9.2: Classifying Angles
Introduction
Moving the Skatepark
Marc and Isaac finished their drawing on time and presented it to the city council. The city council loved their ideas, but did not agree to rebuild the skateboard park in the park. Marc and Isaac were feeling very defeated when they left the meeting.
When they got back to Isaac’s house, there was a message from Principal Fuller at their school. It seems the school has decided to move the soccer field to a bigger space across the street and the boys can submit the design for the new skateboard park right at the school.
“This is great!” Marc says when he hears the news. “Now we can ride before and after school.”
“Yes, but we have to redo our design,” Isaac says. “Let’s get to work.”
The space of the soccer field has been designated by the white lines on the map. To complete their design, the boys will need to count all of the different angles on the map. This way they can figure out where the ramps are going to go and where the rails will also go.
Do you know how to figure out which angles are which? How many right angles can you count? How many 180 degree angles? Are there any supplementary or complementary angles? Use this lesson to figure some things out about angles. When you are finished learning, you can come back and answer these questions using the map.
What You Will Learn
In this lesson you will learn the following skills:
- Classify angles as acute, right, obtuse or straight.
- Classify angle pairs as supplementary or complementary
- Use real-world angle pair diagrams to find unknown angle measures.
Teaching Time
In the last lesson you learned to identify different geometric figures and one of them was the angle. Remember that an angle is formed when two rays connect at a single endpoint. You can measure an angle in degrees. When measuring an angle, you are measuring the space between the two rays.
The arc represents the space of the angle that you are measuring. This angle is very small as you can see by the size of the arc.
This arc is very large. The space between the rays is large so the angle is large too.
We can classify or organize angles according to the size of the angle. Because we measure them in degrees, the angle is classified according to the number of degrees that it has.
What types of angles are there?
1. Right Angle–the first type of angle is a right angle. It is an angle that is easy to recognize because it forms a corner that is straight. Often with a right angle you will see a little box in the corner too.
The corner of this building forms a right angle. You can see buildings like this all the time in the real world. It is an example of a right angle.
2. Acute Angle-an acute angle is an angle that is less than 90 degrees. Here is a picture of an acute angle.
Here is a picture of an acute angle. You can see that it has been labeled \begin{align*}45^\circ\end{align*} to show that it is less than 90 degrees. An acute angle is smaller than a right angle.
3. Straight Angle-a straight angle is the same as a straight line. A straight line is equal to \begin{align*}180^\circ\end{align*}. The angle of a straight line stretches from one side of the line to the other side as indicated by the arc in this picture.
This bike path shows a very straight line in real life.
4. Obtuse angle-an obtuse angle is an angle that is greater than 90 degrees but less than 180 degrees. Here is a picture of an obtuse angle.
This corner forms an obtuse angle. Even if we made a sharp corner from the rounded one, it would still be greater than 90 degrees, but not a straight line, so it is less than 180 degrees.
Now it is time to try a few on your own. Identify the following angles as acute, obtuse, right or straight.
1.
2.
3.
4. \begin{align*}55^\circ\end{align*} angle
Take a few minutes to check your work with a partner.
II. Classify Angle Pairs as Supplementary or Complementary
Sometimes, we can have two angles that are a part of each other or are connected to each other. When we have this happen, we call these two angles angle pairs.
Here we are looking at two special types of angle pairs, supplementary angles or complementary angles.
What are supplementary angles?
Supplementary angles are two angles whose sum is equal to \begin{align*}180^\circ\end{align*}. In other words when we add the measure of one angle in the pair with the other angle in the pair, together they equal 180 degrees.
Let’s look at an example.
Example
These two angles are supplementary because together they form a straight line. We can also tell that they are supplementary because when we add their angle measures the sum is equal to 180 degrees. \begin{align*}120 + 60 = 180^\circ\end{align*}
Here is a real life example of supplementary angles. Notice that the two streets indicated by the arrows are right angles. Two right angles are equal to 180 degrees. Therefore, this intersection is an example of supplementary angles.
What are complementary angles?
Complementary angles are a pair of angles whose sum is \begin{align*}90^\circ\end{align*}. Here is an example of a two complementary angles.
If we add up the two angle measures, the sum is equal to 90 degrees. Therefore, the two angles are complementary.
You can find missing angle measures by using this information about supplementary and complementary angles.
Example
Find the measure of \begin{align*}x\end{align*}.
