# 5.16: Kites

• • Contributed by CK12
• CK12

A kite is a quadrilateral with two distinct sets of adjacent congruent sides. It looks like a kite that flies in the air. Figure $$\PageIndex{1}$$

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. Figure $$\PageIndex{2}$$

1. The non-vertex angles of a kite are congruent. Figure $$\PageIndex{3}$$

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. Figure $$\PageIndex{5}$$

$$\Delta KET$$ 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. Figure $$\PageIndex{6}$$

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.

1. Figure $$\PageIndex{7}$$
2. Figure $$\PageIndex{8}$$

Solution

1. 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}

2. 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. Figure $$\PageIndex{9}$$

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: $$\angle K\cong \angle T$$ Figure $$\PageIndex{10}$$

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.

1. Figure $$\PageIndex{11}$$
2. Figure $$\PageIndex{12}$$
3. Figure $$\PageIndex{13}$$
4. Figure $$\PageIndex{14}$$
5. Figure $$\PageIndex{15}$$
6. Figure $$\PageIndex{16}$$

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

1. Figure $$\PageIndex{17}$$
2. Figure $$\PageIndex{18}$$
3. Figure $$\PageIndex{19}$$
4. Figure $$\PageIndex{20}$$
5. Figure $$\PageIndex{21}$$
1. 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 $$\angle KIT$$ Figure $$\PageIndex{22}$$
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.
1. Fill in the blanks to the proof below.

Given: $$\overline{EK}\cong \overline{ET},\: \overline{KI}\cong \overline{IT}$$

Prove: $$\overline{KT}\perp \overline{EI}$$ Figure $$\PageIndex{23}$$
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.

## Vocabulary

Term Definition
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.

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