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.
-
Figure \(\PageIndex{7}\) -
Figure \(\PageIndex{8}\)
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.
-
Figure \(\PageIndex{11}\) -
Figure \(\PageIndex{12}\) -
Figure \(\PageIndex{13}\) -
Figure \(\PageIndex{14}\) -
Figure \(\PageIndex{15}\)
-
Figure \(\PageIndex{16}\)
For questions 7-11, find the value of the missing variable(s).
-
Figure \(\PageIndex{17}\) -
Figure \(\PageIndex{18}\) -
Figure \(\PageIndex{19}\) -
Figure \(\PageIndex{20}\) -
Figure \(\PageIndex{21}\)
- 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