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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 divides BAD equally, so BACCAD. The proportional relationship is BCCD=ABAD.

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.

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

Solving for an Unknown Value that will make a Proportion True 

Determine the value of x 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

Finding a Missing Variable 

Find the missing variable:

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

12436x=x3=4x=9

Examples

Find the missing variables:

Example 1

Set up a proportion and solve.

20820yy=25y=200=10

Example 2

Set up a proportion and solve. 

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

Example 3

3.

Set up a proportion and solve.

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

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. What did Casey do wrong? Write the correct proportion and solve for a.

Solve for the unknown variable.

Review (Answers)

To view the Review answers, open this PDF file and look for section 7.10. 

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Vocabulary

TermDefinition
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”.

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