# 5.16: Kites

- Page ID
- 5000

Quadrilaterals with two distinct sets of adjacent, congruent sides.

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 \(KITE\) is a kite, then \(\angle K\cong \angle T\).

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

If \(KITE\) is a kite, then \(\angle KEI\cong \angle IET\) and \(\angle KIE\cong \angle EIT\).

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

\( \Delta KET\) and \(\Delta KIT\) are isosceles triangles, so \(\overline{EI}\) is the perpendicular bisector of \(\overline{KT}\) (Isosceles Triangle Theorem).

What if you were told that \(WIND\) 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?

For Examples 1 and 2, use the following information:

\(KITE\) is a kite.

Example \(\PageIndex{1}\)

Find \(m\angle KIS\).

**Solution**

\(m\angle KIS=25^{\circ}\) by the **Triangle Sum Theorem** (remember that \angle KSI is a right angle because the diagonals are perpendicular.)

Example \(\PageIndex{2}\)

Find \(m\angle IST\).

**Solution**

\(m\angle IST=90^{\circ}\) because the diagonals are perpendicular.

Example \(\PageIndex{3}\)

Find the missing measures in the kites below.

**Solution**

- The two angles left are the non-vertex angles, which are congruent.
\( \begin{aligned} 130^{\circ} +60^{\circ} +x+x=360^{\circ} \\ 2x&=170^{\circ} \\ x&=85^{\circ} \qquad Both angles are 85^{\circ} \end{aligned}\)

- The other non-vertex angle is also \(94^{\circ}\). To find the fourth angle, subtract the other three angles from \(360^{\circ}\).
\(\begin{aligned} 90^{\circ} +94^{\circ} +94^{\circ} +x &=360^{\circ} \\ x&=82^{\circ} \end{aligned}\)

Example \(\PageIndex{4}\)

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

**Solution**

Recall that the Pythagorean Theorem says \(a^2+b^2=c^2\), where \(c\) is the hypotenuse. In this kite, the sides are the hypotenuses.

\(\begin{array}{rr}

6^{2}+5^{2}=h^{2} & 12^{2}+5^{2}=j^{2} \\

36+25=h^{2} & 144+25=j^{2} \\

61=h^{2} & 169=j^{2} \\

\sqrt{61}=h & 13=j

\end{array}\)

Example \(\PageIndex{5}\)

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

__Given__: \(KITE\) with \(\overline{KE}\cong \overline{TE}\) and \(\overline{KI}\cong \overline{TI}\)

__Prove__: \(\angle K\cong \angle T\)

**Solution**

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

1. \(\overline{KE}\cong \overline{TE}\) and \(\overline{KI}\cong \overline{TI}\) | 1. Given |

2.\( \overline{EI}\cong \overline{EI}\) | 2. Reflexive PoC |

3. \(\Delta EKI\cong \Delta ETI\) | 3. SSS |

4. \(\angle K\cong \angle T\) | 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__: \(\overline{KE}\cong \overline{TE}\) and \(\overline{KI}\cong \overline{TI}\)

__Prove__: \(\overline{EI}\) is the angle bisector of \(\angle KET\) and \(\angle KIT\)

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

1.\( \overline{KE}\cong \overline{TE} and \overline{KI}\cong \overline{TI}\) | 1. |

2. \(\overline{EI}\cong \overline{EI}\) | 2. |

3. \(\Delta EKI\cong \Delta ETI\) | 3. |

4. | 4. CPCTC |

5. \(\overline{EI} is the angle bisector of \angle KET\) and \angle KIT\) | 5. |

- Fill in the blanks to the proof below.

__Given__: \(\overline{EK}\cong \overline{ET},\: \overline{KI}\cong \overline{IT}\)

__Prove__: \(\overline{KT}\perp \overline{EI}\)

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

1. \(\overline{KE}\cong \overline{TE}\) and \(\overline{KI}\cong \overline{TI}\) | 1. |

2. | 2. Definition of isosceles triangles |

3. \(\overline{EI}\) is the angle bisector of \(\angle KET\) and \(\angle KIT\) | 3. |

4. | 4. Isosceles Triangle Theorem |

5. \(\overline{KT}\perp \overline{EI}\) | 5. |

## Review (Answers)

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

## Vocabulary

Term | Definition |
---|---|

kite |
A quadrilateral with distinct adjacent congruent sides. |

Triangle Sum Theorem |
The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees. |

Vertical Angles |
Vertical angles are a pair of opposite angles created by intersecting lines. |

## Additional Resources

Interactive Element

Video: Kites Principles - Basic

Activities: Kites Discussion Questions

Study Aids: Trapezoids and Kites Study Guide

Practice: Kites

Real World: Go Fly a Kite