Title: READ: Classifying Triangles Reading
Course: Geometry (A)

Classifying Triangles

Learning Objectives

- Define triangles.
- Classify triangles as acute, right, obtuse, or equiangular.
- Classify triangles as scalene, isosceles, or equilateral.

Introduction

By this point, you should be able to readily identify many different types of geometric objects. You have learned about lines, segments, rays, planes, as well as basic relationships between many of these figures. Everything you have learned up to this point is necessary to explore the classifications and properties of different types of shapes. The next two sections focus on two-dimensional shapes—shapes that lie in one plane. As you learn about polygons, use what you know about measurement and angle relationships in these sections.

Defining Triangles

The first shape to examine is the triangle. Though you have probably heard of triangles before, it is helpful to review the formal definition. A triangle is any closed figure made by three line segments intersecting at their endpoints. Every triangle has three vertices (points at which the segments meet), three sides (the segments themselves), and three interior angles (formed at each vertex). All of the following shapes are triangles.

You may have learned in the past that the sum of the interior angles in a triangle is always . Later we will prove this property, but for now you can use this fact to find missing angles. Other important properties of triangles will be explored in later chapters.

Example 1

Which of the figures below are not triangles?

To solve this problem, you must carefully analyze the four shapes in the answer choices. Remember that a triangle has three sides, three vertices, and three interior angles. Choice A fits this description, so it is a triangle. Choice B has one curved side, so its sides are not exclusively line segments. Choice C is also a triangle. Choice D, however, is not a closed shape. Therefore, it is not a triangle. Choices B and D are not triangles.

Example 2

How many triangles are in the diagram below?

To solve this problem, you must carefully count the triangles of different size. Begin with the smallest triangles. There are small triangles.

Now count the triangles that are formed by four of the smaller triangles, like the one below.

There are a total of seven triangles of this size, if you remember to count the inverted one in the center of the diagram.

Next, count the triangles that are formed by nine of the smaller triangles. There are three of these triangles. And finally, there is one triangle formed by smaller triangles.

Now, add these numbers together.

So, there are a total of triangles in the figure shown.

Classifications by Angles

Earlier in this chapter, you learned how to classify angles as acute, obtuse, or right. Now that you know how to identify triangles, we can separate them into classifications as well. One way to classify a triangle is by the measure of its angles. In any triangle, two of the angles will always be acute. This is necessary to keep the total sum of the interior angles at . The third angle, however, can be acute, obtuse, or right.

This is how triangles are classified. If a triangle has one right angle, it is called a right triangle.

If a triangle has one obtuse angle, it is called an obtuse triangle.

If all of the angles are acute, it is called an acute triangle.

The last type of triangle classifications by angles occurs when all angles are congruent. This triangle is called an equiangular triangle.

Example 3

Which term best describes below?

The triangle in the diagram has two acute angles. But, so is an obtuse angle. If the angle measure were not given, you could check this using the corner of a piece of notebook paper or by measuring the angle with a protractor. An obtuse angle will be greater than (the square corner of a paper) and less than (a straight line). Since one angle in the triangle above is obtuse, it is an obtuse triangle.

Classifying by Side Lengths

There are more types of triangle classes that are not based on angle measure. Instead, these classifications have to do with the sides of the triangle and their relationships to each other. When a triangle has all sides of different length, it is called a scalene triangle.

When at least two sides of a triangle are congruent, the triangle is said to be an isosceles triangle.

Finally, when a triangle has sides that are all congruent, it is called an equilateral triangle. Note that by the definitions, an equilateral triangle is also an isosceles triangle.

Example 4

Which term best describes the triangle below?

A. scalene

B. isosceles

C. equilateral

To classify the triangle by side lengths, you have to examine the relationships between the sides. Two of the sides in this triangle are congruent, so it is an isosceles triangle. The correct answer is B.

Lesson Summary

In this lesson, we explored triangles and their classifications. Specifically, we have learned:

- How to define triangles.
- How to classify triangles as acute, right, obtuse, or equiangular.
- How to classify triangles as scalene, isosceles, or equilateral.

These terms or concepts are important in many different types of geometric practice. It is important to have these concepts solidified in your mind as you explore other topics of geometry and mathematics.