What if you made a traditional kite, seen below, by placing two pieces of wood perpendicular to each other (one bisected by the other)? The typical dimensions are included in the picture. If you have two pieces of wood, 36 inches and 54 inches, determine the values of

### Kites

A **kite** is a quadrilateral with two sets of distinct, adjacent congruent sides. A few examples:

From the definition, a kite is the only quadrilateral that we have discussed that could be concave, as with the case of the last kite. If a kite is concave, it is called a ** dart**. The angles between the congruent sides are called

**. The other angles are called**

*vertex angles***. If we draw the diagonal through the vertex angles, we would have two congruent triangles.**

*non-vertex angles*

**Theorem:** The non-vertex angles of a kite are congruent.

**Proof:**

Given:

Prove:

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

1. |
Given |

2. |
Reflexive PoC |

3. |
SSS |

4. |
CPCTC |

**Theorem:** The diagonal through the vertex angles is the angle bisector for both angles.

The proof of this theorem is very similar to the proof above for the first theorem. If we draw in the other diagonal in

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

To prove that the diagonals are perpendicular, look at

#### Measuring Angles

Find the other two angle measures in the kite below.

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

#### Using the Pythagorean Theorem

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

Recall that the Pythagorean Theorem is

#### Finding Missing Angle Measures

Find the other two angle measures in the kite below.

The other non-vertex angle is also

#### Kite Problem Revisited

If the diagonals (pieces of wood) are 36 inches and 54 inches,

The perimeter of the kite would be

### Examples

#### Example 1

m∠KIS

#### Example 2

m∠IST

#### Example 3

m∠SIT

### Interactive Practice

### 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).

- Prove that the long diagonal of a kite bisects its angles.

Given:

Prove:

- Prove the Kite Diagonal Theorem.

Given:

Prove:

Besides a kite and a rhombus, can you find another quadrilateral with perpendicular diagonals? Explain and draw a picture.*Writing*Describe how you would draw or construct a kite.*Writing*

### Review (Answers)

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