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Kites

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? After completing this Concept, you'll be able to find the value of a kite's unknown angles and sides.

Guidance

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.

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.

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

Example A

Find the missing measures in the kites below.

a)

b)

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

$130^\circ + 60^\circ + x + x & = 360^\circ\\2x & = 170^\circ\\x & = 85^\circ \qquad \text{Both angles are}\ 85^\circ.$

b) The other non-vertex angle is also $94^\circ$ . To find the fourth angle, subtract the other three angles from $360^\circ$ .

$90^\circ + 94^\circ + 94^\circ + x & = 360^\circ\\x & = 82^\circ$

Example B

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

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.

$6^2 + 5^2 & = h^2 && 12^2 + 5^2 = j^2\\36 + 25 & = h^2 && 144 + 25 = j^2\\61 & = h^2 && \qquad \ 169 = j^2\\\sqrt{61} & = h && \qquad \quad 13 = j$

Example C

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 : $\angle K \cong \angle T$

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. $\triangle EKI \cong \triangle ETI$ 3. SSS
4. $\angle K \cong \angle T$ 4. CPCTC

Guided Practice

$KITE$ is a kite.

Find:

1. $m \angle KIS$
2. $m \angle IST$
3. $m \angle SIT$

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

2. $m\angle IST=90^\circ$ because the diagonals are perpendicular.

3. $m\angle SIT=25^\circ$ because it is congruent to $\angle KIS$ .

Practice

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).

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 $\angle KIT$

Statement Reason
1. $\overline{KE} \cong \overline{TE}$ and $\overline{KI} \cong \overline{TI}$ 1.
2. $\overline{EI} \cong \overline{EI}$ 2.
3. $\triangle EKI \cong \triangle ETI$ 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 : $\overline{KT} \bot \overline{EI}$

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. $\overline{KT} \bot \overline{EI}$ 5.