Parallelograms

Think of the markings in a zebra crossing. These are closed 4-sided shapes with opposite sides that are equal and parallel. Such shapes are known as parallelograms.

A four-sided plane figure is known as a quadrilateral. In this article, we are going to look at the special type of quadrilateral known as a parallelogram.

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.

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

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.