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# Proportions with Angle Bisectors

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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, $\overrightarrow{AC}$ divides $\angle BAD$ equally, so $\angle BAC \cong \angle CAD$ . The proportional relationship is $\frac{BC}{CD}=\frac{AB}{AD}$ .

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 $x$ .

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

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

#### Example B

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

You can set up this proportion just 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.

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

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.

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

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).

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

Solve for the unknown variable.