### 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 \begin{align*}\triangle BAC \cong \triangle CAD\end{align*}

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.

\begin{align*} \frac{20}{y}&=\frac{15}{28-y}\\ 15y&=20(28-y)\\ 15y&=560-20y\\ 35y&=560\\ y&=16\end{align*}

#### Example 2

Fill in the missing variable:

Set up a proportion and solve.

\begin{align*} \frac{12}{z}&=\frac{15}{9-z}\\ 15z&=12(9-z)\\ 15z&=108=12z\\ 27z&=108\\ z&=4\end{align*}

#### Example 3

Find \begin{align*}x\end{align*}.

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

\begin{align*}\frac{9}{x} &= \frac{21}{14}\\ 21x &= 126\\ x &= 6\end{align*}

#### Example 4

Find the value of \begin{align*}x\end{align*} that would make the proportion true.

You can set up this proportion like the previous example.

\begin{align*}\frac{5}{3} &= \frac{4x+1}{15}\\ 75 &= 3(4x+1)\\ 75 &= 12x+3\\ 72 &= 12x\\ 6 &= x\end{align*}

#### Example 5

Find the missing variable:

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

\begin{align*}\frac{12}{4}&=\frac{x}{3}\\ 36&=4x\\ x&=9\end{align*}

### 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