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? After completing this Concept, you'll be able to solve such problems.

### Watch This

CK-12 Foundation: Chapter7ProportionswithAngleBisectorsA

James Sousa: Triangle Angle Bisector Theorem

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

By definition, \begin{align*}\overrightarrow{AC}\end{align*}

**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.

#### Example A

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

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

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

#### Example B

Determine the value of \begin{align*}x\end{align*}

You can set up this proportion just 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*}

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter7ProportionswithAngleBisectorsB

### 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*}

### Explore More

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 \begin{align*}\frac{5}{a}=\frac{6}{5}\end{align*}

Solve for the unknown variable.

### Answers for Explore More Problems

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