A parallelogram is a special type of quadrilateral, defined by the fact that both pairs of opposite sides are parallel. This gives the parallelogram several other properties, all of which remain true regardless of a particular parallelogram's dimensions. These properties also hold true for special cases that have additional restrictions.

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## The basics

The opposite sides and opposite angles of a parallelogram are congruent. A parallelogram's two diagonals bisect each other, and each of them divides the parallelogram into two congruent triangles. The four angles of a parallelogram, like those of all quadrilaterals, add up to 360 degrees; the sum of two angles adjacent to the same side of a parallelogram is always 180 degrees. This is a result of one pair of sides being transversal lines of the other pair.

## Formulas

The perimeter of a parallelogram is equal to the sum of its sides; its area is the product of one of its sides, called the base, and the corresponding height, defined as the perpendicular distance between the base and its opposite side. Applying the trigonometric law of sines, you can further define the height as the product of the non-base side and the sine of the parallelogram's acute angle. Consider a parallelogram ABCD, its acute angle E and its unknown height H; take CD as the base. The triangle formed by H, the side CA and a small portion of the base is a right triangle, with the right angle opposite CA. Apply the law of sines:

H / sin(E) = CA / sin(90°) H = (sin(E) * CA) / sin(90°)

Because sin(90°) = 1, H = (sin(E) * CA. As a result, you can also find the area of a parallelogram by calculating the product of its sides and the sine of its acute angle.

## Special cases

The parallel could be considered the parent shape of the rhombus, the rectangle and the square, since all three shapes obey all rules of parallelograms with some additional restrictions. More specifically, any parallelogram whose sides are all equal is called a rhombus, while any parallelogram whose angles are all 90 degrees is a rectangle. A square combines the restrictions of both the rhombus and the rectangle, requiring all sides to be equal and all angles to be 90 degrees.

## Inscribed parallelograms

Take any quadrilateral, whether self-intersecting or not, and find the midpoint of each of its sides. If you then draw lines that connect the midpoints sequentially, the resulting shape will always be a parallelogram. This can be easily proven by drawing a diagonal of the original quadrilateral, dividing it into two triangles, and applying the rules that affect the midsegment of a triangle. Consider the quadrilateral ABCD, the shape formed by the midpoints JKLM and the diagonal BD. By definition, the line JM is the midsegment of the triangle ABD, and is therefore parallel to BD and half its length. Similarly, the line LK is the midsegment of the triangle DBC, and is therefore parallel to BD and half its length. Therefore, JM and LK are parallel and congruent and the shape JKLM is a parallelogram.