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7.10: Proportions with Angle Bisectors

Difficulty Level: At Grade Created by: CK-12
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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.

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CK-12 Foundation: Chapter7ProportionswithAngleBisectorsA

James Sousa: Triangle Angle Bisector Theorem

James Sousa: Using the Triangle Angle Bisector Theorem to Determine Unknown Values


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

By definition, \begin{align*}\overrightarrow{AC}\end{align*}AC divides \begin{align*}\angle BAD\end{align*}BAD equally, so \begin{align*}\angle BAC \cong \angle CAD\end{align*}BACCAD. The proportional relationship is \begin{align*}\frac{BC}{CD}=\frac{AB}{AD}\end{align*}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.

Example A

Find \begin{align*}x\end{align*}x.

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

\begin{align*}\frac{9}{x} &= \frac{21}{14}\\ 21x &= 126\\ x &= 6\end{align*}9x21xx=2114=126=6

Example B

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

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

\begin{align*}\frac{5}{3} &= \frac{4x+1}{15}\\ 75 &= 3(4x+1)\\ 75 &= 12x+3\\ 72 &= 12x\\ 6 &= x\end{align*}537575726=4x+115=3(4x+1)=12x+3=12x=x

Example C

Find the missing variable:

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

\begin{align*}\frac{12}{4}&=\frac{x}{3}\\ 36&=4x\\ x&=9\end{align*}12436x=x3=4x=9

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter7ProportionswithAngleBisectorsB


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. Set up a proportion and solve.

\begin{align*} \frac{20}{8}&=\frac{25}{y}\\ 20y&=200 \\ y&=10 \end{align*}20820yy=25y=200=10

2. Set up a proportion and solve.

\begin{align*} \frac{20}{y}&=\frac{15}{28-y}\\ 15y&=20(28-y)\\ 15y&=560-20y\\ 35y&=560\\ y&=16\end{align*}20y15y15y35yy=1528y=20(28y)=56020y=560=16

3. Set up a proportion and solve.

\begin{align*} \frac{12}{z}&=\frac{15}{9-z}\\ 15z&=12(9-z)\\ 15z&=108=12z\\ 27z&=108\\ z&=4\end{align*}12z15z15z27zz=159z=12(9z)=108=12z=108=4


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*}5a=65. What did Casey do wrong? Write the correct proportion and solve for \begin{align*}a\end{align*}a.

Solve for the unknown variable.

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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.


A proportion is an equation that shows two equivalent ratios.


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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Difficulty Level:
At Grade
Date Created:
Jul 17, 2012
Last Modified:
Aug 02, 2016
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