# Quadrilaterals

A quadrilateral \( ABCD \) is a closed shape with four sides (edges) and four vertices. It is also called a polygon with four sides.

The four interior angles of a quadrilateral add up to \( 360^{\circ} \), hence \( \quad \angle A + \angle B + \angle C + \angle D = 360^{\circ} \)

Special quadrilaterals such as parallelograms, trpaezoids, rectangles, squares, kites and rhombuses are presented with their properties.

## Parallelogram

A parallelogram is a quadilateral with the opposite sides parallel.

__Properties__

- Opposite sides are congruent (equal length)
- The diagonals \( AC \) and \( BD \) bisects the at point \( O \) that divides each diagonal into equal parts: \( OA = OC \) and \( OB = OD \)
- The are several pairs of congruent triangles: \( \triangle ADB \) and \( \triangle CBD \) , \( \triangle ADC \) and \( \triangle ABC \) , \( \triangle ADO \) and \( \triangle BCO \) etc ...
- Opposite interior angles are congruent: \( \angle DAB = \angle DCB \) and \( \angle ADC = \angle ABC \)
- Several other angles are congruent: \( \angle ABO = \angle CDO \), \( \angle DAO = \angle BCO \), etc ...
- Area \( A_p \) of a parallelogram is given by: \( A_r = AB \times h\)

## Trapezoid

A trapezoid is a quadilateral with at least two opposite sides parallel.

Area \( A_t \) of a trapezoid is given by: \( A_t = \dfrac{1}{2} h \times (AB+DC) \)

Trapezoid Problems and Calculator

## Rectangle

__Properties__

A rectangle is a parallelogram with interior angles equal to \( 90^{\circ} \). Hence all properties of parallelograms are also property of rectangles and more properties are listed below.

- \( OA = OB = OC = OD \)
- \( \triangle ABC \), \( \triangle BCD \), etc ... are rigth triangles
- Area \( A_{re} \) of a rectangle is given by: \( A_{re} = AB \times BC\)

## Square

A square is a rectangle with all sides equal.

__Properties__

All properties of rectangles are also property of squares and more properties are listed below.

- The diagonals intersect at angle of \( 90^{\circ} \)
- Area \( A_{s} \) of a rectangle is given by: \( A_{s} = AB^2\)

## Kite

A kite is a quadrilateral with two pairs of adjacent sides equal in length. In the diagram below, sides \( BA \) and \( BC \) have equal lengths and sides \( DA \) and \( DC \) have equal lengths.

__Properties__

- The diagonals \( AC \) and \( BD \) bisects the at point \( O \) at an angle of \( 90^{\circ} \).
- The diagonals \( AC \) and \( BD \) bisects the at point \( O \) that divides the diagonal \( AC \) into equal parts: \( OA = OC \)
- \( BOA \), \( BOC\), \( DOA \) and \( DOC\) are right triangles.
- \( \angle ABO = \angle CBO \), \( \angle ADO = \angle CDO \), etc ...
- Triangles \( BOA \) and \( BOC \) are congruent, etc ...
- Area \( A_k \) of a kite is given by: \( A_k = \dfrac{1}{2} \times AC \times BD \)

## Rhombus

A rhombus is a parallelogram will all 4 sides equal in length.

__Properties__

- The diagonals \( AC \) and \( BD \) bisects the at point \( O \) at an angle of \( 90^{\circ} \).
- The diagonals \( AC \) and \( BD \) bisects the at point \( O \) that divides the two diagonals into equal parts: \( OA = OC \) and \( OD = OB \)
- \( BOA \), \( BOC\), \( DOA \) and \( DOC\) are congruent triangles.
- \( \angle ABO = \angle CBO \), \( \angle ADO = \angle CDO \), etc ...
- Area \( A_{ro} \) of a parallelogram is given by: \( A_{ro} = \dfrac{1}{2} \times AC \times BD \)