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High School Trigonometry: Tutoring Solution30 chapters | 201 lessons

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Lesson Transcript

Instructor:
*Elizabeth Often*

Elizabeth has taught high school math for over 10 years, and has a master's in secondary math education.

Have you ever wondered how we classify triangles? In this lesson, we'll learn the definition of a scalene triangle, understand its properties, and look at some examples.

You have probably seen many triangles in your life. Perhaps you've even noticed that there are many different types of triangles. Some of these triangles have all three sides of the same length, some have two sides of the same length, and in some triangles, all three sides are different lengths.

**Scalene triangles** are triangles with three sides of different lengths. The math term for sides of different triangles is noncongruent sides, so you may also see this phrase in your math book. For example, a triangle with side lengths of 2 cm, 3 cm, and 4 cm would be a scalene triangle. A triangle with side lengths of 2 cm, 2 cm, and 3 cm would not be scalene, since two of the sides have the same length.

The most important property of scalene triangles is that they have three sides of different lengths. However, they have some other important properties, too. Like other triangles, all the angles inside a scalene triangle add up to 180 degrees. And just like all the sides of a scalene triangle have different lengths, all the angles of a scalene triangle have different measures.

Let's take a look at some examples of triangles that we can classify as scalene or not scalene by their angle measures:

- 40 degrees - 50 degrees - 90 degrees is a scalene triangle since all the angle measures are different.
- 60 degrees - 60 degrees - 60 degrees is not a scalene triangle since the angle measures are not all different.
- 120 degrees - 10 degrees - 50 degrees is a scalene triangle since all the angle measures are different.

The triangle on the left is scalene because it has three different angles. The triangle on the right is NOT scalene because it has two angles of the same size.

In addition, there are some other properties that you might find useful as you encounter scalene triangles in math problems.

- The longest side of the triangle is opposite the largest angle. This means that in the 120-10-50 triangle above, the longest side of the triangle is across from the 120-degree angle.
- The shortest side of the triangle is opposite the smallest angle. This means that in the 120-10-50 triangle, the shortest side is located across from the 10-degree angle.

This picture shows these concepts more clearly. In the picture, the bright green side, labeled longest side, is across from the largest angle, *B*. The shortest side, in black, is across from the smallest angle, angle *C*.

Triangles are used in many places in the construction industry because they are very stable. For example, you may see a scalene triangle in a roof truss.

You can also see a scalene triangle if you cut a piece of printer paper from one corner to the other, or if you find the diagonal of your television or computer screen. You may also see scalene triangles when looking at a street map of a city.

Apart from their real-world applications, you are likely to see scalene triangles in your math classes as well. The 30-60-90 triangle is very important in geometry and trigonometry. The relationship between the sides of a 30-60-90 triangle make it useful in problem solving, and this triangle has earned the name 'special right triangle' because of its many applications. It is one of the first triangles you'll study in a trigonometry class.

But even if you aren't interested in trigonometry, problems in your math text may specify that a triangle is scalene and you will need to know the properties we discussed to solve the problem. Additionally, remembering that the longest side is opposite the largest angle can help you with checking your work to see that your answers in math class make sense.

The **scalene triangle** is a type of triangle with sides of three different lengths and angles of three different measurements. In every scalene triangle, the shortest side is opposite the smallest angle and the longest side is opposite the largest angle. This is an important triangle in geometry, trigonometry, and many other fields.

For a triangle to be considered scalene, it needs to have:

- three different angle measures that add up to 180 degrees
- three sides of different lengths, or noncongruent sides

When you are finished, you should be able to:

- Recall the properties of a scalene triangle
- Identify a scalene triangle

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High School Trigonometry: Tutoring Solution30 chapters | 201 lessons

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