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# Proportions with Angle Bisectors

## Angle bisectors divide triangles proportionally.

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Practice Proportions with Angle Bisectors
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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.

### Guidance

When an angle within a triangle is bisected, the bisector divides the triangle proportionally

By definition, \begin{align*}\overrightarrow{AC}\end{align*} divides \begin{align*}\angle BAD\end{align*} equally, so \begin{align*}\angle BAC \cong \angle CAD\end{align*}. The proportional relationship is \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 \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:

#### Example B

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

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

#### Example C

Find the missing variable:

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

Watch this video for help with the Examples above.

### Vocabulary

Pairs of numbers are proportional if they are in the same ratio. An angle bisector is a ray that divides an angle into two congruent angles.

### Guided Practice

Find the missing variables:

1.

2.

3.

1. Set up a proportion and solve.

2. Set up a proportion and solve.

3. Set up a proportion and solve.

### Practice

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

Solve for the unknown variable.

### 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”.