In
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, a
parallelogram is a
convex'The word convex means curving out or bulging outward, as opposed to concave. Convex or convexity may refer to:Mathematics:* Convex set, a set of points containing all line segments between each pair of its points...
quadrilateralIn Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
with two pairs of
parallelParallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean Parallel Postulate and neither condition can be proven without appealing to the Euclidean Parallel Postulate or one of its equivalent formulations. The three-dimensional counterpart of a parallelogram is a
parallelepipedIn geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...
.
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In
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, a
parallelogram is a
convex'The word convex means curving out or bulging outward, as opposed to concave. Convex or convexity may refer to:Mathematics:* Convex set, a set of points containing all line segments between each pair of its points...
quadrilateralIn Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
with two pairs of
parallelParallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean Parallel Postulate and neither condition can be proven without appealing to the Euclidean Parallel Postulate or one of its equivalent formulations. The three-dimensional counterpart of a parallelogram is a
parallelepipedIn geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...
.
The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects the definition.
Characterizations
A
convex'The word convex means curving out or bulging outward, as opposed to concave. Convex or convexity may refer to:Mathematics:* Convex set, a set of points containing all line segments between each pair of its points...
quadrilateralIn Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
is a parallelogram
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
any one of the following statements is true:
- Each diagonal
A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
divides the quadrilateral into two congruent triangles with the same orientation.
- The opposite sides are equal in length.
- The diagonals bisect each other.
- The opposite angles are equal in measure.
- The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law
In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals...
.)
- It possesses rotational symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
.
- One pair of opposite sides are parallel
-Mathematics and science:* Parallel , an imaginary east-west line circling a globe* Parallel circuits, as opposed to series* Parallel * Parallel evolution* Parallel transport* Parallel of declination, used in astronomy-Computing:...
and equal in length.
- Adjacent
Adjacent is an adjective meaning contiguous, adjoining or abuttingIn geometry, adjacent is when sides meet to make an angle.In graph theory adjacent nodes in a graph are linked by an edge....
angles are supplementarySupplementary angles are pairs of angles that add up to 180 degrees. Thus the supplement of an angle of x degrees is an angle of degrees....
.
Properties
- Opposite sides of a parallelogram are parallel (by definition) and so 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
Adjacent is an adjective meaning contiguous, adjoining or abuttingIn geometry, adjacent is when sides meet to make an angle.In graph theory adjacent nodes in a graph are linked by an edge....
sides.
- Any line through the midpoint of a parallelogram bisects the area.
- Any non-degenerate affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
takes a parallelogram to another parallelogram.
There is an infinite number of affine transformations which take any given parallelogram to a squareIn geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
.
- A parallelogram has rotational symmetry
Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...
of order 2 (through 180°). If it also has two lines of reflectional symmetry then it must be a rhombus or an oblong.
- The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent sides.
Types of parallelogram
- rhomboid
Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are oblique.A parallelogram with sides of equal length is a rhombus but not a rhomboid....
- A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angleIn geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...
s
- Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
- A parallelogram with four angles of equal size
- Rhombus
In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...
- A parallelogram with four sides of equal length.
- Square
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
- A parallelogram with four sides of equal length and four angles of equal size (right angles).
Area formulas
- The area K 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
-
- and the area of a single orange triangle is
-
- Therefore, the area of the parallelogram is
-
- Another area formula, for two sides B and C and angle
, is
-
- The area of a parallelogram with sides B and C (B ≠ C) and angle
at the intersection of the diagonals is given by
-
The area on coordinate system
Let vectors 
and let 
denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to 
.
Let vectors 
and let 
Then the area of the parallelogram generated by a and b is equal to 
.
Let points 
. 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:
Proof that diagonals bisect each other
To prove that the diagonals of a parallelogram bisect each other, we will use congruentIn geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
triangles:
(alternate interior angles are equal in measure)
(alternate interior angles are equal in measure).
(since these are angles that a transversal makes with parallel linesParallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
and 
).
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,
Since the diagonals 
and 
divide each other into segments of equal length, the diagonals bisect each other.
Separately, since the diagonals 
and 
bisect each other at point 
, point 
is the midpoint of each diagonal.
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