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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 use the Angle Bisector Theorem to solve such problems.

### Watch This

First watch this video.

Now watch this video.

### 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 $\triangle BAC \cong \triangle CAD$ , then $\frac{BC}{CD} = \frac{AB}{AD}$ .

#### Example A

Find $x$ .

The ray is the angle bisector and it splits the opposite side in the same ratio as the other two sides. The proportion is:

$\frac{9}{x} &= \frac{21}{14}\\21x &= 126\\x &= 6$

#### Example B

Find the value of $x$ that would make the proportion true.

You can set up this proportion like the previous example.

$\frac{5}{3} &= \frac{4x+1}{15}\\75 &= 3(4x+1)\\75 &= 12x+3\\72 &= 12x\\6 &= x$

#### Example C

Find the missing variable:

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

$\frac{12}{4}&=\frac{x}{3}\\ 36&=4x\\ x&=9$

### Guided Practice

Find the missing variables:

1.

2.

3.

1. Set up a proportion and solve.

$\frac{20}{8}&=\frac{25}{y}\\ 20y&=200 \\ y&=10$

2. Set up a proportion and solve.

$\frac{20}{y}&=\frac{15}{28-y}\\ 15y&=20(28-y)\\ 15y&=560-20y\\ 35y&=560\\ y&=16$

3. Set up a proportion and solve.

$\frac{12}{z}&=\frac{15}{9-z}\\ 15z&=12(9-z)\\ 15z&=108=12z\\ 27z&=108\\ z&=4$

### Practice

Find the value of the missing variable(s).

Solve for the unknown variable.

### Vocabulary Language: English Spanish

angle bisector

angle bisector

A ray that divides an angle into two congruent angles.
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”.