** Parallelogram vs Quadrilateral
**

Quadrilaterals and parallelograms are polygons found in Euclidean Geometry. Parallelogram is a special case of the quadrilateral. Quadrilaterals can be either planar (2D) or 3 Dimensional while parallelograms are always planar.

**Quadrilateral**

Quadrilateral is a polygon with four sides. It has four vertices, and the sum of the internal angles is 3600 (2π rad). Quadrilaterals are classified into self-intersecting and simple quadrilateral categories. The self-intersecting quadrilaterals have two or more sides crossing each other, and smaller geometric figures (such as triangles are formed inside the quadrilateral).

The simple quadrilaterals are also divided into convex and concave quadrilaterals. Concave quadrilaterals have adjacent sides forming reflex angles inside the figure. The simple quadrilaterals that don’t have reflex angles internally are convex quadrilaterals. The convex quadrilaterals can always have tessellations.

A major part of the geometry of quadrilaterals at the initial levels concerns the convex quadrilaterals. Some quadrilaterals are very familiar to us from the days of elementary schools. Following is a diagram showing different convex quadrilaterals.

**Parallelogram**

Parallelogram can be defined as the geometric figure with four sides, with opposite sides parallel to each other. More precisely it is a quadrilateral with two pairs of parallel sides. This parallel nature gives many geometric characteristics to the parallelograms.

A quadrilateral is a parallelogram if following geometric characteristics are found.

*• Two pairs of opposing sides are equal in length. (AB = DC, AD = BC)*

*• Two pairs of opposing angles are equal in size. ([latex]D\hat{A}B = B\hat{C}D, A\hat{D}C = A\hat{B}C[/latex])*

*• If the adjacent angles are supplementary [latex]D\hat{A}B + A\hat{D}C = A\hat{D}C + B\hat{C}D = B\hat{C}D + A\hat{B}C = A\hat{B}C + D\hat{A}B = 180^{\circ} = \pi rad[/latex]*

*• A pair of sides, which are opposing each other, is parallel and equal in length. ( AB = DC & AB∥DC)*

*• The diagonals bisect each other (AO = OC, BO = OD)*

*• Each diagonal divides the quadrilateral into two congruent triangles. (∆ADB ≡ ∆BCD, ∆ABC ≡ ∆ADC)*

Further, the sum of the squares of the sides is equal to the sum of the squares of diagonals. This is sometimes referred to as the * parallelogram law* and has widespread applications in physics and engineering. (AB

^{2 }+ BC

^{2 }+ CD

^{2 }+ DA

^{2 }= AC

^{2 }+ BD

^{2})

Each of the above characteristics can be used as properties, once it’s established that the quadrilateral is a parallelogram.

Area of the parallelogram can be calculated by the product of the length of one side and the height to the opposite side. Therefore, area of the parallelogram can be stated as

*Area of parallelogram = base × height = AB×h*

The area of the parallelogram is independent of the shape of individual parallelogram. It is dependent only on the length of base and the perpendicular height.

If the sides of a parallelogram can be represented by two vectors, the area can be obtained by the magnitude of the vector product (cross product) of the two adjacent vectors.

If sides AB and AD are represented by the vectors ([latex]\overrightarrow{AB}[/latex]) and ([latex]\overrightarrow{AD}[/latex]) respectively, the area of the parallelogram is given by [latex]\left | \overrightarrow{AB}\times \overrightarrow{AD} \right | = AB\cdot AD \sin \alpha [/latex], where α is the angle between [latex]\overrightarrow{AB}[/latex] and [latex]\overrightarrow{AD}[/latex].

Following are some advanced properties of the parallelogram;

*• The area of a parallelogram is twice the area of a triangle created by any of its diagonals.*

*• The area of the parallelogram is divided in half by any line passing through the midpoint.*

*• Any non-degenerate affine transformation takes a parallelogram to another parallelogram*

*• A parallelogram has rotational symmetry of order 2*

*• The sum of the distances from any interior point of a parallelogram to the sides is independent of the location of the point*

**What is the difference between Parallelogram and Quadrilateral?**

• Quadrilaterals are polygons with four sides (sometimes called tetragons) while parallelogram is a special type of a quadrilateral.

• Quadrilaterals can have their sides in different planes (in 3d space) while all sides of the parallelogram lies on the same plane (planar/ 2dimensional).

• Interior angles of the quadrilateral can take any value (including reflex angles) such that they add up to 3600. Parallelograms can only have obtuse angles as the maximum type of angle.

• Four sides of the quadrilateral can be of different lengths while the opposite sides of the parallelogram are always parallel to each other and equal in length.

• Any diagonal divides the parallelogram into two congruent triangles, while the triangles formed by the diagonal of a general quadrilateral are not necessarily congruent.

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