### Isosceles Triangles

An isosceles triangle is a triangle that has ** at least** two congruent sides. The congruent sides of the isosceles triangle are called the

**legs**. The other side is called the

**base**. The angles between the base and the legs are called

**base angles**. The angle made by the two legs is called the

**vertex angle**. One of the important properties of isosceles triangles is that their base angles are always congruent. This is called the

**Base Angles Theorem.**

For

Another important property of isosceles triangles is that the angle bisector of the vertex angle is also the perpendicular bisector of the base. This is called the **Isosceles Triangle Theorem**. (*Note this is ONLY true of the vertex angle.*) The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true as well.

**Base Angles Theorem Converse:** If two angles in a triangle are congruent, then the sides opposite those angles are also congruent. So for

**Isosceles Triangle Theorem Converse:** The perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex angle. So for isosceles

What if you were presented with an isosceles triangle and told that its base angles measure *x* and *y*?

### Examples

#### Example 1

Find the value of

The two angles are equal, so set them equal to each other and solve for

Substitute

#### Example 2

True or false: Base angles of an isosceles triangle can be right angles.

This statement is false. Because the base angles of an isosceles triangle are congruent, if one base angle is a right angle then both base angles must be right angles. It is impossible to have a triangle with two right (

#### Example 3

Which two angles are congruent?

This is an isosceles triangle. The congruent angles are opposite the congruent sides. From the arrows we see that

#### Example 4

If an isosceles triangle has base angles with measures of

Draw a picture and set up an equation to solve for the vertex angle,

#### Example 5

If an isosceles triangle has a vertex angle with a measure of

Draw a picture and set up and equation to solve for the base angles,

### Review

Find the measures of

Determine if the following statements are true or false.

- Base angles of an isosceles triangle are congruent.
- Base angles of an isosceles triangle are complementary.
- Base angles of an isosceles triangle can be equal to the vertex angle.
- Base angles of an isosceles triangle are acute.

Fill in the proofs below.

- Given: Isosceles
△CIS , with base angles∠C and∠S IO¯¯¯¯¯¯ is the angle bisector of∠CIS Prove:IO¯¯¯¯¯¯ is the perpendicular bisector ofCS¯¯¯¯¯¯¯

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

1. | 1. Given |

2. | 2. Base Angles Theorem |

3. |
3. |

4. | 4. Reflexive PoC |

5. |
5. |

6. |
6. |

7. | 7. CPCTC |

8. |
8. |

9. | 9. Congruent Supplements Theorem |

10. |
10. |

- Given: Isosceles
△ICS with∠C and∠S IO¯¯¯¯¯¯ is the perpendicular bisector of \begin{align*}\overline{CS}\end{align*} Prove: \begin{align*}\overline{IO}\end{align*} is the angle bisector of \begin{align*}\angle CIS\end{align*}

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

1. | 1. |

2. \begin{align*}\angle C \cong \angle S\end{align*} | 2. |

3. \begin{align*}\overline{CO} \cong \overline{OS}\end{align*} | 3. |

4. \begin{align*}m\angle IOC = m\angle IOS = 90^\circ\end{align*} | 4. |

5. | 5. |

6. | 6. CPCTC |

7. \begin{align*}\overline{IO}\end{align*} is the angle bisector of \begin{align*}\angle CIS\end{align*} | 7. |

On the \begin{align*}x-y\end{align*} plane, plot the coordinates and determine if the given three points make a scalene or isosceles triangle.

- (-2, 1), (1, -2), (-5, -2)
- (-2, 5), (2, 4), (0, -1)
- (6, 9), (12, 3), (3, -6)
- (-10, -5), (-8, 5), (2, 3)
- (-1, 2), (7, 2), (3, 9)

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.10.