What if you were told that a ray was an angle bisector of a triangle? How would you use this fact to find unknow values regarding the triangle's side lengths? After completing this Concept, you'll be able to use the Angle Bisector Theorem to solve such problems.

### Watch This

CK-12 Foundation: Proportions with Angle Bisectors

First watch this video.

James Sousa: Triangle Angle Bisector Theorem

Now watch this video.

James Sousa: Using the Triangle Angle Bisector Theorem to Determine Unknown Values

### Guidance

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*}, then \begin{align*}\frac{BC}{CD} = \frac{AB}{AD}\end{align*}.

#### Example A

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 B

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 C

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*}

CK-12 Foundation: Proportions with Angle Bisectors

### Guided Practice

Find the missing variables:

1.

2.

3.

**Answers:**

1. Set up a proportion and solve.

\begin{align*} \frac{20}{8}&=\frac{25}{y}\\ 20y&=200 \\ y&=10 \end{align*}

2. 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*}

3. 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*}

### Practice

Find the value of the missing variable(s).

Solve for the unknown variable.