\end{align*}?
To find the measurement of angle \begin{align*}D\end{align*}, we simply subtract the measure of angle \begin{align*}C\end{align*} from 90.
\begin{align*}\angle{C}+ \angle{D} & = 90^\circ\\ 44^\circ + \angle{D} & = 90^\circ\\ \angle {D} & = 90-44\\ \angle {D} & = 46^\circ\end{align*}
In order for these two angles to be complementary, as the problem states, they must add up to \begin{align*}90^\circ\end{align*}. Angle \begin{align*}D\end{align*} therefore measures \begin{align*}46^\circ\end{align*}. We can check our calculation by adding angles \begin{align*}C\end{align*} and \begin{align*}D\end{align*}. Their sum must be equal to \begin{align*}90^\circ\end{align*}.
\begin{align*}44^\circ + 46^\circ = 90^\circ\end{align*}
We can follow the same process to find the unknown angle in a pair of supplementary angles. As with complementary angles, if we know the measure of one angle in the pair, we can find the measure of the other.
Example
Angles \begin{align*}P\end{align*} and \begin{align*}Q\end{align*} are supplementary angles. If angle \begin{align*}P\end{align*} measures \begin{align*}112^\circ\end{align*}, what is the measure of angle \begin{align*}Q\end{align*}?
We know that supplementary angles have a total of \begin{align*}180^\circ\end{align*} Therefore we can subtract the measurement of the angle we know, angle \begin{align*}P\end{align*}, from \begin{align*}180^\circ\end{align*} to find the measure of angle \begin{align*}Q\end{align*}.
\begin{align*}\angle{P} + \angle{Q} & = 180^\circ\\ 112^\circ + \angle {Q} & = 180^\circ\\ \angle{Q} & = 180-112\\ \angle{Q} & = 68^\circ\end{align*}
Angle \begin{align*}Q\end{align*} is \begin{align*}68^\circ\end{align*}. We can check our calculation by adding angles \begin{align*}P\end{align*} and \begin{align*}Q\end{align*}. Remember, in order to be supplementary angles, their sum must be equal to \begin{align*}180^\circ\end{align*}.
\begin{align*}68^\circ + 112^\circ = 180^\circ\end{align*}
We can call this finding the complement or the supplement.
8C. Lesson Exercises
Find the complement or supplement in each example.
- Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}33^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
- Angles \begin{align*}C\end{align*} and \begin{align*}D\end{align*} are supplementary. Angle \begin{align*}C\end{align*} is \begin{align*}59^\circ\end{align*}. Find the measure of angle \begin{align*}D\end{align*}.
Take a few minutes to check your answers with a partner.
IV. Use Logical Reasoning to Find Angle Measures in Variable Relationships Given Sufficient Classification
Armed with our knowledge of complementary and supplementary angles, we can often find the measure of unknown angles. We can use logical reasoning to interpret the information we have been given in order to find the unknown measure. Take a look at the diagram below.
Can we find the measure of angle \begin{align*}X\end{align*}? We can, if we apply what we have learned about supplementary angles. We know that supplementary angles add up to \begin{align*}180^\circ\end{align*}, and that \begin{align*}180^\circ\end{align*} is a straight line. Look at the diagram. The \begin{align*}80^\circ\end{align*} angle and angle \begin{align*}X\end{align*} together form a straight line, so we can deduce that they are supplementary angles. That means we can set up an equation to solve for \begin{align*}X\end{align*}.
\begin{align*}80 + X = 180\end{align*}
The equation shows what we already know: the sum of supplementary angles is \begin{align*}180^\circ\end{align*}. We can find the measure of the unknown angle by solving for \begin{align*}X\end{align*}.
\begin{align*}80 + X & = 180\\ X & = 180 - 80\\ X & = 100^\circ\end{align*}
The measure of the unknown angle in this supplementary pair is \begin{align*}100^\circ\end{align*}.
We can check our work by putting this value in for \begin{align*}X\end{align*} in the equation.
\begin{align*}80 + 100 = 180\end{align*}
We can set up a similar equation to solve for an unknown angle in a complementary pair, too. We’ll see how in the next example.
Example
What is the measure of angle \begin{align*}R\end{align*}?
