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Proportions with Angle Bisectors

Angle bisectors divide triangles proportionally.

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Proportions with Angle Bisectors

Angle Bisector Theorem

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 BACCAD, then BCCD=ABAD.

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?

Examples

Example 1

Fill in the missing variable:

Set up a proportion and solve.

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

Example 2

Fill in the missing variable:

Set up a proportion and solve.

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

Example 3

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:

9x21xx=2114=126=6

Example 4

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

You can set up this proportion like the previous example.

537575726=4x+115=3(4x+1)=12x+3=12x=x

Example 5

Find the missing variable:

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

12436x=x3=4x=9

Review

Find the value of the missing variable(s).

Solve for the unknown variable.

Review (Answers)

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

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Vocabulary

angle bisector

A ray that divides an angle into two congruent angles.

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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