9.3: Classifying Triangles
Introduction
The Bermuda Triangle
Isaac and Marc are continuing their work on their skate park. They are both fascinated by the Bermuda Triangle and have decided to name one of the parts of their design after this triangle.
The Bermuda Triangle is located in an area of water right around Bermuda. There have been many mysteries surrounding this area of the ocean. Many ships have been lost there as well!!
Since they love the idea of building a challenging rail, they have decided to name it the Bermuda Triangle. The triangle will be connected to a ramp on each side of the triangle, so that students will come down the ramp onto the rails. There they will either succeed or be lost at sea!
Marc drew the following picture of the triangle.
The triangle has three angles and the boys want to reproduce these angles in their structure. The first angle \begin{align*}B\end{align*}
Marc can’t remember the measure of angle \begin{align*}A\end{align*}. He thinks there is a way to figure this out, but he can’t remember what it is. Do you know? Pay attention in this lesson so you can help Marc figure this out. There is way to do it without looking up the answer!!
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills:
- Classify triangles by angles.
- Classify triangles by sides.
- Draw specified triangles using a ruler and a protractor.
- Find unknown angle measures in given triangles.
Teaching Time
This next lesson is all about triangles, the prefix “tri” means three-triangle means three angles.
In this lesson you will learn how to classify or organize triangles in a couple of different ways. The first way that we are going to classify triangles is according to their angles.
I. Classify Triangles by Angles
When we classify a triangle according to its angles, we look at the angles inside the triangle. We will be using the number of degrees in these angles to classify the triangle. Let’s look at a picture of a triangle to explain.
Here is a triangle. We can look at the measure of each angle inside the triangle to figure out what kind of triangle it is. There are four types of triangles based on angle measures.
What are the four kinds of triangles?
The first type of triangle is a right triangle. A right triangle is a triangle that has one right angle and two acute angles. One of the angles in the triangle measures \begin{align*}90^\circ\end{align*} and the other two angles are less than 90. Here is a picture of a right triangle.
Can you figure out which angle is the \begin{align*}90^\circ\end{align*} one just by looking at it?
Sure, you can see that the 90 degree angle is the one in the bottom left corner. You can even draw in the small box to identify it as a 90 degree angle. If you look at the other two angles you cans see that those angles are less than 90 degrees and are acute.
Here we have one \begin{align*}90^\circ\end{align*} angle and two \begin{align*}45^\circ\end{align*} angles. We can find the sum of the three angles.
\begin{align*}90 + 45 + 45 = 180^\circ\end{align*}
The sum of the three angles of a triangle is equal to \begin{align*}180^\circ\end{align*}.
The second type of triangle is an equiangular triangle. If you look at the word “equiangular” you will see that the word “equal” is right in the word. This means that all three of the angles in a equiangular triangle are equal.
The three angles of this triangle are equal. This is an equiangular triangle.
In an equiangular triangle, all of the angle measures are the same. We know that the sum of the three angles is equal to \begin{align*}90^\circ\end{align*}, therefore, for all three angles to be equal, each angle must be equal to \begin{align*}60^\circ\end{align*}.
\begin{align*}60 + 60 + 60 = 180^\circ\end{align*}
The sum of the angles is equal to \begin{align*}180^\circ\end{align*}.
The next type of triangle is an acute triangle. The definition of an acute triangle is in the name “acute.” All three angles of the triangle are less than 90 degrees. Here is an example of an acute triangle.
All three of these angles measure less than 90 degrees.
\begin{align*}33 + 80 + 67 = 180^\circ\end{align*}
The sum of the angles is equal to \begin{align*}180^\circ\end{align*}.
The last type of triangle that we are going to learn about is called an obtuse triangle. An obtuse triangle has one angle that is obtuse or greater than 90 and two angles that are less than 90 or are acute.
\begin{align*}130 + 25 + 25 = 180^\circ\end{align*}
The sum of the angles is equal to \begin{align*}180^\circ\end{align*}.
