# Parallelogram. Formulas and Properties of a Parallelogram

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Definition of a parallelogram
Characterizations of a parallelogram
The basic properties of a parallelogram
The sides of a parallelogram
The diagonal of a parallelogram
The perimeter of a parallelogram
The area 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.

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:

AC

^{2}+ BD

^{2}= AB

^{2}+ BC

^{2}+ CD

^{2}+ AD

^{2}

## 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 = | d_{1} |

2 | |

BO = DO = | d_{2} |

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:

AC

^{2}+ BD

^{2}= 2AB

^{2}+ 2BC

^{2}

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 = | √d_{1}^{2} + d_{2}^{2} - 2d_{1}d_{2}·cosγ | = | √d_{1}^{2} + d_{2}^{2} + 2d_{1}d_{2}·cosδ |

2 | 2 |

b = | √d_{1}^{2} + d_{2}^{2} + 2d_{1}d_{2}·cosγ | = | √d_{1}^{2} + d_{2}^{2} - 2d_{1}d_{2}·cosδ |

2 | 2 |

2. Formula of parallelogram sides in terms of diagonals and other side:

a = | √2d_{1}^{2} + 2d_{2}^{2} - 4b^{2} |

2 |

b = | √2d_{1}^{2} + 2d_{2}^{2} - 4a^{2} |

2 |

3. Formula of parallelogram sides in terms of altitude (height) and sine of an angle:

a = | h_{b} |

sin α |

b = | h_{a} |

sin α |

4. Formula of parallelogram sides in terms of area and altitude (height):

a = | A |

h_{a} |

b = | A |

h_{b} |

## 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 d

_{1}, and shorter - d_{2}### Diagonal of a parallelogram formulas:

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

d
d

_{1}= √a

^{2}+ b

^{2}- 2ab·cosβ

_{2}= √a

^{2}+ b

^{2}+ 2ab·cosβ

2. Formula of parallelogram diagonal in terms of sides and cosine α (cosine theorem)

d
d

_{1}= √a

^{2}+ b

^{2}+ 2ab·cosα

_{2}= √a

^{2}+ b

^{2}- 2ab·cosα

3. Formula of parallelogram diagonal in terms of two sides and other diagonal:

d

d

_{1}= √2a

^{2}+ 2b

^{2}- d

_{2}

^{2}

_{2}= √2a

^{2}+ 2b

^{2}- d

_{1}

^{2}

4. Formula of parallelogram diagonal in terms of area, other diagonal and angles between diagonals:

d_{1} = | 2A | = | 2A |

d_{2}·sinγ | d_{2}·sinδ |

d_{2} = | 2A | = | 2A |

d_{1}·sinγ | d_{1}·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 + √2d

P = 2b + √2d

_{1}

^{2}+ 2d

_{2}

^{2}- 4a

^{2}

_{1}

^{2}+ 2d

_{2}

^{2}- 4b

^{2}

3. Formula of parallelogram perimeter in terms of side, height and sine of an angle:

P = | 2(b + | h_{b} | ) |

sin α |

P = | 2(a + | h_{a} | ) |

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 · h

A = b · h

_{a}

A = b · h

_{b}

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 = | 1 | d_{1}d_{2} sin γ |

2 |

A = | 1 | d_{1}d_{2} sin δ |

2 |

Geometry formulas
Square. Formulas and Properties of a Square
Rectangle. Formulas and Properties of a Rectangle
Parallelogram. Formulas and Properties of a Parallelogram
Rhombus. Formulas and Properties of a Rhombus
Circle, disk, segment, sector. Formulas and properties
Ellipse. Formulas and properties of ellipse
Cylinder. Formulas and properties of a cylinder
Cone. Formulas, characterizations and properties of a cone
Area. Formulas of area
Perimeter. Formulas of perimeter
Volume. Formulas of volume
Surface Area Formulas

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