### Kites

A **kite** is a quadrilateral with two distinct sets of adjacent congruent sides. It looks like a kite that flies in the air.

From the definition, a kite could be concave. If a kite is concave, it is called a **dart.** The word **distinct** in the definition means that the two pairs of congruent sides have to be different. This means that a square or a rhombus is not a kite.

The angles between the congruent sides are called **vertex angles.** The other angles are called **non-vertex angles.** If we draw the diagonal through the vertex angles, we would have two congruent triangles.

#### Facts about Kites

1. The non-vertex angles of a kite are congruent.

If is a kite, then .

2. The diagonal through the vertex angles is the angle bisector for both angles.

If is a kite, then and .

3. **Kite Diagonals Theorem:** The diagonals of a kite are perpendicular.

and are isosceles triangles, so is the perpendicular bisector of (Isosceles Triangle Theorem).

What if you were told that is a kite and you are given information about some of its angles or its diagonals? How would you find the measure of its other angles or its sides?

### Examples

For Examples 1 and 2, use the following information:

is a kite.

#### Example 1

Find .

by the Triangle Sum Theorem (remember that is a right angle because the diagonals are perpendicular.)

#### Example 2

Find .

because the diagonals are perpendicular.

#### Example 3

Find the missing measures in the kites below.

The two angles left are the non-vertex angles, which are congruent.

The other non-vertex angle is also . To find the fourth angle, subtract the other three angles from .

#### Example 4

Use the Pythagorean Theorem to find the lengths of the sides of the kite.

Recall that the Pythagorean Theorem says , where is the hypotenuse. In this kite, the sides are the hypotenuses.

#### Example 5

Prove that the non-vertex angles of a kite are congruent.

Given: with and

Prove:

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

1. and | 1. Given |

2. | 2. Reflexive PoC |

3. | 3. SSS |

4. | 4. CPCTC |

### Review

For questions 1-6, find the value of the missing variable(s). All figures are kites.

For questions 7-11, find the value of the missing variable(s).

- Fill in the blanks to the proof below.

Given: and

Prove: is the angle bisector of and

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

1. and | 1. |

2. | 2. |

3. | 3. |

4. | 4. CPCTC |

5. is the angle bisector of and | 5. |

- Fill in the blanks to the proof below.

Given:

Prove:

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

1. and | 1. |

2. | 2. Definition of isosceles triangles |

3. is the angle bisector of and | 3. |

4. | 4. Isosceles Triangle Theorem |

5. | 5. |

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.7.