<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Skip Navigation
Our Terms of Use (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use.

Proportions with Angle Bisectors

Angle bisectors divide triangles proportionally.

Atoms Practice
Estimated5 minsto complete
Practice Proportions with Angle Bisectors
Estimated5 minsto complete
Practice Now
Turn In
Proportions with Angle Bisectors

Proportions with Angle Bisectors 

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.












Solving for Unknown Values 

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

Solving for an Unknown Value that will make a Proportion True 

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







Finding a Missing Variable 

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








Find the missing variables:

Example 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

Example 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

Example 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.

Review (Answers)

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

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More


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”.

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Proportions with Angle Bisectors.
Please wait...
Please wait...