# 9.7: Triangle Classification by Angles

**At Grade**Created by: CK-12

**Practice**Triangle Classification by Angles

Do you know about different types of triangles? Well, Cassie is learning all about them.

"As if protractors weren't bad enough," Cassie complained sitting at the kitchen table.

"What's the matter?" her brother Kyle asked.

"Well, look at this," Cassie said showing him the book. "I have to identify these triangles."

"That's not so bad if you know what to look for," Kyle explained.

What is Kyle talking about? Cassie is puzzled about this.

**But Kyle is right. There are things to look for when classifying triangles. Pay attention to this Concept and you will know what Kyle is talking about by the end of it.**

### Guidance

This next Concept is all about triangles; the prefix “tri” means three-triangle means three 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.

#### Example A

A triangle with angles that are all 60 degrees is called _________________.

**Solution: An Equiangular Triangle**

#### Example B

A triangle with one angle that is 90 degrees is called _________________.

**Solution: A Right Triangle**

#### Example C

A triangle with one angle that is 120 degrees is called _______________.

**Solution: An Obtuse Triangle**

Now back to Cassie and the triangles. Here is the original problem once again.

"As if protractors weren't bad enough," Cassie complained sitting at the kitchen table.

"What's the matter?" her brother Kyle asked.

"Well, look at this," Cassie said showing him the book. "I have to identify these triangles."

"That's not so bad if you know what to look for," Kyle explained.

What is Kyle talking about? Cassie is puzzled about this.

To identify each triangle by angles, Kyle knows that Cassie needs to look at the interior angles of each triangle. Let's use the information that you just learned in this Concept to classify each triangle.

The first one has three angles less than 90, so this is an acute triangle.

The second one has one right angle, therefore it is a right triangle.

The third triangle has one angle greater than 90, so it is an obtuse triangle.

The last triangle has one angle greater than 90, so it is also an obtuse triangle.

### Vocabulary

Here are the vocabulary words in this Concept.

- 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

### Guided Practice

Here is one for you to try on your own.

True or false. An acute triangle can also be an equiangular triangle.

**Answer**

This is true. Because all of the angles in an acute triangle are less than 90 and all of the angles in an equiangular triangle are 60 degrees, an acute triangle can also be an equiangular triangle.

### Video Review

James Sousa, Angle Relationships and Types of Triangles

### Practice

Directions: Classify each triangle according to its angles.

1.

2.

3.

4.

5.

Directions: Classify the following triangle by looking at the sum of the angle measures.

6. \begin{align*}40 + 55 + 45 = 180^\circ\end{align*}

7. \begin{align*}20 + 135 + 25 = 180^\circ\end{align*}

8. \begin{align*}30 + 90 + 60 = 180^\circ\end{align*}

9. \begin{align*}60 + 60 + 60 = 180^\circ\end{align*}

10. \begin{align*}110 + 15 + 55 = 180^\circ\end{align*}

11. \begin{align*}105 + 65 + 10 = 180^\circ\end{align*}

12. \begin{align*}80 + 55 + 45 = 180^\circ\end{align*}

13. \begin{align*}70 + 45 + 65 = 180^\circ\end{align*}

14. \begin{align*}145 + 20 + 15 = 180^\circ\end{align*}

15. \begin{align*}60 + 80 + 40 = 180^\circ\end{align*}

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Acute Triangle

An acute triangle has three angles that each measure less than 90 degrees.Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are the same length.Obtuse Triangle

An obtuse triangle is a triangle with one angle that is greater than 90 degrees.Right Triangle

A right triangle is a triangle with one 90 degree angle.Triangle

A triangle is a polygon with three sides and three angles.### Image Attributions

Here you'll learn how to classify triangles by identifying angles.