Parallelogram

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Different parallel types

In the field of geometry, a parallelogram —or parallelogram in Chile— is a quadrilateral whose pairs of opposite sides are equal and parallel two by two. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean Parallel Postulate, and no condition can be proved without appeal to the Euclidean Parallel Postulate or one of its equivalent formulations.

The three-dimensional counterpart of a parallelogram is a parallelepiped.

The etymology (Greek παραλληλ-όγραμμον, parallēl-ógrammon, a form of "parallel lines") reflects the definition.

Classes of parallelograms

The square, which has all its sides of equal length, and all its angles are straight.
The rombo, which has all its sides of equal length, and only two pairs of congruent angles. Any parallelogram that is neither a rectangle nor a rombo was traditionally called romboid, but this term is not used in modern mathematics.
The rectangle, which has only its opposite sides of equal length, and all its angles are straight.
The romboid, which has only opposite sides of equal length and only two pairs of congruent angles.

Properties

By definition of parallelogram:

Every quadrilateral has four vertices, four sides. Four inner angles.
The opposite sides of a parallelogram are parallel.

Properties of parallelograms deductible from their definition:

It inherits all the properties of quadrilateral:
The sum of the interior angles of all parallelogram is always equal to 360°.
The opposite sides are of equal length, (congruent).
The internal angles in two contiguous vertices are additional (Suman 180°).
The opposite internal angles are equal to measure.
The area of a parallelogram is the double of the area of a triangle formed by any of its diagonals and the contiguous sides of the figure.
Any secant straight cut to parallelogram at no more than two points.
All parallels are convex.
The diagonals of a parallelogram bise with each other in the "centre" of the parallelogram.
The "center" of the parallelogram is also the baricentro of the same.
Any dry straight that passes through the "centre" of a parallelogram divides its surface into two equal parts.
Any coplanar line that passes through the "baricentro" of a parallelogram is also "transversal of gravity" of it.

Properties caused by different applications:

Any undegenerated aphin transformation transforms a parallelogram into another parallelogram.
There is an infinite number of similar transformations that transform to a parallelogram given in a square.
A homeomorphism can be established between a parallelogram and a circumference.
A translation, a rotation of a parallelogram retains shape and size.

Given a parallelogram constructed using vectors:

The area of a parallelogram is equal to the magnitude (modulum) of the two contiguous vector product, considered as vectors. The opposite sides of a parallelogram are of equal length (congruent). The opposite angles of a parallelogram are equal to measure. The angles of two contiguous vertices are any additional (up to 180°). The sum of the interior angles of all parallelogram is always equal to 360°.

The «square» parallelogram has rotation symmetry of order 4 (45°) The parallelograms «rhomboid», «rhombus» and «rectangle», have rotation symmetry of order 2 (90°) If it has no reflection axis of symmetry, then it is a "rhomboid" parallelogram. If it has 2 diagonal reflection axes of symmetry, then it is a "rhombus" parallelogram. If it has 2 reflection axes of symmetry perpendicular to its sides, then it is a "rectangle" parallelogram. If it has 4 axes of reflection symmetry, then it is a "square" parallelogram.

Cases of symmetry for various classes of parallelograms

The parallelogram “square”, has symmetry of rotation of order 4 (90°).
The parallels are "romboid", "rombo" and "rectangle", with symmetry of rotation of order 2 (180°).
If you do not have any axis of reflection symmetry, then it is a parallelogram "romboid".
If you have 2 axes of diagonal reflection symmetry, then it is a parallelogram «rombo».
If you have 2 axes of symmetry of perpendicular reflection on your sides, then it is a parallelogram "rectangle".
If you have 4 axes of reflection symmetry, then it is a "square" parallelogram.

