### Proportions with Angle Bisectors

When an angle within a triangle is bisected, the bisector divides the triangle proportionally

By definition,

**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.

#### Solving for Unknown Values

Find

Because the ray is the angle bisector it splits the opposite side in the same ratio as the sides. So, the proportion is:

#### Solving for an Unknown Value that will make a Proportion True

Determine the value of

You can set up this proportion just like the previous example.

#### Finding a Missing Variable

Find the missing variable:

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

### Examples

Find the missing variables:

#### Example 1

Set up a proportion and solve.

#### Example 2

Set up a proportion and solve.

#### Example 3

3.

Set up a proportion and solve.

### Review

Find the value of the missing variable(s).

Find the value of each variable in the pictures below.

Find the unknown lengths.

*Error Analysis*

Casey attempts to solve for a in the diagram using the proportion

Solve for the unknown variable.

### Review (Answers)

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