- Angle Bisector Definition
- How to Construct an Angle Bisector
- Angle Bisector Theorem
- Ratios & Proportions
- Angle Bisector Examples
- Angle Bisector of a Triangle

The **Angle Bisector Theorem** helps you find unknown lengths of sides of triangles, because an angle bisector divides the side opposite that angle into two segments that are proportional to the triangle's other two sides.

Draw $\u25b3ABC$ on a piece of paper. Interior angles $A,B,C$ have opposite sides $a,b,c$.

Get some linear object and put one endpoint at $\angle A$. Allow its other end to cross side $a$. Divide $\angle A$ into two equal angles. You **bisected** $\angle A$.

The linear object is an **angle bisector**. When it crossed side $a$ opposite $\angle A$, it divided $\u25b3ABC$ into two smaller triangles and divided side $a$ in two.

Replace your object with a drawn line segment or ray. Where the angle bisector crosses side $a$, label that point $D$. The angle bisector is now line segment $AD$ and creates two smaller triangles, $\u25b3ACD$ and $\u25b3ABD$. Side $a$ is now two line segments, $CD$ and $DB$.

Here is one version of the Angle Bisector Theorem:

**An angle bisector of a triangle** divides the interior angle's opposite side into two segments that are proportional to the other two sides of the triangle.

Angle bisector $AD$ cuts side $a$ into two line segments, $CD$ and $DB$. $CD$ and $DB$ relate to sides $b$ ($CA$) and $c$ ($BA$) in the same proportion as $CA$ and $BA$ relate to each other. $\u25b3ACD$ and $\u25b3ABD$, created by the angle bisector, are also similarly proportional.

Ratios compare values. You can establish ratios between sides $CA$ and $BA$, and line segments $CD$ and $DB$. **Proportions** compare ratios; you can learn if two ratios are equal.

For $\u25b3ABC$ with angle bisector $AD$, sides $CA$ and $BA$, and side $a$ divided into $CD$ and $DB$, we can set up ratios between the sides and line segments and compare them:

$\frac{CD}{DB}=\frac{CA}{BA}$

Line segment $CD$ (from angle bisector $AD$) has the same ratio to line segment $DB$ as the triangle's side $CA$ has to side $BA$.

Be careful to set up the ratios correctly. $CD$ is the smaller of the two line segments, so it is the numerator of our ratio. $DB$ is the longer line segment. So $CD$ is to $DB$ as the triangle's shorter side $CA$, is to the triangle's longer side $BA$.

Other ratios emerge, too. Compare line segments of the side divided by the angle bisector to the remaining sides:

$\frac{CD}{CA}=\frac{DB}{BA}$

The relationship of the two sides of the new, smaller $\u25b3CDA$ is the same relationship as the two sides of new, smaller $\u25b3DBA$. They are in the same proportions.

How do you know a line segment extending from an interior angle is an angle bisector? **Check the ratios**.

Suppose we are told a line segment $AD$ divides side $a$ into $CD$ and $DB$, of lengths 10 cm and 30 cm. We are also told side $CA$ is 30 cm and side $BA$ is 90 cm. See if the ratios are proportional to each other:

$\frac{CD}{DB}=\frac{CA}{BA}$

$\frac{10}{30}=\frac{30}{90}$

We see $\frac{10}{30}$ is the same ratio as $\frac{30}{90}$, so $AD$ is an angle bisector.

If we know the length of original sides $a$ and $b$, we can use the Angle Bisector Theorem to find the unknown length of side $c$. The angle bisector divides side $a$ into $CD$ and $DB$ (the total length of side $a$, $CB$).

Assume these lengths:

$CD=10$

$DB=20$

$CA=25$

Recall our ratios and substitute the values:

$\frac{CD}{CA}=\frac{DB}{BA}$

$\frac{10}{25}=\frac{20}{BA}$

Using cross multiplication ($25\times 20=500$) and then division ($\frac{500}{10}$) , we get $BA$ = 50 meters.

You are now able to define the Angle Bisector Theorem, use ratios and proportions to verify an angle is a bisector, use the Angle Bisector Theorem to find the unknown lengths of sides of triangles, and identify an angle bisector by evaluating the lengths of the sides of the triangle.

Once you view the video, examine the graphics, and read the lesson, you will be able to:

- Define the Angle Bisector Theorem
- Use ratios and proportion to verify angle bisectors
- Identify angle bisectors using the lengths of the sides of triangles
- Find the unknown lengths of sides of triangles

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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