Some common metric properties

The perimeter of a parallelogram is 2 (a + bWhere a and b are the lengths of two contiguous sides any.
The sum of the squares of the sides is equal to the sum of the squares of the diagonals (see the rule of the parallelgram).
To calculate the area of a parallelogram, it can be considered as a figure composed of two congruent triangles and a rectangle, drawing heights of the vertices of the obtusous angles. And the triangle has the most measures

Formulas

All of the area formulas for general convex quadrilaterals apply to parallelograms. Other formulas are specific to parallelograms:

A parallelogram with base b and height h can be divided into a trapezoid and a right triangle, and rearranged into a rectangle, as shown in figure A on left. This means that the area of a parallelogram is equal to that of a rectangle with the same base and height:

{displaystyle K=bh.}

The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram

The base area × height formula can also be derived using the figure to the right. The area K of the parallelogram on the right (the blue area) is the total area of the rectangle minus the area of the two orange triangles. The area of the rectangle is

{displaystyle K_{text{rect}}=(B+A)times H,}

and the area of a single triangle is

{displaystyle K_{text{tri}}={frac {A}{2}}times H.,}

Therefore, the area of the parallelogram is

{displaystyle K=K_{text{rect}}-2times K_{text{tri}}=((B+A)times H)-(Atimes H)=Btimes H.}

Another area formula, for two sides B and C and angle θ, is

{displaystyle K=Bcdot Ccdot sin theta.,}

The area of a parallelogram with sides B and C (B I was. C) and angle $gamma$ in the intersection of the diagonals is given by

{displaystyle K={frac {|tan gamma |}{2}}cdot left|B^{2}-C^{2}right|.}

When the parallelogram is specified from the lengths B and C of two adjacent sides together with the length D₁ of any of the diagonals, then the area can be found from Heron's formula. specifically is

{displaystyle K=2{sqrt {S(S-B)(S-C)(S-D_{1})}}}

where ${displaystyle S=(B+C+D_{1})/2}$ and the main factor 2 comes from the fact that the chosen diagonal divides the parallelogram into "two" congruent triangles.

Parallelgram Formula
Area	${displaystyle A,=,acdot h_{a}=bcdot h_{b}=,left\|left\|,{overrightarrow {AB}},times ,{overrightarrow {AD}},right\|right\|}$ ${displaystyle A,=,acdot bcdot sin alpha =acdot bcdot sin beta ={frac {ecdot fcdot sin theta }{2}}}$
Altitude of a	${displaystyle h_{a},=,bcdot sin alpha =bcdot sin beta ={frac {A}{a}}}$
Height of b	${displaystyle h_{b},=,acdot sin alpha =acdot sin beta ={frac {A}{b}}}$
Diagonal (theorem of the cosine)	${displaystyle f={sqrt {a^{2}+b^{2}-2cdot acdot bcdot cos(alpha)}}}$ ${displaystyle e={sqrt {a^{2}+b^{2}+2cdot acdot bcdot cos(alpha)}}}$
Angles	${displaystyle alpha =gamma ;;;;;beta =delta ;;;;;beta =180^{circ }-alpha }$

Parallelogram Rule

Main article: Rule of parallelogram

The four sides of a parallelogram (AB, BC, CD and DA),
the four vertices (A, B, C and D) and their two diagonals (AC and BD).

There is a geometric identity that relates the sides of a parallelogram to its diagonals, called the parallelogram rule. This says that the sum of the squares of the lengths of the four sides of any parallelogram is equal to the sum of the squares of the lengths of the two diagonals. In mathematical notation, it is represented by the following formula:

{displaystyle (AB)^{2}+(BC)^{2}+(CD)^{2}+(DA)^{2}=(AC)^{2}+(BD)^{2}.,}

where A, B, C, and D are the vertices of the parallelogram.

Since the sides are equal two by two, the formula is usually represented simplified:

{displaystyle 2cdot ((AB)^{2}+(BC)^{2})=2cdot ((CD)^{2}+(DA)^{2})=(AC)^{2}+(BD)^{2}.,}

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