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# Triangle Classification

## Categories of triangles based on angle measurements or the number of congruent sides.

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Triangle Classification
License: CC BY-NC 3.0

Kevin is constructing a model. The model has several pieces that are shaped like triangles. He notices that triangles have a lot of different sizes and variations in appearance. He decides to make a pile for each different classification of triangles. He finds one triangle that has a 110o\begin{align*}110^o\end{align*} angle. How should he classify the triangle?

In this concept, you will learn how to classify triangles.

### Classifying Triangles

The angles in a triangle can vary a lot in size and shape, but they always total 180\begin{align*}180^\circ\end{align*}. The angles are used to classify triangles. A triangle can either be acute, obtuse or right.

Acute triangles have three angles that measure less than 90\begin{align*}90^\circ\end{align*}. Below are a few examples of acute triangles.

Notice that each angle in the triangles above is less than 90\begin{align*}90^\circ\end{align*}, but the total for each triangle is still 180\begin{align*}180^\circ\end{align*}.

A triangle that has an obtuse angle is classified as an obtuse triangle. This means that one angle in the triangle measures more than 90\begin{align*}90^\circ\end{align*}. Here are three examples of obtuse triangles.

You can see that obtuse triangles have one wide angle that is greater than 90\begin{align*}90^\circ\end{align*}. Still, the three angles in obtuse triangles always add up to 180\begin{align*}180^\circ\end{align*}. Only one angle must be obtuse to make it an obtuse triangle.

The third kind of triangle is a right triangle. Right triangles have one right angle that measures exactly 90\begin{align*}90^\circ\end{align*}. Often, a small box in the corner tells you when an angle is a right angle. Let’s examine a few right triangles.

Once again, even with a right angle, the three angles still total 180\begin{align*}180^\circ\end{align*}.

One short cut is to compare the angles to 90\begin{align*}90^\circ\end{align*}. If an angle is exactly 90\begin{align*}90^\circ\end{align*}, the triangle must be a right triangle. If any angle is more than 90\begin{align*}90^\circ\end{align*}, the triangle must be an obtuse triangle. If there are no right or obtuse angles, the triangle must be an acute triangle

Let's look at an example.

Label the triangle as acute, obtuse or right.

License: CC BY-NC 3.0

First, look at the angles and compare the angles to 90o\begin{align*}90^o\end{align*}.

All of the angles are less than 90o\begin{align*}90^o\end{align*}.

Next, list the classifications of triangles.

Triangles can be acute, right or obtuse.

Then, select the classification that fits the criteria.

Acute.

The answer is that the triangle is an acute triangle.

Triangles can also be classified by the lengths of their sides.

A triangle with three equal sides is an equilateral triangle. It doesn’t matter how long the sides are, as long as they are all congruent, or equal. Here are a few examples of equilateral triangles.

An isosceles triangle has two congruent sides. It doesn’t matter which two sides, any two will do. Let’s look at a few examples of isosceles triangles.

The third type of triangle is a scalene triangle. In a scalene triangle, none of the sides are congruent.

Let's look at an example.

Classify the triangle as equilateral, isosceles, or scalene.

License: CC BY-NC 3.0

First, examine the lengths of the sides to see if any sides are congruent.

Two sides are 7 meters long, but the third side is shorter.

Then, classify the triangle.

This triangle is an isosceles triangle.

### Examples

#### Example 1

Earlier, you were given a problem about Kevin's model.

He has one triangle that has a 110o\begin{align*}110^o\end{align*} angle. Classify the triangle as acute, obtuse or right.

First, look at the given angle and compare it to 90o\begin{align*}90^o\end{align*}.

The angle is larger than 90o\begin{align*}90^o\end{align*}.

Next, list the classifications of triangles.

Triangles can be acute, right or obtuse.

Then, select the classification that fits the criteria.

Obtuse.

The answer is that the triangle is an obtuse triangle.

#### Example 2

Classify the triangle as equilateral, isosceles or scalene.

License: CC BY-NC 3.0

First, examine the lengths of the sides to see if any sides are congruent.

All three sides equal 4.5 inches.

Then, determine if the the triangle is equilateral, isosceles or scalene.

Equilateral.

The answer is that the triangle is equilateral.

#### Example 3

Use the angles to classify the triangle.

License: CC BY-NC 3.0

First, look at the angles and compare the angles to 90o\begin{align*}90^o\end{align*}.

One of the angles is equal to 90o\begin{align*}90^o\end{align*}.

Next, list the classifications of triangles.

Triangles can be acute, right or obtuse.

Then, select the classification that fits the criteria.

Right.

The answer is that the triangle is a right triangle.

#### Example 4

Use the angles to classify the triangle.

License: CC BY-NC 3.0

First, look at the angles and compare the angles to 90o\begin{align*}90^o\end{align*}.

None of the angles are 90o\begin{align*}90^o\end{align*} . One of the angles is larger than 90o\begin{align*}90^o\end{align*}.

Next, list the classifications of triangles.

Triangles can be acute, right or obtuse.

Then, select the classification that fits the criteria.

Obtuse.

The answer is that the triangle is an obtuse triangle.

#### Example 5

Identify the triangle as equilateral, isosceles or scalene.

License: CC BY-NC 3.0

First, examine the lengths of the sides to see if any sides are congruent.

None of the sides are congruent.

Then, determine if the the triangle is equilateral, isosceles or scalene.

Scalene.

The answer is that the triangle is scalene.

### Review

Find the measure of angle H\begin{align*}H\end{align*} in each figure below.

Identify each triangle as right, acute, or obtuse.

Identify each triangle as equilateral, isosceles, or scalene.

Use what you have learned to answer each question.

1. True or false. An acute triangle has three sides that are all different lengths.
2. True or false. A scalene triangle can be an acute triangle as well.
3. True or false. An isosceles triangle can also be a right triangle.
4. True or false. An equilateral triangle has three equal sides.
5. True or false. An obtuse triangle can have multiple obtuse angles.
6. True or false. A scalene triangle has three angles less than 90 degrees.
7. True or false. A triangle with a 100\begin{align*}100^\circ\end{align*} angle must be an obtuse triangle.
8. True or false. The angles of an equilateral triangle are also equal in measure.

To see the Review answers, open this PDF file and look for section 8.8.

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### Vocabulary Language: English

TermDefinition
Acute Triangle An acute triangle has three angles that each measure less than 90 degrees.
Congruent Congruent figures are identical in size, shape and measure.
Exterior angles An exterior angle is the angle formed by one side of a polygon and the extension of the adjacent side.
Interior angles Interior angles are the angles inside a figure.
Isosceles Triangle An isosceles triangle is a triangle in which exactly two sides are the same length.
Obtuse Triangle An obtuse triangle is a triangle with one angle that is greater than 90 degrees.
Right Angle A right angle is an angle equal to 90 degrees.
Scalene Triangle A scalene triangle is a triangle in which all three sides are different lengths.
Triangle A triangle is a polygon with three sides and three angles.
Equilateral A polygon is equilateral if all of its sides are the same length.
Equiangular A polygon is equiangular if all angles are the same measure.

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