The kite

Definition: a kite is a quadrilateral with two pairs of equal adjacent sides.

In the kite $\displaystyle ABCD$ in the figure, $\displaystyle {AB}={AD}$ and $\displaystyle {CB}={CD}$:

• Equal sides meet at the vertices $\displaystyle A$ and $\displaystyle C$. The diagonal $\displaystyle AC$, which joins them, is the main diagonal.
• The diagonal $\displaystyle BD$ is the secondary diagonal.

Theorem: in the kite $\displaystyle ABCD$, where $\displaystyle {AB}={AD}$ and $\displaystyle {CB}={CD}$:

• The angles $\displaystyle B$ and $\displaystyle D$ are equal: $\displaystyle {\angle ABC}={\angle ADC}$. At each of them two unequal sides meet.
• The main diagonal bisects the angles $\displaystyle A$ and $\displaystyle C$: $\displaystyle {\angle BAC}={\angle DAC}$ and $\displaystyle {\angle BCA}={\angle DCA}$.
• The main diagonal is the perpendicular bisector of the secondary diagonal: it is perpendicular to it, $\displaystyle AC\perp BD$, and passes through its midpoint.

Given: $\displaystyle {AB}={AD}$ and $\displaystyle {CB}={CD}$.
Prove: the three properties.

The main diagonal divides the kite into triangles $\displaystyle ABC$ and $\displaystyle ADC$. The proof below shows that they are congruent by SSS, then draws the secondary diagonal $\displaystyle BD$ and uses the isosceles triangle $\displaystyle ABD$:

The figure shows the quadrilateral $\displaystyle ABCD$ in a coordinate system.

• The lengths of the sides, by Pythagoras:
  • $\displaystyle {AB}={\sqrt{2^{2}+3^{2}}}={\sqrt{13}}$, and also $\displaystyle {AD}={\sqrt{2^{2}+3^{2}}}={\sqrt{13}}$;
  • $\displaystyle {CB}={\sqrt{4^{2}+3^{2}}}={5}$, and also $\displaystyle {CD}={\sqrt{4^{2}+3^{2}}}={5}$.
• So $\displaystyle ABCD$ is a kite, and its main diagonal is $\displaystyle AC$.
• $\displaystyle A$ and $\displaystyle C$ have the same $\displaystyle y$-coordinate, so $\displaystyle AC$ is horizontal. $\displaystyle B$ and $\displaystyle D$ have the same $\displaystyle x$-coordinate, so $\displaystyle BD$ is vertical. So $\displaystyle AC\perp BD$.
• The diagonals cross at the point $\displaystyle (3,3)$, the midpoint of $\displaystyle BD$.

In the kite $\displaystyle ABCD$, $\displaystyle {AB}={AD}$ and $\displaystyle {CB}={CD}$.
Find the angle asked for, and give the reason.

$\displaystyle {\angle BAD}={70^{\circ}}$ and $\displaystyle {\angle BCD}={50^{\circ}}$.
Find $\displaystyle \angle ABC$.

$\displaystyle {\angle ABC}={110^{\circ}}$ and $\displaystyle {\angle BAC}={40^{\circ}}$.
Find $\displaystyle \angle BCD$.

In the kite $\displaystyle ABCD$, $\displaystyle {AB}={AD}$ and $\displaystyle {CB}={CD}$.
The point $\displaystyle E$ is on the main diagonal $\displaystyle AC$.
Prove: $\displaystyle {EB}={ED}$.

In the quadrilateral $\displaystyle ABCD$, the diagonal $\displaystyle AC$ bisects the angle $\displaystyle A$ and the angle $\displaystyle C$.
Prove: $\displaystyle ABCD$ is a kite.

The figure shows the quadrilateral $\displaystyle KLMN$ in a coordinate system.

Find the lengths of the four sides, and show that $\displaystyle KLMN$ is a kite.

Which diagonal is the main diagonal? Show that it is the perpendicular bisector of the secondary diagonal.

The diagonal $\displaystyle KM$ cuts the kite into two triangles, $\displaystyle KLM$ and $\displaystyle KNM$.
Find the area of the kite.