# Parallelogram: Wikis

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# Encyclopedia

Parallelogram

This parallelogram is a rhomboid as its angles are oblique.
Edges and vertices 4
Symmetry group C2 (2)

In geometry, a parallelogram is a quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are equal. The three-dimensional counterpart of a parallelogram is a parallelepiped.

The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects the definition.

## Properties

• Opposite sides of a parallelogram are equal in length.
• Opposite angles of a parallelogram are equal in measure.
• The area, A, of a parallelogram is A = bh, where b is the base of the parallelogram and h is its height.
• Opposite sides of a parallelogram will never intersect.
• The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
• The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.
• The diagonals of a parallelogram bisect each other.
• Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
There is an infinite number of affine transformations which take any given parallelogram to a square.'
• A general parallelogram (that is not a rhombus or a rectangle) has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

## Types of parallelogram

• Rhomboid - A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles
• Rectangle - A parallelogram with four angles of equal size (right angles).
• Rhombus - A parallelogram with four sides of equal length.
• Square - A parallelogram with four sides of equal length and four angles of equal size (right angles).

## Proof that diagonals bisect each other

To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles:

$\angle ABE \cong \angle CDE$ (alternate interior angles are equal in measure)
$\angle BAE \cong \angle DCE$ (alternate interior angles are equal in measure).

(since these are angles that a transversal makes with parallel lines AB and DC ).

Also, side AB is equal in length to side DC, since opposite sides of a parallelogram are equal in length.

Therefore triangles ABE and CDE are congruent (ASA postulate, two corresponding angles and the included side).

Therefore, [[Media:Media:Example.ogg[[Media:Example.ogg[[File:Insertformulahere]]]]]]

AE = CE
BE = DE.

Since the diagonals AC and BD divide each other into segments of equal length, the diagonals bisect each other.

Separately, since the diagonals AC and BD  bisect each other at point E, point E  is the midpoint of each diagonal.

## The area formula

Area of the parallelogram is in blue

The area formula,

$A = B \times H,\,$

can be derived as follows:

The area of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is

$A_\text{rect} = (B+A) \times H\,$

and the area of a single orange triangle is

$A_\text{tri} = \frac{1}{2} A \times H\,$ or $S_\text{tri} = \frac{1}{2} bh.$

Therefore, the area of the parallelogram is

$A = A_\text{rect} - 2 \times A_\text{tri} = \left( (B+A) \times H \right) - \left( A \times H \right) = B \times H.\,$

## Computing the area of a parallelogram

Let $a,b\in\R^2$ and let $V=[a\ b]\in\R^{2\times2}$ denote the matrix with columns a and b. Then the area of the parallelogram generated by a and b is equal to | det(V) |

Let $a,b\in\R^n$ and let $V=[a\ b]\in\R^{n\times2}$ Then the area of the parallelogram generated by a and b is equal to $\sqrt{\det(V^T V)}$

Let $a,b,c\in\R^2$. Then the area of the parallelogram with vertices at a, b and c is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:

$V = \left| \det \begin{bmatrix} a_1 & a_2 & 1 \ b_1 & b_2 & 1 \ c_1 & c_2 & 1 \end{bmatrix} \right|.$

# Simple English

[[File:|right|250px]] A parallelogram is a polygon with four sides. 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),

$AE = CE$
$BE = DE.$

## References

1. "Online Etymology Dictionary". etymonline.com. Retrieved 10 January 2011.