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# Kites

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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 x\begin{align*}x\end{align*} and 2x\begin{align*}2x\end{align*}. Then, determine how large a piece of canvas you would need to make the kite (find the perimeter of the kite). After completing this Concept, you'll be able to answer these questions using your knowledge of kites.

### 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 non-vertex angles. If we draw the diagonal through the vertex angles, we would have two congruent triangles.

Theorem: The non-vertex angles of a kite are congruent.

Proof:

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

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

Statement Reason
1. KE¯¯¯¯¯¯¯¯TE¯¯¯¯¯¯¯\begin{align*}\overline{KE} \cong \overline{TE}\end{align*} and KI¯¯¯¯¯¯¯TI¯¯¯¯¯¯\begin{align*}\overline{KI} \cong \overline{TI}\end{align*} Given
2. EI¯¯¯¯¯¯EI¯¯¯¯¯¯\begin{align*}\overline{EI} \cong \overline{EI}\end{align*} Reflexive PoC
3. EKIETI\begin{align*}\triangle EKI \cong \triangle ETI\end{align*} SSS
4. KT\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 KITE\begin{align*}KITE\end{align*} we find that the two diagonals are perpendicular.

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

To prove that the diagonals are perpendicular, look at KET\begin{align*}\triangle KET\end{align*} and KIT\begin{align*}\triangle KIT\end{align*}. Both of these triangles are isosceles triangles, which means EI¯¯¯¯¯¯\begin{align*}\overline{EI}\end{align*} is the perpendicular bisector of KT¯¯¯¯¯¯¯¯\begin{align*}\overline{KT}\end{align*} (the Isosceles Triangle Theorem). Use this information to help you prove the diagonals are perpendicular in the practice questions.

#### Example A

Find the other two angle measures in the kite below.

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

130+60+x+x2xx=360=170=85Both angles are 85.\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 a2+b2=c2\begin{align*}a^2+b^2=c^2\end{align*}, where c\begin{align*}c\end{align*} is the hypotenuse. In this kite, the sides are all hypotenuses.

62+5236+256161=h2=h2=h2=h 122+52=j2144+25=j2169=j2 13=j\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 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*}

Watch this video for help with the Examples above.

#### Concept Problem Revisited

If the diagonals (pieces of wood) are 36 inches and 54 inches, \begin{align*}x\end{align*} is half of 36, or 18 inches. Then, \begin{align*}2x\end{align*} is 36. To determine how large a piece of canvas to get, find the length of each side of the kite using the Pythagorean Theorem.

\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*} inches or 11 ft, 10.5 in.

### 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 non-vertex angles.

If a kite is concave, it is called a dart.

### Guided Practice

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

Find:

1. \begin{align*}m \angle KIS\end{align*}
2. \begin{align*}m \angle IST\end{align*}
3. \begin{align*}m \angle SIT\end{align*}

1. \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.)

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

3. \begin{align*} m\angle SIT=25^\circ\end{align*} because it is congruent to \begin{align*}\angle KIS\end{align*}.

### 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. Prove that the long diagonal of a kite bisects its angles.

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*}

1. 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*}

1. Writing Besides a kite and a rhombus, can you find another quadrilateral with perpendicular diagonals? Explain and draw a picture.
2. Writing Describe how you would draw or construct a kite.

### Vocabulary Language: English

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.