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Parallelogram. Formulas and Properties of a Parallelogram

Definition.
Parallelogram is a quadrilateral whose opposite sides are parallel and pairwise equal(lie on parallel lines)..
Parallelograms differ in size of an adjacent sides and angles but opposite angles is equal.
Image of a parallelogram Image of a parallelogram
Fig.1 Fig.2

Characterizations of a parallelogram

Quadrilateral ABCD is a parallelogram, if at least one of the following conditions:
1. Quadrilateral has two pairs of parallel sides:
AB||CD, BC||AD
2. Quadrilateral has a pair of parallel sides with equal lengths:
AB||CD, AB = CD (или BC||AD, BC = AD)
3. Opposite sides are equal in the quadrilateral:
AB = CD, BC = AD
4. Opposite angles are equal in the quadrilateral:
∠DAB = ∠BCD, ∠ABC = ∠CDA
5. Diagonals bisect the intersection point in the quadrilateral:
AO = OC, BO = OD
6. The sum of the quadrilateral angles adjacent to any side is 180°:
∠ABC + ∠BCD = ∠BCD + ∠CDA = ∠CDA + ∠DAB = ∠DAB + ∠DAB = 180°
7. The sum of the diagonals squares equals the sum of the sides squares in the quadrilateral:
AC2 + BD2 = AB2 + BC2 + CD2 + AD2

The basic properties of a parallelogram

Square, rectangle and rhombus is a parallelogram.
1. Opposite sides of a parallelogram have the same length:

AB = CD, BC = AD
2. Opposite sides of a parallelogram are parallel:

AB||CD,   BC||AD
3. Opposite angles of a parallelogram are equal:

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

∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°
5. The sum of the parallelogram angles adjacent to any sides is 180°:
∠ABC + ∠BCD = ∠BCD + ∠CDA = ∠CDA + ∠DAB = ∠DAB + ∠DAB = 180°
6. Each diagonal divides the parallelogram into two equal triangle
7. Two diagonals is divided parallelogram into two pairs of equal triangles
8. The diagonals of a parallelogram intersect and intersection point separating each one in half:

AO = CO =  d1
2
BO = DO =  d2
2
9. Intersection point of the diagonals is called a center of parallelogram symmetry
10. Sum of the diagonals squares equals the sum of sides squares in parallelogram:
AC2 + BD2 = 2AB2 + 2BC2
11. Bisectors of parallelogram opposite angles are always parallel
12. Bisectors of parallelogram adjacent angles always intersect at right angles (90°)

The sides of a parallelogram

Sides of a parallelogram formulas:

1. Formula of parallelogram sides in terms of diagonal and angle between the diagonals:

a = d12 + d22 - 2d1d2·cosγ = d12 + d22 + 2d1d2·cosδ
22
b = d12 + d22 + 2d1d2·cosγ = d12 + d22 - 2d1d2·cosδ
22
2. Formula of parallelogram sides in terms of diagonals and other side:

a = 2d12 + 2d22 - 4b2
2
b = 2d12 + 2d22 - 4a2
2
3. Formula of parallelogram sides in terms of altitude (height) and sine of an angle:
a = hb
sin α
b = ha
sin α
4. Formula of parallelogram sides in terms of area and altitude (height):
a = A
ha
b = A
hb

The diagonal of a parallelogram

Definition.
The diagonal of a parallelogram is any segment that connects two vertices of a parallelogram opposite angles.
Parallelogram has two diagonally - a longer let be d1, and shorter - d2

Diagonal of a parallelogram formulas:

1. Formula of parallelogram diagonal in terms of sides and cosine β (cosine theorem)

d1 = √a2 + b2 - 2ab·cosβ
d2 = √a2 + b2 + 2ab·cosβ
2. Formula of parallelogram diagonal in terms of sides and cosine α (cosine theorem)

d1 = √a2 + b2 + 2ab·cosα
d2 = √a2 + b2 - 2ab·cosα
3. Formula of parallelogram diagonal in terms of two sides and other diagonal:

d1 = √2a2 + 2b2 - d22

d2 = √2a2 + 2b2 - d12
4. Formula of parallelogram diagonal in terms of area, other diagonal and angles between diagonals:

d1 = 2A = 2A
d2·sinγd2·sinδ
d2 = 2A = 2A
d1·sinγd1·sinδ

The perimeter of a parallelogram

Definition.
The perimeter of a parallelogram is the sum of the all parallelogram sides lengths.

Perimeter of a parallelogram formulas:

1. Formula of parallelogram perimeter in terms of sides:

P = 2a + 2b = 2(a + b)
2. Formula of parallelogram perimeter in terms of one side and diagonals:

P = 2a + √2d12 + 2d22 - 4a2

P = 2b + √2d12 + 2d22 - 4b2
3. Formula of parallelogram perimeter in terms of side, height and sine of an angle:
P = 2(b + hb)
sin α
P = 2(a + ha)
sin α

The area of a parallelogram

Definition.
The area of a parallelogram the space is restricted parallelogram sides or within the perimeter of a parallelogram.

Area of a parallelogram formulas:

1. Formula of parallelogram area in terms of side and height:

A = a · ha
A = b · hb
2. Formula of parallelogram area in terms of sides and sine of an angle between this sides:

A = ab sinα

A = ab sinβ
3. Formula of parallelogram area in terms of diagonals and sine of an angle between diagonals:

A = 1d1d2 sin γ
2
A = 1d1d2 sin δ
2

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