** Parallelogram vs Trapezoid **

Parallelogram and trapezoid (or trapezium) are two convex quadrilaterals. Even though these are quadrangles, the geometry of the trapezoid differs significantly from the parallelograms.

**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*

**Trapezoid**

Trapezoid (or **Trapezium** in British English) is a convex quadrilateral where at least two sides are parallel and unequal in length. The parallel sides of the trapezoid are known as the bases and the other two sides are called the legs.

Following are main characteristics of trapezoids;

*• If the adjacent angles are not on the same base of the trapezoid, they are supplementary angles. i.e. they add up to 180° ([latex]B\hat{A}D+A\hat{D}C = A\hat{B}C+B\hat{C}D = 180^{\circ}[/latex])*

*• *Both diagonals of a trapezium intersect at the same ratio (ratio between the section of the diagonals are equal).

*• *If a and b are bases and c,d are legs, the lengths of the diagonals are given by

[latex]\sqrt{\frac{ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}[/latex]

and

[latex]\sqrt{\frac{ab^{2}-a^{2}b-ac^{2}+bc^{2}}{b-a}}[/latex]

Area of the trapezoid can be calculated using following formula

*Area of trapezoid = *[latex]\frac{a+b}{2}\times h[/latex]

**What is the difference between Parallelogram and Trapezoid (Trapezium)?**

*• *Both parallelogram and trapezoid are convex quadrilaterals.

*• *In a parallelogram, both pairs of the opposing sides are parallel while, in a trapezoid, only a pair is parallel.

*• *The diagonals of the parallelogram bisect each other (1:1 ratio) while the diagonals of the trapezoid intersects with a constant ratio between the sections.

*• *The area of the parallelogram depends on the height and the base while the area of the trapezoid depends on the height and the mid-segment.

*• *The two triangles formed by a diagonal in a parallelogram are always congruent while the triangles of the trapezoid can either be congruent or not.

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