6.7: Kites
What if you made a traditional kite, seen below, by placing two pieces of wood perpendicular to each other (one bisected by the other)? The typical dimensions are included in the picture. If you have two pieces of wood, 36 inches and 54 inches, determine the values of \begin{align*}x\end{align*}
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CK12 Foundation: Chapter6KitesA
Guidance
A kite is a quadrilateral with two sets of distinct, adjacent congruent sides. A few examples:
From the definition, a kite is the only quadrilateral that we have discussed that could be concave, as with the case of the last kite. If a kite is concave, it is called a dart. The angles between the congruent sides are called vertex angles. The other angles are called nonvertex angles. If we draw the diagonal through the vertex angles, we would have two congruent triangles.
Theorem: The nonvertex angles of a kite are congruent.
Proof:
Given: \begin{align*}KITE\end{align*}
Prove: \begin{align*}\angle K \cong \angle T\end{align*}
Statement  Reason 

1. \begin{align*}\overline{KE} \cong \overline{TE}\end{align*} 
Given 
2. \begin{align*}\overline{EI} \cong \overline{EI}\end{align*} 
Reflexive PoC 
3. \begin{align*}\triangle EKI \cong \triangle ETI\end{align*} 
SSS 
4. \begin{align*}\angle K \cong \angle T\end{align*} 
CPCTC 
Theorem: The diagonal through the vertex angles is the angle bisector for both angles.
The proof of this theorem is very similar to the proof above for the first theorem. If we draw in the other diagonal in \begin{align*}KITE\end{align*}
Kite Diagonals Theorem: The diagonals of a kite are perpendicular.
To prove that the diagonals are perpendicular, look at \begin{align*}\triangle KET\end{align*}
Example A
Find the other two angle measures in the kite below.
The two angles left are the nonvertex angles, which are congruent.
\begin{align*}130^\circ + 60^\circ + x + x & = 360^\circ\\
2x & = 170^\circ\\
x & = 85^\circ \qquad \quad \text{Both angles are}\ 85^\circ.\end{align*}
Example B
Use the Pythagorean Theorem to find the length of the sides of the kite.
Recall that the Pythagorean Theorem is \begin{align*}a^2+b^2=c^2\end{align*}
\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 \ 13=j\end{align*}
Example C
Find the other two angle measures in the kite below.
The other nonvertex angle is also \begin{align*}94^\circ\end{align*}
\begin{align*}90^\circ + 94^\circ + 94^\circ + x & = 360^\circ\\
x & = 82^\circ\end{align*}
Watch this video for help with the Examples above.
CK12 Foundation: Chapter6KitesB
Concept Problem Revisited
If the diagonals (pieces of wood) are 36 inches and 54 inches, \begin{align*}x\end{align*}
\begin{align*}18^2+18^2& =s^2 && 18^2+36^2=t^2\\
324& =s^2 && \qquad 1620=t^2\\
18\sqrt{2} & \approx 25.5 \approx s && \quad \ \ 18\sqrt{5} \approx 40.25 \approx t\end{align*}
The perimeter of the kite would be \begin{align*}25.5 + 25.5 + 40.25 + 40.25 = 131.5\end{align*}
Vocabulary
A kite is a quadrilateral with two distinct sets of adjacent congruent sides. The angles between the congruent sides are called vertex angles. The other angles are called nonvertex angles.
If a kite is concave, it is called a dart.
Guided Practice
\begin{align*}KITE\end{align*}
Find:

\begin{align*}m \angle KIS\end{align*}
m∠KIS 
\begin{align*}m \angle IST\end{align*}
m∠IST 
\begin{align*}m \angle SIT\end{align*}
m∠SIT
Answers:
1. \begin{align*}m\angle KIS=25^\circ\end{align*}
2. \begin{align*}m\angle IST=90^\circ\end{align*}
3. \begin{align*} m\angle SIT=25^\circ\end{align*}
Practice
For questions 16, find the value of the missing variable(s). All figures are kites.
For questions 711, find the value of the missing variable(s).
 Prove that the long diagonal of a kite bisects its angles.
Given: \begin{align*}\overline{KE} \cong \overline{TE}\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*}
 Prove the Kite Diagonal Theorem.
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*}
 Writing Besides a kite and a rhombus, can you find another quadrilateral with perpendicular diagonals? Explain and draw a picture.
 Writing Describe how you would draw or construct a kite.
Image Attributions
Here you'll learn the properties of kites and how to apply them.