### Angle Bisector Theorem

When an angle within a triangle is bisected, the bisector divides the triangle proportionally. This idea is called the **Angle Bisector Theorem**.

**Angle Bisector Theorem:** If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides.

If

What if you were told that a ray was an angle bisector of a triangle? How would you use this fact to find unknown values regarding the triangle's side lengths?

### Examples

#### Example 1

Fill in the missing variable:

Set up a proportion and solve.

#### Example 2

Fill in the missing variable:

Set up a proportion and solve.

#### Example 3

Find

The ray is the angle bisector and it splits the opposite side in the same ratio as the other two sides. The proportion is:

#### Example 4

Find the value of

You can set up this proportion like the previous example.

#### Example 5

Find the missing variable:

Set up a proportion and solve like in the previous examples.

### Review

Find the value of the missing variable(s).

Solve for the unknown variable.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.10.

### Resources