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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson and learn how you can use the angle bisector theorem to help you find the lengths of missing sides. Also learn how you can identify an angle bisector just by looking at the lengths of the sides of a triangle.

Picture a triangle. Now picture one of the triangle's angles being split into two equal smaller triangles. That line that was used to cut the angle in half is called the angle bisector. When you do this to a triangle, it actually divides the triangle's sides in a unique way. The angle bisector theorem tells us what that way is.

The **angle bisector theorem** tells us that the angle bisector divides the triangle's sides proportionally. When you have an angle bisector, you also have two smaller triangles. It is these two smaller triangles that are proportional.

We can write the angle bisector theorem using ratios. We can then use this ratio to help us solve problems.

So, if we have triangle ABC, where line segment AD is the angle bisector of angle A, then the angle bisector theorem tells us that the ratio of the length of BD to the length of DC is equal to the ratio of the length of AB to the length of AC.

The first way we can use the angle bisector theorem is in checking whether or not a line segment is really an angle bisector or not. To do this, we plug in the numbers that we are given for the sides to see if they work in the ratio. Are the ratios equal to each other? If they are, then yes, the line segment is an angle bisector.

So say, for instance, we have a triangle with a line segment AD that seems to cut angle A in half. We can check to see if it really does by plugging in the lengths of the sides into our ratio. We are given side BD = 3, side DC = 1, side AB = 9, and side AC = 3. Let's plug these numbers into our angle bisector theorem ratio to see if they check out.

3/1 = 9/3

3/1 = 3/1

They do check out since 9/3 simplifies to 3/1. Both the 9 and the 3 can be divided by a 3. The two sides are equal, and so yes, line AD is an angle bisector.

Another way in which we can use the angle bisector theorem is when we need to find the length of a missing side. If we know the lengths of three of the sides mentioned in the angle bisector theorem ratio, we can find the length of the fourth side using the ratio.

Let's see how this works.

If we have a triangle ABC with angle bisector AD, we can find the missing length of side AB if we are given the lengths of sides BD, DC, and AC. We are given side BD = 5, DC = 10, and AC = 20. We can plug in these values into our ratio, and we will put an *x* for our side AB.

5/10 = x/20

Looking at this ratio, we see that we can use algebra to help us solve for our missing length. So we go ahead and multiply both sides by 20 to get *x* by itself.

(5/10)*20 = (*x*/20)*20

100/10 = *x*

10 = *x*

We have our answer. The missing length of side AB is 10.

What have we learned? We've learned that the **angle bisector theorem** tells us the angle bisector divides a triangle into two proportional triangles whose sides are proportional. If we have a triangle ABC with angle bisector AD bisecting angle A, the proportion can be written in ratio form as BD/DC = AB/AC where BD, DC, AB, and AC are all line segments. We can use the angle bisector theorem and its ratio to help us check whether a line segment is an angle bisector or not and to help us find missing lengths.

After completing this lesson, you should be able to:

- Define the angle bisector theorem
- Use ratios confidently to confirm the angle bisector theorem

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Geometry: High School15 chapters | 160 lessons

- Ratios and Proportions: Definition and Examples 5:17
- Geometric Mean: Definition and Formula 5:15
- Angle Bisector Theorem: Definition and Example 4:58
- Similar Polygons: Definition and Examples 8:00
- The Transitive Property of Similar Triangles 4:50
- Triangle Proportionality Theorem 4:53
- Constructing Similar Polygons 4:59
- Properties of Right Triangles: Theorems & Proofs 5:58
- The Pythagorean Theorem: Practice and Application 7:33
- The Pythagorean Theorem: Converse and Special Cases 5:02
- Similar Triangles & the AA Criterion 5:07
- What is a Polygon? - Definition, Shapes & Angles 6:08
- Go to High School Geometry: Similar Polygons

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