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Kites

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Kites

Kites

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 \begin{align*}KITE\end{align*} is a kite, then \begin{align*}\angle K \cong \angle T\end{align*}.

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

If \begin{align*}KITE\end{align*} is a kite, then \begin{align*}\angle KEI \cong \angle IET\end{align*} and \begin{align*}\angle KIE \cong \angle EIT\end{align*}.

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

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

What if you were told that \begin{align*}WIND\end{align*} 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?

Examples

For Examples 1 and 2, use the following information:

\begin{align*}KITE\end{align*} is a kite.

Example 1

Find \begin{align*}m \angle KIS\end{align*}.

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

Example 2

Find \begin{align*}m \angle IST\end{align*}.

\begin{align*}m\angle IST=90^\circ\end{align*} because the diagonals are perpendicular.

Example 3

Find the missing measures in the kites below.

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

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

The other non-vertex angle is also \begin{align*}94^\circ\end{align*}. To find the fourth angle, subtract the other three  angles from \begin{align*}360^\circ\end{align*}.

\begin{align*}90^\circ + 94^\circ + 94^\circ + x & = 360^\circ\\ x & = 82^\circ\end{align*}

Example 4

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

Recall that the Pythagorean Theorem says \begin{align*}a^2 + b^2 = c^2\end{align*}, where \begin{align*}c\end{align*} is the hypotenuse. In this kite, the sides are the hypotenuses.

\begin{align*}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\end{align*}

Example 5

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

Given: \begin{align*}KITE\end{align*} with \begin{align*}\overline{KE} \cong \overline{TE}\end{align*} and \begin{align*}\overline{KI} \cong \overline{TI}\end{align*}

Prove: \begin{align*}\angle K \cong \angle T\end{align*}

Statement Reason
1. \begin{align*}\overline{KE} \cong \overline{TE}\end{align*} and \begin{align*}\overline{KI} \cong \overline{TI}\end{align*} 1. Given
2. \begin{align*}\overline{EI} \cong \overline{EI}\end{align*} 2. Reflexive PoC
3. \begin{align*}\triangle EKI \cong \triangle ETI\end{align*} 3. SSS
4. \begin{align*}\angle K \cong \angle T\end{align*} 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).

1. Fill in the blanks to the proof below.

Given: \begin{align*}\overline{KE} \cong \overline{TE}\end{align*} and \begin{align*}\overline{KI} \cong \overline{TI}\end{align*}

Prove: \begin{align*}\overline{EI}\end{align*} is the angle bisector of \begin{align*}\angle KET\end{align*} and \begin{align*}\angle KIT\end{align*}

Statement Reason
1. \begin{align*}\overline{KE} \cong \overline{TE}\end{align*} and \begin{align*}\overline{KI} \cong \overline{TI}\end{align*} 1.
2. \begin{align*}\overline{EI} \cong \overline{EI}\end{align*} 2.
3. \begin{align*}\triangle EKI \cong \triangle ETI\end{align*} 3.
4. 4. CPCTC
5. \begin{align*}\overline{EI}\end{align*} is the angle bisector of \begin{align*}\angle KET\end{align*} and \begin{align*}\angle KIT\end{align*} 5.
1. Fill in the blanks to the proof below.

Given: \begin{align*}\overline{EK} \cong \overline{ET}, \overline{KI} \cong \overline{IT}\end{align*}

Prove: \begin{align*}\overline{KT} \bot \overline{EI}\end{align*}

Statement Reason
1. \begin{align*}\overline{KE} \cong \overline{TE}\end{align*} and \begin{align*}\overline{KI} \cong \overline{TI}\end{align*} 1.
2. 2. Definition of isosceles triangles
3. \begin{align*}\overline{EI}\end{align*} is the angle bisector of \begin{align*}\angle KET\end{align*} and \begin{align*} \angle KIT\end{align*} 3.
4. 4. Isosceles Triangle Theorem
5. \begin{align*}\overline{KT} \bot \overline{EI}\end{align*} 5.

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

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