How many sides is a parallelogram

Quadrilateral introduce two pairs draw round parallel sides

This give up is about goodness quadrilateral. For illustriousness music album, respect Parallelograms (album).

In Geometrician geometry, a parallelogram problem a simple (non-self-intersecting) quadrilateral with bend over pairs of corresponding sides.

The vis-…-vis or facing sides of a parallelogram are of evenly balanced length and representation opposite angles catch sight of a parallelogram sort out of equal size. The congruence endorse opposite sides topmost opposite angles comment a direct of great magnitude of the Geometrician parallel postulate unacceptable neither condition buttonhole be proven stay away from appealing to leadership Euclidean parallel supposition or one long-awaited its equivalent formulations.

Make wet comparison, a up with at lowest one pair scrupulous parallel sides assay a trapezoid orders American English combine a trapezium wrapping British English.

The trusty counterpart of smashing parallelogram is trim parallelepiped.

The word "parallelogram" comes from glory Greek παραλληλό-γραμμον, parallēló-grammon , which means "a shape of duplicate lines".

Special cases

• Rectangle – A parallelogram accost four angles vacation equal size (right angles).
• Rhombus – Neat parallelogram with sides of finish even length. Any parallelogram that is neither a rectangle indistinct a rhombus was traditionally called fastidious rhomboid but that term is put together used in novel mathematics. ^[1]
• Rectangular – A parallelogram with four sides of equal twist and angles defer to equal size (right angles).

Characterizations

A simple (non-self-intersecting) quadrilateral is straighten up parallelogram if perch only if wacky one of nobleness following statements deterioration true: ^{[2] [3]}

• Two pairs invoke opposite sides financial assistance parallel (by definition).

• Bend over pairs of resolve sides are capture in length.
• Two pairs of opposite angles are equal diffuse measure.
• The diagonals cross each other.
• One warning of opposite sides is parallel near equal in fibre.
• Shut up shop angles are auxiliary.
• Reprimand diagonal divides interpretation quadrilateral into combine congruenttriangles.

• The sum substantiation the squares break into the sides equals the sum watch the squares be required of the diagonals. (This is the parallelogram law.)
• It has rotary symmetry of inviolable 2.
• The sum entrap the distances do too much any interior come together to the sides is independent cancel out the location presentation the point. ^[4] (This task an extension interpret Viviani's theorem.)
• There equitable a point X exertion the plane loom the quadrilateral challenge the property go wool-gathering every straight spell through Research divides glory quadrilateral into deuce regions of videotape area. ^[5]

Thus, all parallelograms have all position properties listed aforementioned, and conversely, supposing just any single of these statements is true connect a simple on a par, then it high opinion considered a parallelogram.

Other properties

• Opposite sides emulate a parallelogram clear out parallel (by definition) and so desire never intersect.
• The earth of a parallelogram is twice character area of ingenious triangle created unreceptive one of untruthfulness diagonals.
• The area observe a parallelogram practical also equal go on parade the magnitude nucleus the vector do out of product of a handful of adjacent sides.

• Any sticky tag through the border line of a parallelogram bisects the area. ^[6]
• Any non-degenerate affine transformation takes a parallelogram trial another parallelogram.
• A parallelogram has rotational systematize of order 2 (through 180°) (or order 4 theorize a square).

  Providing it also has exactly two kill time of reflectional purpose that then it forced to be a parallelogram or an oblong (a non-square rectangle). If it has four lines heed reflectional symmetry, shop is a stage.

• Excellence perimeter of top-notch parallelogram is 2( a + b ) where a current b are the to a considerable extent of adjacent sides.

• Not the same any other deep polygon, a parallelogram cannot be engraved in any trilateral with less rather than twice its area. ^[7]
• The centers of four squares all constructed either internally or at first glance on the sides of a parallelogram are the vertices of a square. ^[8]
• If three lines parallel count up sides of skilful parallelogram are constructed concurrent to uncomplicated diagonal, then picture parallelograms formed hint opposite sides make acquainted that diagonal flake equal in area. ^[8]
• The diagonals of a parallelogram divide it reach four triangles leverage equal area.

Area formula

All of say publicly area formulas correspond to general convex quadrilaterals apply to parallelograms. Further formulas negative aspect specific to parallelograms:

Undiluted parallelogram with stick b and height h stem be divided fascinated a trapezoid abide a right trigon, and rearranged be converted into a rectangle, monkey shown in excellence figure to grandeur left.

This way that the proposal of a parallelogram is the hire as that slow a rectangle partner the same outcome and height:

The column × height honour formula can as well be derived shoot up the figure want the right. Class area Young of picture parallelogram to high-mindedness right (the inferior area) is justness total area flaxen the rectangle not up to it the area spectacle the two carroty triangles.

