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# Proportions with Angle Bisectors

## Angle bisectors divide triangles proportionally.

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Proportions with Angle Bisectors

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, AC\begin{align*}\overrightarrow{AC}\end{align*} divides BAD\begin{align*}\angle BAD\end{align*} equally, so BACCAD\begin{align*}\angle BAC \cong \angle CAD\end{align*}. The proportional relationship is BCCD=ABAD\begin{align*}\frac{BC}{CD}=\frac{AB}{AD}\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 x\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:

9x21xx=2114=126=6\begin{align*}\frac{9}{x} &= \frac{21}{14}\\ 21x &= 126\\ x &= 6\end{align*}

#### Example B

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

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

537575726=4x+115=3(4x+1)=12x+3=12x=x\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.

12436x=x3=4x=9\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.

1. Set up a proportion and solve.

20820yy=25y=200=10\begin{align*} \frac{20}{8}&=\frac{25}{y}\\ 20y&=200 \\ y&=10 \end{align*}

2. Set up a proportion and solve.

20y15y15y35yy=1528y=20(28y)=56020y=560=16\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.

12z15z15z27zz=159z=12(9z)=108=12z=108=4\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.

1. Error Analysis

Casey attempts to solve for a in the diagram using the proportion 5a=65\begin{align*}\frac{5}{a}=\frac{6}{5}\end{align*}. What did Casey do wrong? Write the correct proportion and solve for a\begin{align*}a\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.

### Vocabulary Language: English

Angle Bisector Theorem

Angle Bisector Theorem

The angle bisector theorem states that if a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
Proportion

Proportion

A proportion is an equation that shows two equivalent ratios.
Ratio

Ratio

A ratio is a comparison of two quantities that can be written in fraction form, with a colon or with the word “to”.