Now it is time to practice. Identify each type of triangle according to its angles.
- A triangle with angles that are all 60 degrees is called _________________.
- A triangle with one angle that is 90 degrees is called _________________.
- A triangle with one angle that is 120 degrees is called _______________.
Take a few minutes to check your work with a friend.
II. Classify Triangles by Sides
You just learned how to look inside the triangle at its angles to help classify a triangle. Well, we can also look at the lengths of the sides to help us classify triangles.
No they aren’t the same. Let’s look at how we can classify triangles according to side length.
The first triangle to think about is an equilateral triangle. An equilateral triangle has side lengths that are the same. Let’s look at an example.
These little lines let you know that the side lengths are the same. Sometimes you will see these and sometimes you won’t. You may have to figure it out on your own or by measuring with a ruler.
The second type of triangle is a scalene triangle. A scalene triangle is a triangle where the lengths of all three sides are different. Here is an example of a scalene triangle.
Here you can see that all three sides of the triangle are different lengths. This is called a scalene triangle.
The third type of triangle is an isosceles triangle. An isosceles triangle has two side lengths that are the same and one side length that is different. Here is an example of an isosceles triangle.
Classify these three triangles on your own. Classify them according to their side lengths.
1.
2.
3.
Take a few minutes to check your work with a peer.
III. Draw Specified Triangles Using a Ruler and a Protractor
In this lesson you have been working with triangles that have already been drawn for you. You have learned how to classify triangles according to their angles and according to their sides. You also know what the sum of the angles inside or interior angles of a triangle is equal to. This information is going to be important as you begin drawing your own triangles.
How can we draw a specific triangle?
We can draw a specific type of triangle using a ruler and a protractor. Remember in the last lesson that a protractor is used to measure angles. We can use the protractor to figure out the measure of an angle and then draw in the rest of the triangle with the ruler.
Let’s look at an example.
Example
Draw an obtuse triangle.
Remember that an obtuse triangle has one angle that is greater than 90 degrees.
To draw an obtuse triangle, begin by drawing the obtuse angle. Use your protractor to measure an angle that is greater than 90 degrees.
Here is a protractor where an angle that is 105 degrees has been drawn. We can call this an obtuse angle because it is greater than 90 degrees but less than 180 degrees. Next, we can draw in the rest of the triangle using our ruler.
Here is our obtuse triangle.
Next we can draw an acute triangle. An acute triangle has all three angles that are smaller than 90 degrees. We will need to be careful as we draw and measure this triangle. Let’s begin with one angle.
The first angle that I have drawn here is \begin{align*}30^\circ\end{align*}. I can draw in the other two angles by making sure that they are less than 90 degrees.
Here is an acute triangle.
You can draw any type of triangle that you wish by using a protractor and a ruler.
Practice drawing a few of these on your own.
- Draw an acute triangle.
- Draw an equiangular triangle.
- Draw an obtuse triangle.
Take a few minutes to check your work with a peer.
IV. Find Unknown Angle Measures in Given Triangles
You know that the sum of the angles of a triangle is equal to \begin{align*}180^\circ\end{align*}. What happens if you know two but not all three of the measures of a triangle? How can you figure out the measure of the missing angle?
What does this look like?
To understand how to do this better, let’s look at an example.
Example
Now we can tell that this is a right triangle and that one of the angles is equal to 90 degrees. To figure out the measure of the missing angle, we have used a variable to represent the unknown quantity. Here is our equation.
\begin{align*}55 + 90 + x & = 180 \\ 145 + x & = 180 \\ 180 - 145 & = x \\ x & = 35^\circ .\end{align*}
Our answer is \begin{align*}35^\circ\end{align*}.
Practice finding the missing angle in the following triangles.
1.
2.
Take a few minutes to check your answers.
Real Life Example Completed
The Bermuda Triangle
Now it is time for you to apply what you have learned. Reread the problem and underline the important information. Then apply what you have learned about triangles and the measure of their angles to help Marc figure out his missing angle measure.
