Parallelogram

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

Contents 1 Properties 2 Vector spaces 3 Proof that diagonals bisect each other 4 See also 5 External links

[edit] Properties

• The two parallel sides are of equal length.
• The area, <math>A</math>, of a parallelogram is <math>A = BH</math>, where <math>B</math> is the base of the parallelogram and <math>H</math> is its height.
• The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
• The area is also equal to the magnitude of the vector cross product of two adjacent sides.
• The diagonals of a parallelogram bisect each other.
• It is possible to create a tessellation of a plane with any parallelogram.
• The parallelogram is a special case of the trapezoid.
• The rectangle is a special case of the parallelogram.
• The rhombus is a special case of the parallelogram.

[edit] Vector spaces

In a vector space, addition of vectors is usually defined using the parallelogram law. The parallelogram law distinguishes Hilbert spaces from other Banach spaces.

[edit] Proof that diagonals bisect each other

To prove that the diagonals of a parallelogram bisect each other, first note a few pairs of equivalent angles:

<math>\angle ABE \cong \angle CDE</math>

<math>\angle BAE \cong \angle DCE</math>

Since they are angles that a transversal makes with parallel lines <math>AB</math> and <math>DC</math>.

Also, <math>\angle AEB \cong \angle CED</math> since they are a pair of vertical angles.

Therefore, <math>\triangle ABE \sim \triangle CDE</math> since they have the same angles.

From this similarity, we have the ratios

<math>{AB \over CD} = {AE \over CE} = {BE \over DE}</math>

Since <math>AB = DC</math>, we have

<math>{AB \over CD} = 1</math>.

Therefore,

<math>AE = CE</math>

<math>BE = DE</math>

<math>E</math> bisects the diagonals <math>AC</math> and <math>BD</math>.

[edit] See also

• Fundamental parallelogram
• Parallelogram of force
• Rhombus
• Synthetic geometry
• Gnomon (figure)

[edit] External links

• Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
• Eric W. Weisstein, Parallelogram at MathWorld.
• Interactive Parallelogram --sides, angles and slope
• Area of Parallelogram at cut-the-knot
• National Parallogram Dealers WebsiteNational Parallelogram Dealers Association
• Equilateral Triangles On Sides of a Parallelogram at cut-the-knot
• Varignon and Wittenbauer Parallelograms by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
• Van Aubel's theorem Quadrilateral with four squares by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
• Parallelogram Quiz
• Definition and properties of a parallelogram with animated applet
• Interactive applet showing parallelogram area calculation interactive appletar:متوازي أضلاع

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