First, we can identify that these two angles are supplementary. They form a straight line. The total number of degrees in a straight line is 180. Therefore, we can write the following equation to solve.
\begin{align*}130 + x & = 180 \\ x & = 50^\circ\end{align*}
Our missing angle is equal to \begin{align*}50^\circ\end{align*}.
Example
Here the two angles are complementary. Therefore the sum of the two angles is equal to 90 degrees. We can write an equation and solve for the missing angle measure.
\begin{align*}75 + x & = 90 \\ x & = 15^\circ\end{align*}
Our missing angle measure is equal to \begin{align*}50^\circ\end{align*}.
It is time to practice. Write whether each pair is complementary or supplementary.
- If the sum of the angles is equal to 180 degrees.
- If one angle is 60 degrees and the other angle is 120 degrees.
- If the sum of the angle measures is 90 degrees.
Real Life Example Completed
Moving the Skatepark
Now it is time to reconsider this map and figure out where the different types of angle are located. Begin by underlining the important information. Be sure to reread the problem!!
Marc and Isaac finished their drawing on time and presented it to the city council. The city council loved their ideas, but did not agree to rebuild the skateboard park in the park. Marc and Isaac were feeling very defeated when they left the meeting.
When they got back to Isaac’s house, there was a message from Principal Fuller at their school. It seems the school has decided to move the soccer field to a bigger space across the street and the boys can submit the design for the new skateboard park right at the school.
“This is great!” Marc says when he hears the news. “Now we can ride before and after school.”
“Yes, but we have to redo our design,” Isaac says. “Let’s get to work.”
The space of the soccer field has been designated by the white lines on the map. To complete their design, the boys will need to count all of the different angles on the map. This way they can figure out where the ramps are going to go and where the rails will also go.
Do you know how to figure out which angles are which? How many right angles can you count? How many 180 degree angles? Are there any supplementary or complementary angles?
By drawing arrows on the school map, we can see where all of the right angles are located. There are eight right angles located on the outside border of the plan for the soccer field. Two of the pairs of right angles in the middle of the field add together to form straight lines.Therefore, we can say that there are two pairs of supplementary angles.
Isaac and Marc can use these angles to create the perfect design for the new school skatepark.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Acute angle
- an angle less than 90 degrees.
- Right angle
- an angle equal to 90 degrees.
- Obtuse angle
- an angle greater than 90 degrees but less than 180 degrees.
- Straight angle
- a straight line equal to 180 degrees
- Supplementary angles
- two angles whose sum is 180 degrees.
- Complementary angles
- two angles whose sum is 90 degrees.
Technology Integration
James Sousa, Introduction to Angles
Other Videos:
- http://www.teachertube.com/members/viewVideo.php?video_id=25730&title=ROCK_SONG_3_Kinds_of_Angles – A song about the three kinds of angles
- http://www.mathplayground.com/mv_using_protractor.html – Brightstorm video on using a protractor, calculating and classifying angles according to their measures.
Time to Practice
Directions: Classify each angle as acute, right, obtuse or straight.
1.
2.
3.
4.
5. An angle measuring \begin{align*}88^\circ\end{align*}
6. An angle measuring \begin{align*}90^\circ\end{align*}
7. An angle measuring \begin{align*}180^\circ\end{align*}
8. An angle measuring \begin{align*}105^\circ\end{align*}
9. An angle measuring \begin{align*}118^\circ\end{align*}
10. An angle measuring \begin{align*}5^\circ\end{align*}
11. An angle measuring \begin{align*}17^\circ\end{align*}
12. An angle measuring \begin{align*}35^\circ\end{align*}
Directions: Identify each angle pair as supplementary or complementary angles.
13.
14.
15.
16.
Directions: Use what you have learned about complementary and supplementary angles to answer the following questions.
17. If two angles are complementary, then their sum is equal to _________ degrees.
18. If two angles are supplementary, then their sum is equal to ________ degrees.
19. True or false. If one angle is \begin{align*}120^\circ\end{align*}, then the second angle must be equal to \begin{align*}90^\circ\end{align*} for the angles to be supplementary.
20. True or false. If the angles are supplementary and one angle is equal to \begin{align*}100^\circ\end{align*}, then the other angle must be equal to \begin{align*}80^\circ\end{align*}.
21. True or false. The sum of complementary angles is \begin{align*}180^\circ\end{align*}.
22. True or false. The sum of supplementary angles is \begin{align*}90^\circ\end{align*}.
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