Isaac and Marc are continuing their work on their skate park. They are both fascinated by the Bermuda Triangle and have decided to name one of the parts of their design after this triangle.
The Bermuda Triangle is located in an area of water right around Bermuda. There have been many mysteries surrounding this area of the ocean. Many ships have been lost there as well!!
Since they love the idea of building a challenging rail, they have decided to name it the Bermuda Triangle. The triangle will be connected to a ramp on each side of the triangle, so that students will come down the ramp onto the rails. There they will either succeed or be lost at sea!
Marc drew the following picture of the triangle.
The triangle has three angles and the boys want to reproduce these angles in their structure. The first angle \begin{align*}B\end{align*} is equal to \begin{align*}62^\circ\end{align*}, the second angle \begin{align*}C\end{align*} is equal to \begin{align*}63^\circ\end{align*}.
Marc can’t remember the measure of angle \begin{align*}A\end{align*}. He thinks there is a way to figure this out, but he can’t remember what it is.
The sum of the interior angles of a triangle is equal to \begin{align*}180^\circ\end{align*}.
Marc knows the measure of two of the angles of the triangle. Therefore, he can write an equation to figure out the measure of the third angle.
\begin{align*}63 + 62 + x = 180\end{align*}
The variable \begin{align*}x\end{align*} is used to represent the measure of angle \begin{align*}A\end{align*}. Marc is working to find the measure of angle \begin{align*}A\end{align*}.
\begin{align*}125 + x & = 180 \\ 180 - 125 & = 55\end{align*}
The measure of Angle \begin{align*}A\end{align*} is \begin{align*}55^\circ\end{align*}
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Triangle
- a three sided figure with three angles. The prefix “tri”means three.
- Acute Triangle
- all three angles are less than 90 degrees.
- Right Triangle
- One angle is equal to 90 degrees and the other two are acute angles.
- Obtuse Triangle
- One angle is greater than 90 degrees and the other two are acute angles.
- Equiangular Triangle
- all three angles are equal
- Scalene Triangle
- all three side lengths are different
- Isosceles Triangle
- two side lengths are the same and one is different
- Equilateral Triangle
- all three side lengths are the same
- Protractor
- A tool used to measure angles
- Interior angles
- the angles inside a figure
Technology Integration
James Sousa, Angle Relationships and Types of Triangles
Time to Practice
Directions: Classify each triangle according to its angles.
1.
2.
3.
4.
5.
Directions: Answer the following questions using what you have learned about triangles their angles and side lengths.
6. If a triangle is a right triangle, then how many angles are acute?
7. How many angles in a right triangle are right angles?
8. How many degrees are there in a right triangle?
9. What is an obtuse angle?
10. How many obtuse angles are in an obtuse triangle?
11. If there is one obtuse angle, how many angles are acute?
12. If a triangle is equiangular, what is the measure of all three angles?
13. What does the word “interior angle” mean?
14. True or false. The side lengths of a scalene triangle are all equal.
15. True or false. The side lengths of a scalene triangle are all different.
16. True or false. The side lengths of an equilateral triangle are all equal.
17. True or false. An isosceles triangle has two side lengths the same and one different.
18. True or false. A scalene triangle can also be an isosceles triangle.
Directions: Critical Thinking - Each question combines information about the angles and side lengths. Answer each question carefully.
19. True or false. If a triangle is equiangular, it can also be equilateral.
20. True or false. A scalene triangle can not be an equilateral triangle.
21. True or false. The word “equiangular” applies to side lengths.
22. True or false. An isosceles triangle can be an obtuse or acute triangle.
23. A ___________________ is a tool used to measure angles.
24. A __________ angle is equal to 90 degrees.
25. A __________angle is equal to 180 degrees.
26. An ________ angle is less than 90 degrees.
27. An _________ angle is greater than 90 but less than 180 degrees.
28. The prefix “tri” means ______________.
29. How many angles are there in a triangle?
30. What is the sum of the interior angles of a triangle?