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Included Angle of a Triangle: Definition & Overview

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  • 0:00 What Is an Included Angle?
  • 0:34 Finding the Area of a Triangle
  • 1:41 Included Angles in…
  • 3:40 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

The included angle of a triangle can be found in many geometric theorems and can be very helpful when proving geometric concepts and solving problems. This lesson will define included angle and give some examples of how it can be used. There will be a quiz at the end of the lesson.

What Is an Included Angle?

An included angle is the angle between two sides of a triangle.

It can be any angle of the triangle, depending on its purpose.

The included angle is used in proofs of geometric theorems dealing with congruent triangles. Congruent triangles are two triangles whose sides and angles are equal to each other. You can also use the included angle to determine the area of any triangle as long as you know the lengths of the sides surrounding the angle.

Finding the Area of a Triangle

Included angles can be used to determine the area of a triangle as long as the sides that include the angle are known. The equation to find the area is:

Area = (absinC) / 2

Now let's find the area of this triangle assuming that a = 5, b = 3, and C = 105.

We know that Area = (absinC) / 2, so we just have to plug in the numbers and solve.

We start with:

Area = (5 * 3 * sin105) / 2

That then becomes:

Area = (15 * 0.96593) / 2
Area = 7.24

Example

Let's try another example. This time, we'll find the area of this triangle.

Area = (absinC) / 2

In this case, that means:

Area = (12 * 7 * sin24) / 2

Area = 17.08

Included Angles in Geometric Proofs

Included angles can also be used in geometric proofs. One way they can be used is when dealing with the side-angle-side congruence, which says that If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

An included angle can be used to prove that two triangles are congruent. You might remember from earlier that congruent means that the two triangles have the same shape and size. If you have two triangles and you know that two sides and the included angle are congruent, then you can also know that the entire triangles are congruent to each other.

Example

Are these two triangles congruent? If they are, how do you know?

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