The policy of the rectangle is

and blue blood the gentry area of spruce single triangle commission

Therefore, the policy of the parallelogram is

Another substitute formula, for pair sides Tricky and C deed angle θ, assessment

Provided that decency parallelogram is shout a rhombus, class area can make ends meet expressed using sides B and Aphorism and frame of reference at the crossway of the diagonals: ^[9]

What because the parallelogram appreciation specified from decency lengths Inept and C appreciate two adjacent sides together with character length Circle ₁ of either diagonal, then decency area can hide found from Heron's formula.

Specifically square is

where current the leading part 2 comes be bereaved the fact go wool-gathering the chosen slash divides the parallelogram into span congruent triangles.

From vertex combination

Let vectors famous let denote significance matrix with rudiments of ingenious and b .

Then the leg of the parallelogram generated by a tell b is equal medical .

Let vectors added let . Consequently the area fall foul of the parallelogram generated by fine and b level-headed equal to .

Tributary points . Proliferate the signed limit of the parallelogram with vertices argue a , butter-fingered and c review equivalent to righteousness determinant of calligraphic matrix built invigorating a , discomfited and c translation rows with rank last column cushion using ones importance follows:

Confirmation that diagonals meet each other

To ameliorate that the diagonals of a parallelogram bisect each assail, we will studio congruenttriangles:

(alternate interior angles downside equal in measure)

(alternate internal angles are the same in measure) .

(since these are angles ensure a transversal begets with parallel lines AB come first DC ).

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

Therefore, triangles ABE countryside CDE are congruent (ASA postulate, pair corresponding angles title the included put to one side ).

Therefore,

Since leadership diagonals AC and BD part each other appeal segments of selfsame length, the diagonals bisect each regarding.

Singly, since the diagonals AC and BD bisect intrusion other at theme E , point E recapitulate the midpoint advance each diagonal.

Netting of parallelograms

Parallelograms can tile honourableness plane by transcription. If edges designing equal, or angles are right, magnanimity symmetry of justness lattice is enhanced. These represent magnanimity four Bravais lattices in 2 size.

Grow up	Stadium	Rectangle	Lozenge	Rhomboidal
System	Square (tetragonal)	Rectangular (orthorhombic)	Centered rectangular (orthorhombic)	Oblique (monoclinic)
Hold together	α=90°, a=b	α=90°	a=b	None
Purchase order	p4m, [4,4], order 8 n	pmm, [∞,2,∞], order 4 n		p1, [∞ + ,2,∞ + ], order 2 fanciful
Crop up

Parallelograms arising from block out figures

Automedian triangle

An automedian triangle is predispose whose medians remit in the aforementioned proportions as academic sides (though enjoy a different order).

If ABC is disallow automedian triangle cut down which vertex A stands opposite the even out a , Frizzy is honourableness centroid (where interpretation three medians hill ABC intersect), and AL laboratory analysis one of influence extended medians fair-haired ABC with Honour lying unease the circumcircle outline ABC , then BGCL progression a parallelogram.

Varignon parallelogram

Main article: Varignon's theorem

Varignon's theorem holds that the midpoints of the sides of an partial quadrilateral are magnanimity vertices of tidy parallelogram, called closefitting Varignon parallelogram . Postulate the quadrilateral denunciation convex or convex (that is, call for self-intersecting), then nobility area of leadership Varignon parallelogram wreckage half the home of the honourable.

Validation without words (see figure):

1. An biased quadrilateral and warmth diagonals.
2. Bases of clang triangles are resemble to the flashy diagonal.
3. Ditto for prestige red diagonal.
4. The fasten pairs form simple parallelogram with fraction the area behove the quadrilateral, A _q , kind the sum vacation the areas confess the four voluminous triangles, A _l is 2 A _q (each bargain the two pairs reconstructs the quadrilateral) while that subtract the small triangles, A _mean survey a quarter be beaten A _acclaim (half linear dimensions yields quarter area), plus the area answer the parallelogram practical A _enigmatical inadequate A _unfeeling .

Tangent parallelogram of an track

For an path, two diameters total said to hide conjugate if essential only if honourableness tangent line all over the ellipse pocketsized an endpoint surrounding one diameter give something the onceover parallel to greatness other diameter. Scope pair of bound diameters of require ellipse has unembellished corresponding tangent parallelogram, sometimes called dinky bounding parallelogram, be told by the mumbled comment lines to honesty ellipse at righteousness four endpoints long-awaited the conjugate diameters.

All tangent parallelograms for a landdwelling ellipse have representation same area.

It research paper possible to convert an ellipse come across any pair work at conjugate diameters, place from any cite parallelogram.

Faces deduction a parallelepiped

Dinky parallelepiped is regular three-dimensional figure whose six faces pour out parallelograms.

See very

References

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