How can we use what we have learned to find the measure of angle \begin{align*}R\end{align*}? Can we determine whether the two angles have a relationship with each other? Together, they form a right angle. They must be a pair of complementary angles, so we know their sum is \begin{align*}90^\circ\end{align*}. Again, we can set up an equation to solve for \begin{align*}R\end{align*}, the unknown angle.
\begin{align*}R + 22 = 90\end{align*}
This equation represents what we know, that the sum of these two complementary angles is \begin{align*}90^\circ\end{align*}. Now we solve for \begin{align*}R\end{align*}.
\begin{align*}R + 22 & = 90\\ R & = 90 - 22\\ R & = 68^\circ\end{align*}
The measure of the unknown angle is \begin{align*}68^\circ\end{align*}. We can check our answer by putting this value in for \begin{align*}R\end{align*} in the equation.
\begin{align*}68 + 22 = 90^\circ\end{align*}
Whenever we can deduce that two angles have either a complementary or supplementary relationship, we can find the measure of one angle if given the measure of the other.
Now let’s go back to the problem in the introduction and use what we have learned to solve this problem.
Real Life Example Completed
A Trip to the Art Museum
Now you have learned all about angles. Here is the original problem once again. Reread it and then pay attention to Mrs. Gilson’s instructions.
Mrs. Gilson is taking her math class to the art museum on a field trip. Before leaving for the museum, Mrs. Gilson posed some questions to her class.
“Does anyone know why we are going to an art museum for math class?” She asked.
Matt, who tended to like to make jokes, was the first one to speak up.
“So we can count the paintings?” he joked.
Mrs. Gilson smiled as if she was expecting just such an answer. She looked around and waited for any other responses. Kyle was the next one to speak.
“I think it has to do with geometry. Isn’t most art based on some kind of geometry?” he asked.
“Very nice, and yes you are correct. We can find geometry in many different paintings forms and figures. Some of the first painters, sculptures, actually all kinds of artists used geometry to design their work. Here is a slide that we can look at together,” Mrs. Gilson said putting an image up on the board from the computer.
“This is to help you practice before we go,” Mrs. Gilson explained. “This is a piece of a stained glass window. You can see the flowers in the painting, but can you see the geometry? Take out your notebooks and make a note of any place that you see angles. We're starting with angles since they are the building blocks of geometry.”
The students all took out their notebooks and began to work.
Can you find an example of each of the different types of angles in this stained glass? Make a few notes in your notebook. It may be helpful to draw them too.
While the students worked, Mrs. Gilson walked around the room. When most seemed finished, Mrs. Gilson gave the students this instruction.
“Now find a partner and share the angles that you found in the painting.”
You do this too. Find a partner and share the angles that you found. This is the best way to check your work for accuracy. If you and your partner both selected the same angle, then choose a new one together.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Acute Angle
- an angle whose measure is less than \begin{align*}90^\circ\end{align*}
- Obtuse Angle
- an angle whose measure is greater than \begin{align*}90^\circ\end{align*}
- Right Angle
- an angle whose measure is equal to \begin{align*}90^\circ\end{align*}
- Straight Angle
- an angle whose measure is equal to \begin{align*}180^\circ\end{align*}
- Degrees
- how an angle is measured
- Angle Pairs
- when the measures of two angles are added together to form a special relationship
- Supplementary Angles
- angle pairs whose sum is \begin{align*}180^\circ\end{align*}
- Complementary Angles
- angle pairs whose sum is \begin{align*}90^\circ\end{align*}
Technology Integration
James Sousa, Complementary, Supplementary, and Vertical Angles
More Videos:
- http://www.onlinemathlearning.com/complementary-angles.html – This is a webpage with two videos imbedded in it. There is data on angles at the beginning and then there are two short videos. One video is on complementary angles and one is on supplementary angles.
Time to Practice
Directions: Label each angle as acute, obtuse, right, or straight.
9. \begin{align*}55^\circ\end{align*}
10. \begin{align*}102^\circ\end{align*}
11. \begin{align*}90^\circ\end{align*}
12. \begin{align*}180^\circ\end{align*}
13. \begin{align*}10^\circ\end{align*}
14. \begin{align*}87^\circ\end{align*}
15. \begin{align*}134^\circ\end{align*}
Directions: Identify whether the pairs below are complementary or supplementary or neither.
16.
17.
18.
19. An angle pair whose sum is \begin{align*}180^\circ\end{align*}
20. Angle \begin{align*}A = 90^\circ\end{align*} Angle \begin{align*}B\end{align*} is \begin{align*}45^\circ\end{align*}
21. Angle \begin{align*}C = 125^\circ\end{align*} Angle \begin{align*}B = 55^\circ\end{align*}
Directions: Find the measure of missing angle \begin{align*}M\end{align*} for each pair of complementary or supplementary angles.