What if you were told that is the angle bisector of ? How would you find the length of given the length of ? After completing this Concept, you'll be able to use the Angle Bisector Theorem to solve problems like this one.

### Watch This

First watch this video.

James Sousa: Introduction to Angle Bisectors

Next watch this video.

James Sousa: Proof of the Angle Bisector Theorem

Then watch this video.

James Sousa: Proof of the Angle Bisector Theorem Converse

Finally, watch this video.

James Sousa: Solving For Unknown Values Using Angle Bisectors

### Guidance

An **angle bisector** cuts an angle exactly in half. One important property of angle bisectors is that if a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. This is called the **Angle Bisector Theorem.**

In other words, if bisects , and, then .

The converse of this theorem is also true.

**Angle Bisector Theorem Converse:** If a point is in the interior of an angle and equidistant from the sides, then it lies on the bisector of that angle.

When we construct angle bisectors for the angles of a triangle, they meet in one point. This point is called the **incenter** of the triangle.

#### Example A

Is on the angle bisector of ?

If is on the angle bisector, then and both segments need to be perpendicular to the sides of the angle. From the markings we know and . Second, . So, yes, is on the angle bisector of .

#### Example B

is the angle bisector of . Find the measure of .

by the Angle Bisector Theorem.

#### Example C

is the angle bisector of . Solve for the missing variable.

by the Angle Bisector Theorem, so we can set up and solve an equation for .

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### Guided Practice

1. is the angle bisector of . Solve for the missing variable.

2. Is there enough information to determine if is the angle bisector of ? Why or why not?

3. A angle is bisected. What are the measures of the resulting angles?

**Answers:**

1. by the Angle Bisector Theorem, so .

2. No because is not necessarily equidistant from and . We do not know if the angles in the diagram are right angles.

3. We know that to bisect means to cut in half, so each of the resulting angles will be half of . The measure of each resulting angle is .

### Explore More

For questions 1-4, is the angle bisector of . Solve for the missing variable.

Is there enough information to determine if is the angle bisector of ? Why or why not?

- In what type of triangle will all angle bisectors pass through vertices of the triangle?
- What is another name for the angle bisectors of the vertices of a square?
- Draw in the angle bisectors of the vertices of a square. How many triangles do you have? What type of triangles are they?
- Fill in the blanks in the Angle Bisector Theorem Converse.

Given: , such that and are the shortest distances to and

Prove: bisects

Statement |
Reason |
---|---|

1. | 1. |

2. | 2. The shortest distance from a point to a line is perpendicular. |

3. and are right angles | 3. |

4. | 4. |

5. | 5. |

6. | 6. |

7. | 7. CPCTC |

8. bisects | 8. |