Parallelograms

Definition of a parallelogram

We know that a quadrilateral has 4 sides. In a parallelogram, these 4 sides consist of 2 pairs of opposite parallel sides.

The following diagram illustrates a parallelogram.

Fig. 1: Parallelogram illustration.

In the above figure:

• AB // CD
• AC // BD

Properties of parallelograms

In addition to the above, we can identify various properties of parallelograms.

We will use the following parallelogram ABDC with diagonals d₁=BC and d₂=AD to illustrate the properties.

Fig. 2: Parallelogram with diagonals, d1 and d2.

• In a parallelogram, the opposite side are equal.

  This means that in the above parallelogram, AB=CD and AC=BD.

• In a parallelogram, the opposite angles are equal.

  This means that in the above parallelogram, ∠CAB=∠CDB and ∠ACD=∠ABD

• In a parallelogram, consecutive angles are supplementary.

  In any parallelogram, you can identify 4 pairs of consecutive angles. These are always supplementary (which means the angles sum up to 180 degrees). In the above parallelogram:

  ∠CAB + ∠ABD = 180,

  ∠ABD + ∠BDC = 180,

  ∠BDC + ∠DCA = 180,

  ∠DCA + ∠CAB = 180.

• If any angle in a parallelogram is a right angle, that means all 4 internal angles are right angles.

  This is a direct consequence of the above property. If any angle in a parallelogram is a right angle, then the adjacent angle is 180-90=90 (according to the above property). In turn, the next adjacent angle will be a right angle and so on. Therefore, in any parallelogram, if you identify any angle as a right angle, you can directly conclude that all 4 angles are right angles.

• The diagonals of a parallelogram bisect each other.

  In the above parallelogram, the point O is the mid-point of both the diagonals d₁ and d₂.

• Each diagonal of a parallelogram separates the parallelogram into two congruent triangles.

  In the above parallelogram, the diagonal d₁ would divide the parallelogram into two congruent triangles, ΔABC and ΔBCD. Similarly, the diagonal d₂ would divide the parallelogram into two congruent triangles, ΔABD and ΔACD.

Area of parallelograms

Fig. 3: Parallelogram with base b and height h.

The area of a parallelogram is given by the formula:

Area = b × h

where b = base, h = height

Now the value, b, is the length of the side AB, which is considered the base here. Conventionally, one of the longer sides of the parallelogram is taken to be the base. The value h is the height of the parallelogram. It is also sometimes called the altitude. The height is the length of the line drawn from the base to its opposite side. The height is perpendicular to the base.

Parallelogram: Example problems

In this section we explore examples of math problems that you may encounter about parallelograms and their properties.

Different types of parallelogram shapes

In this section, we will identify 3 special types of parallelograms, each with its own characteristics and properties:

1. Rhombus

2. Rectangle

3. Square

Rhombus

A rhombus is a quadrilateral with all 4 of its sides equal in length (equilateral). It turns out that the opposite pairs of sides of a rhombus are always parallel. This makes every rhombus a parallelogram. Conversely, we can say that an equilateral parallelogram is a rhombus. The diagonals of a rhombus always bisect each other at right angles.

Since a rhombus is a special type of parallelogram, a rhombus exhibits all the properties of a parallelogram as well.

Rectangle

A rectangle is a parallelogram with all of its internal angles as right angles. Since all angles in a rectangle are equal, it is equiangular.

Since a rectangle is a special type of parallelogram, it exhibits all the properties of a parallelogram as well.

Square

A square is a quadrilateral with all 4 of its sides equal and with all of its angles as right angles. This makes a square a type of parallelogram, a type of rhombus, and a type of rectangle! Thus, a square demonstrates all the properties of parallelograms, rhombuses, and rectangles.