### Angle Bisector Theorem

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

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.

What if you were told that

### Examples

#### Example 1

Is there enough information to determine if

No because

#### Example 2

A

We know that to bisect means to cut in half, so each of the resulting angles will be half of

#### Example 3

Is

If

#### Example 4

#### Example 5

### Review

For questions 1-4,

Is there enough information to determine if

- 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:

Prove:

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

1. | 1. |

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

3. |
3. |

4. |
4. |

5. |
5. |

6. \begin{align*}\triangle ABD \cong \triangle CBD\end{align*} | 6. |

7. | 7. CPCTC |

8. \begin{align*}\overrightarrow{B D}\end{align*} bisects \begin{align*}\angle ABC\end{align*} | 8. |

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.3.