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High School Geometry: Help and Review13 chapters | 162 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

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.

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.

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 = (*ab*sin*C*) / 2

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

We know that Area = (*ab*sin*C*) / 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 = (*ab*sin*C*) / 2

In this case, that means:

Area = (12 * 7 * sin24) / 2

Area = 17.08

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?**

The answer is yes, the triangles are congruent. You know this because of the side-angle-side theorem. Since two sides and the included angle are congruent, the triangles are congruent.

There is also a **side-angle-side theorem for triangle similarity**. It states If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar. This theorem states that two triangles are **similar**, meaning they are proportional to each other, if there is congruence between one set of angles and the sides that contain those angles are proportional to each other, then the triangles are similar.

**Example**

**Are these triangles similar? If yes, how can it be proven?**

Yes, the triangles are similar. It can be proven by using the side-angle-side theorem for similarity, which says that if two triangles have a congruent angle and the sides surrounding that angle are proportional, then the triangles are similar. These two triangles have a congruent angleâ€”the 90 degree angleâ€”and the sides surrounding that angle are proportionalâ€”the sides of the larger triangle are double that of the smaller triangle. Therefore, the triangles are similar.

An **included angle** is the angle between two sides of a triangle. The included angle is important because it can be used to prove that two triangles are congruent or similar. It also can be used to determine the area of a triangle using the equation:

Area = (*ab*sin*C*) / 2

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High School Geometry: Help and Review13 chapters | 162 lessons

- Area of Triangles and Rectangles 5:43
- Perimeter of Triangles and Rectangles 8:54
- How to Identify Similar Triangles 7:23
- Angles and Triangles: Practice Problems 7:43
- Triangles: Definition and Properties 4:30
- Classifying Triangles by Angles and Sides 5:44
- Interior and Exterior Angles of Triangles: Definition & Examples 5:25
- Constructing the Median of a Triangle 4:47
- Median, Altitude, and Angle Bisectors of a Triangle 4:50
- Constructing Triangles: Types of Geometric Construction 5:59
- Properties of Concurrent Lines in a Triangle 6:17
- How to Find the Height of a Triangle 4:41
- Hypotenuse: Definition & Formula 4:38
- Included Angle of a Triangle: Definition & Overview 4:06
- Interior Angle Theorem: Definition & Formula 4:37
- Median of a Triangle: Definition & Formula 3:12
- Midsegment: Theorem & Formula 4:18
- Percent of Change: Definition, Formula & Examples
- Go to Properties of Triangles: Help and Review

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