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# Proportions with Angle Bisectors

## Angle bisectors divide triangles proportionally.

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Proportions with Angle Bisectors

### Proportions with Angle Bisectors

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.

#### Solving for Unknown Values

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*}

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

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*}

#### Finding a Missing Variable

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*}

### Examples

Find the missing variables:

#### Example 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*}

#### Example 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*}

#### Example 3

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*}

### Review

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.

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### Vocabulary Language: English

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

A proportion is an equation that shows two equivalent ratios.

Ratio

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