Angle Bisector Theorem: Definition and Example

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  • 0:08 Angle Bisector Theorem
  • 0:51 The Ratio
  • 1:22 Is It an Angle Bisector?
  • 2:42 Finding a Missing Side
  • 4:07 Lesson Summary
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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.

Angle Bisector Theorem

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.

The Ratio

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.

Is It an Angle Bisector?

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.

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