Parallelogram
A parallelogram is a polygon with four sides (a quadrilateral). It has two pairs of parallel sides (sides which never meet) and four edges. The opposite sides of a parallelogram have the same length (they are equally long). The word "parallelogram" comes from the Greek word "parallelogrammon" (bounded by parallel lines).[1] Rectangles, rhombuses, and squares are all parallelograms.
As shown in the picture on the right, because triangles ABE and CDE are congruent (have the same shape and size),
- [math]\displaystyle{ AE = CE }[/math]
- [math]\displaystyle{ BE = DE. }[/math]
In all Parallelogram's opposite angles are equal to each other. Angles which are not opposite in the Parallelogram will add up to 180 degrees.
Characterizations
A simple (non self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:[2][3]
- Two pairs of opposite sides are equal in length
- Two pairs of opposite angles are equal in measure
- The diagonals bisect each other
- One pair of opposite sides are parallel and equal in length
- Adjacent angles are supplementary
- Each diagonal divides the quadrilateral into two congruent triangles
- The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law)
- It has rotational symmetry of order 2
- It has two lines of symmetry
Properties
- Opposite sides of parallelogram are parallel.
- Any line through the midpoint of a parallelogram bisects the area.
- Parallelograms are quadrilaterals.
References
Other websites
- Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
- Interactive Parallelogram --sides, angles and slope
- ↑ Template:Cite web
- ↑ Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, pp. 51-52.
- ↑ Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 22.