Angle Bisector Theorem & Definition

Table of Contents

1. Angle Bisector Definition
2. How to Construct an Angle Bisector
3. Angle Bisector Theorem
4. Ratios & Proportions
5. Angle Bisector Examples
6. Angle Bisector of a Triangle

Angle Bisector Definition

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.

How to Construct an Angle Bisector

Draw $△ABC$ 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 $△ABC$ 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, $△ACD$ and $△ABD$. Side $a$ is now two line segments, $CD$ and $DB$.

Angle Bisector Theorem

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. $△ACD$ and $△ABD$, created by the angle bisector, are also similarly proportional.

Ratios & Proportions

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 $△ABC$ 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 $△CDA$ is the same relationship as the two sides of new, smaller $△DBA$. They are in the same proportions.

Angle Bisector Examples

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.

Angle Bisector of a Triangle

Using the Angle Bisector Theorem to Find An Unknown Side

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.

Lesson Summary

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.

Next Lesson:

How To Solve Proportions