Parallelogram
A parallelogram is a special type of quadrilateral that has equal and parallel opposite sides.
The perimeter of a parallelogram:
1. Formula of parallelogram perimeter in terms of sides:
2. Formula of parallelogram perimeter in terms of one side and diagonals:
3. Formula of parallelogram perimeter in terms of side, height and sine of an angle:
The area of a parallelogram
1. Formula of parallelogram area in terms of side and height:
2. Formula of parallelogram area in terms of sides and sine of an angle between this sides:
3. Formula of parallelogram area in terms of diagonals and sine of an angle between diagonals:
Diagonal of a parallelogram:
1. Formula of parallelogram diagonal in terms of sides and cosine β (cosine theorem)
2. Formula of parallelogram diagonal in terms of sides and cosine α (cosine theorem)
3. Formula of parallelogram diagonal in terms of two sides and other diagonal:
4. Formula of parallelogram diagonal in terms of area, other diagonal and angles between diagonals:
Characterizations of a parallelogram:
Quadrilateral ABCD is a parallelogram, if at least one of the following conditions:

 Quadrilateral has two pairs of parallel sides:
ABCD, BCAD

 Quadrilateral has a pair of parallel sides with equal lengths:
ABCD, AB = CD (или BCAD, BC = AD)

 Opposite sides are equal in the quadrilateral:
AB = CD, BC = AD

 Opposite angles are equal in the quadrilateral:
∠DAB = ∠BCD, ∠ABC = ∠CDA

 Diagonals bisect the intersection point in the quadrilateral:
AO = OC, BO = OD

 The sum of the quadrilateral angles adjacent to any side is 180°:
∠ABC + ∠BCD = ∠BCD + ∠CDA = ∠CDA + ∠DAB = ∠DAB + ∠DAB = 180°

 The sum of the diagonals squares equals the sum of the sides squares in the quadrilateral:
AC^{2} + BD^{2} = AB^{2} + BC^{2} + CD^{2} + AD^{2}
The basic properties of a parallelogram:

 Opposite sides of a parallelogram have the same length:
AB = CD, BC = AD

 Opposite sides of a parallelogram are parallel:
ABCD, BCAD

 Opposite angles of a parallelogram are equal:
∠ABC = ∠CDA, ∠BCD = ∠DAB

 Sum of the parallelogram angles is equal to 360°:
∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°

 The sum of the parallelogram angles adjacent to any sides is 180°:
∠ABC + ∠BCD = ∠BCD + ∠CDA = ∠CDA + ∠DAB = ∠DAB + ∠DAB = 180°

 Each diagonal divides the parallelogram into two equal triangle
 Two diagonals is divided parallelogram into two pairs of equal triangles
 The diagonals of a parallelogram intersect and intersection point separating each one in half:
AO = CO = d_{1}/2
BO = DO = d_{2}/2

 Intersection point of the diagonals is called a center of parallelogram symmetry
 Sum of the diagonals squares equals the sum of sides squares in parallelogram:
AC^{2} + BD^{2} = 2AB^{2} + 2BC^{2}
 Bisectors of parallelogram opposite angles are always parallel
 Bisectors of parallelogram adjacent angles always intersect at right angles (90°)