# 7.13: Proportions and Angle Bisectors

- Page ID
- 5918

Angle bisectors divide triangles proportionally.

## 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 \(\Delta BAC\cong \Delta CAD\), then \(\dfrac{BC}{CD}=\dfrac{AB}{AD}\).

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?

Example \(\PageIndex{1}\)

Fill in the missing variable:

**Solution**

Set up a **proportion** and solve.

\(\begin{aligned} \dfrac{20}{y}&=\dfrac{15}{28−y} \\ 15y&=20(28−y) \\ 15y&=560−20y \\ 35y&=560 \\ y&=16\end{aligned}\)

Example \(\PageIndex{2}\)

Fill in the missing variable:

**Solution**

Set up a proportion and solve.

\(\begin{aligned}\dfrac{12}{z}&=\dfrac{15}{9−z} \\ 15z&=12(9−z) \\ 15z&=108-12z \\ 27z&=108 \\ z&=4\end{aligned}\)

Example \(\PageIndex{3}\)

Find \(x\).

**Solution**

The ray is the angle bisector and it splits the opposite side in the same **ratio** as the other two sides. The proportion is:

\(\begin{aligned} \dfrac{9}{x}&=\dfrac{21}{14} \\ 21x&=126 \\ x&=6\end{aligned}\)

Example \(\PageIndex{4}\)

Find the value of \(x\) that would make the proportion true.

**Solution**

You can set up this proportion like the previous example.

\(\begin{aligned} \dfrac{5}{3}&=\dfrac{4x+1}{15} \\ 75&=3(4x+1) \\ 75&=12x+3 \\ 72&=12x \\ 6&=x\end{aligned}\)

Example \(\PageIndex{5}\)

Find the missing variable:

**Solution**

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

\(\begin{aligned}\dfrac{12}{4}&=\dfrac{x}{3} \\ 36&=4x \\ x&=9\end{aligned}\)

## 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.

## Resources

## Vocabulary

Term | Definition |
---|---|

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”. |

## Additional Resources

Interactive Element

Video: Using the Properties of the Triangle Angle Bisector Theorem to Determine Unknown Values

Activities: Proportions with Angle Bisectors Discussion Questions

Study Aids: Proportionality Relationships Study Guide

Real World: Triangle Proportionality