Quadrilateral is a geometric form that is composed of 4 points (vertices) sequentially joined by right line segments (sides). We uncover the etymology of the word in S. Schwartzman"s *The indigenous of Mathematics*:

quadrilateral (noun, adjective): the first element is indigenous Latin quadri- "four" native the Indo-European source kwetwer- "four." The second element is from Latin latus, stem later-, "side," of unknown front origin. A quadrilateral is a four-sided polygon. The Latin hatchet is a partial translate in of Greek tetragon, literally "four angles," because a closed number with 4 angles additionally has four sides. Although we usage words prefer pentagon and also polygon, the term square has completely replaced tetragon.

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The seldom supplied term quadrangle has precisely the same meaning as quadrilateral, but the two related terms -- complete quadrangle and also complete square -- explain essentially various configurations.

A quadrilateral may be convex or concave (see the diagram below.) A quadrilateral the is concave has an angle exceeding 180o. In one of two people case, the quadrilateral is simple, which means that the 4 sides of the quadrilateral only satisfy at the vertices, two at a time. For this reason that two non-adjacent sides do not cross. A quadrilateral the is not simple is additionally known as self-intersecting to indicate that a pair of his non-adjacent sides intersect.

The point of intersection that the political parties is not considered a crest of the quadrilateral.

The shapes of elementary school geometry space invariably convex. Starting with the most continual quadrilateral, namely, the square, us shall specify other shapes by relaxing its properties.

A square is a quadrilateral with all political parties equal and also all angles likewise equal. Angles in any kind of quadrilateral include up come 360°. It adheres to that, in a square, all angles measure 90°. An equiangular quadrilateral, i.e. The one with all angle equal is a rectangle. All angles of a rectangle equal 90°. An it is provided quadrilateral, i.e. The one with all political parties equal, is a rhombus.

In a square, rectangle, or rhombus, the opposite next lines room parallel. A quadrilateral with the opposite next lines parallel is recognized as a parallelogram. If only one pair of opposite political parties is forced to it is in parallel, the shape is a trapezoid. A trapezoid, in i beg your pardon the non-parallel sides space equal in length, is called isosceles. A quadrilateral through two different pairs the equal adjacent sides is generally called a kite. However, if the dragon is concave, a dart is a an ext appropriate term. Kite and also dart are instances of orthodiagonal quadrilaterals, i.e. Quadrilaterals v perpendicular diagonals. A square and a rhombus are also details cases of this class.

The 4 vertices that a quadrilateral may be concyclic, i.e., lie on the same circle. In this case, the square is well-known as circumscritptible or, simpler, cyclic. If a quadrilateral admits one incircle that touches all 4 of its sides (or more generally, next lines), that is well-known as inscriptible. A quadrilateral, both cyclic and inscriptible, is bicentric.

The diagram listed below (which is a alteration of one native wikipedia.org) summarize the relationship between various kinds of quadrilaterals:

The applet listed below illustrates the nature of miscellaneous quadrilaterals. In the applet, one can drag the vertices and the sides of the quadrilateral. You can display its diagonals, edge bisectors and the perpendicular bisectors of its sides. V these props, it"s a basic matter to observe every single kind of quadrilateral, through a feasible exception that bicentric. Which, too, is not overly difficult if you first get an isosceles trapezoid.)

### This applet calls for Sun"s Java VM 2 which your internet browser may perceive as a popup. Which it is not. If you desire to view the applet work, visit Sun"s website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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As in the classification of triangles, the meanings may be either inclusive or exclusive. For example, trapezoid might be identified inclusively together a quadrilateral through a pair that parallel the contrary sides, or solely as a quadrilateral with precisely one such pair. In the former case, parallelogram is a trapezoid, in the latter, the is not. Similarly, a square might or may not be a rectangle or a rhombus. My choice is through the inclusive approach. For, I"d like to think of a square as a rhombus with right angles, or as a rectangle v all four sides equal.

Here is a perform of every the nature of quadrilaterals the we have mentioned in addition to the classes of the quadrilaterals that possess those properties:

Property | Quadrilaterals | |

Orthodiagonal | Kite, Dart, Rhombus, Square | |

Cyclic | Square, Rectangle, Isosceles Trapezoid | |

Inscriptible | Kite, Dart, Rhombus, Square | |

Having two parallel sides | Rhombus, Square, Rectangle, Parallelogram, Trapezoid | |

Having 2 pairs of parallel sides | Rhombus, Square, Rectangle, Parallelogram | |

Equilateral | Rhombus, Square | |

Equiangular | Rectangle, Square |

Orthodiagonal or inscriptible parallel is a rhombus; cyclic parallel is a rectangle. In particular, a parallelogram v equal diagonals is necessarily a rectangle. And not come forget, every simple quadrilateral tiles the plane.

A simple quadrilateral with two bag of same opposite angle is a parallelogram. (Because then the the contrary sides room parallel.) A straightforward quadrilateral through two pairs of same opposite political parties is a parallelogram. (Because the SSS when you attract one that the diagonals.)There is a simple quadrilateral with two bag of same sides: a kite (or a dart). It does have actually a pair that opposite equal angles.

Nathan Bowler said a general building and construction of a quadrilateral through a pair of same opposite sides and a pair of same opposite angle which is not necessarily a parallelogram (there is a dynamic illustration):

Let abc be isosceles with abdominal muscle = AC. Pick D ~ above BC. Let C" it is in the have fun of C in the perpendicular bisector the AD. ABDC" has two opposite political parties the same length and two opposite angles equal but is no a parallelogram if D isn"t the midpoint that AB. This building and construction gives every such quadrilaterals.

For one isosceles trapezoid ABCD with abdominal = CD, the square ABDC has actually a pair of same opposite sides and two pairs of equal opposite angles.

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## References

S. Schwartzman,*The words of Mathematics*, MAA, 1994

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