Properties of Parallelogram

The properties of a parallelogram help us to identify a parallelogram from a given set of figures easily and quickly. Before we learn about the properties, let us first know about parallelograms. It is a four-sided closed figure with equal and parallel opposite sides and equal opposite angles. Let us learn more about the properties of parallelograms in detail in this article.

A parallelogram is a type of quadrilateral in which the opposite sides are parallel and equal. There are four angles in a parallelogram at the vertices. Understanding the properties of parallelograms helps to easily relate its angles and sides. Also, the properties are helpful for calculations in problems relating to the sides and angles of a parallelogram.

Parallelogram Angle Properties

The important properties of parallelograms related to angles are as follows:

1. The opposite angles of a parallelogram are equal, i.e., ∠A = ∠C, and ∠B = ∠D.
2. All the angles of a parallelogram add up to 360°, i.e., ∠A + ∠B + ∠C + ∠D = 360°.
3. The consecutive angles of a parallelogram are supplementary, i.e.,
   ∠A + ∠B = 180°
   ∠B + ∠C = 180°
   ∠C + ∠D = 180°
   ∠D + ∠A = 180°

Parallelogram Side Properties

The opposite sides of a parallelogram are equal and parallel to each other.

Observe the following figure to understand the properties of a parallelogram.

All the above properties hold true for all types of parallelograms, but now let us also learn about the individual properties of some special parallelograms. The three different parallelograms are square, rectangle, and rhombus which are different from each other because of their properties yet they all come under the category of parallelograms.

Properties of a Square

• All four sides of a square are equal.
• All four angles are equal and of 90 degrees each.
• The diagonals of a square bisect its angles.
• Both the diagonals of a square have the same length.
• The opposite sides of a square are equal and parallel to each other.

Properties of a Rectangle

• The opposite sides of a rectangle are equal and parallel.
• All four angles of a rectangle are equal and measure 90° each.
• Both the diagonals of a rectangle are of the same length.

Properties of Rhombus

• All sides of a rhombus are equal in length.
• The diagonals of a rhombus bisect each other at 90°.
• The sum of any two adjacent interior angles is 180°.
• The opposite sides of a rhombus are equal and parallel to each other.

Now, let us expand our knowledge by learning about the properties of diagonals of parallelograms in the following section..

First, we will recall the meaning of a diagonal. Diagonals are line segments that join the non-adjacent vertices of any polygon. In parallelogram ABCD (refer to the figure given above), AC and BD are the diagonals. Let us assume that O is the intersecting point of the diagonals AC and BD. The properties of diagonals of a parallelogram are as follows:

• The diagonals of a parallelogram bisect each other, i.e., OB = OD and OA = OC.
• Each diagonal divides the parallelogram into two congruent triangles, i.e., ΔCDA ≅ ΔABC and ΔBAD ≅ ΔDCB.
• Parallelogram Law: The sum of the squares of the sides is equal to the sum of the squares of the diagonals, i.e., AB² + BC² + CD² + DA² = AC² + BD²

The theorems on properties of a parallelogram are helpful to define the rules for working across the problems on parallelograms. The properties relating to the sides and angles of a parallelogram can all be easily understood and applied to solve various problems. Further, these theorems are also supportive of understanding the concepts in other quadrilaterals. Four important theorems related to the properties of a parallelogram are given below:

• The opposite sides of a parallelogram are equal.
• The opposite angles of a parallelogram are equal.
• The diagonals of a parallelogram bisect each other.
• If one pair of opposite sides is equal and parallel in a quadrilateral then it is a parallelogram.

Theorem 1: In a Parallelogram the Opposite Sides are Equal.

Proof:

Given: ABCD is a parallelogram.

To Prove: The opposite sides are equal, AB = CD and BC = AD.

In parallelogram ABCD, compare triangles ABC and CDA. In these triangles:

• AC = CA (common side)
• ∠BAC = ∠DCA (alternate interior angles)
• ∠BCA = ∠DAC (alternate interior angles)

Hence by the ASA criterion, both the triangles are congruent and the corresponding sides are equal. Therefore we have AB = CD, and BC = AD.

Converse of Theorem 1: If the opposite sides in a quadrilateral are equal, then it is a parallelogram.

Proof:

Given: The opposite sides in a quadrilateral ABCD are equal, AB = CD, and BC = AD.

To Prove: ABCD is a parallelogram.

n the quadrilateral ABCD we are given that AB = CD and AD = BC. Now compare the two triangles ABC, and CDA. Here we have

• AC = AC (Common sides)
• AB = CD (since alternate interior angles are equal)
• AD = BC (given).

Thus by the SSS criterion, both the triangles are congruent, and the corresponding angles are equal. Hence we can conclude that ∠BAC = ∠DCA, and ∠BCA = ∠DAC.

Therefore AB || CD, BC || AD, and ABCD is a parallelogram.

Theorem 2: In a Parallelogram, the Opposite Angles are Equal.

Proof:

Given: ABCD is a parallelogram, and ∠A, ∠B, ∠C, ∠D are the four angles.

To Prove: ∠A = ∠C and ∠B = ∠D

Let us assume that ABCD is a parallelogram. Now compare triangles ABC and CDA. Here we have,

• AC = CA (common side)
• ∠1 = ∠4 (alternate interior angles)
• ∠2 = ∠3 (alternate interior angles)

Thus, by ASA, the two triangles are congruent, which means that ∠B = ∠D. Similarly, we can show that ∠A = ∠C. This proves that opposite angles in any parallelogram are equal.

Converse of Theorem 2: If the opposite angles in a quadrilateral are equal, then it is a parallelogram.

Proof:

Given: ∠A = ∠C and ∠B = ∠D in the quadrilateral ABCD.

To Prove: ABCD is a parallelogram.

Assume that ∠A = ∠C and ∠B = ∠D in the parallelogram ABCD given above. We have to prove that ABCD is a parallelogram. We have:

∠A + ∠B + ∠C + ∠D = 360º

2(∠A + ∠B) =360º

∠A + ∠B = 180º.

This must mean that AD || BC. Similarly, we can show that AB || CD. Hence, AD || BC, and AB || CD. Therefore ABCD is a parallelogram.

Theorem 3: Diagonals of a Parallelogram Bisect Each Other.

Proof:

Given: PQTR is a parallelogram. PT and QR are the diagonals of the parallelogram.

To Prove: The diagonals PT and RQ bisect each other, i.e., PE = ET and ER = EQ.

First, let us assume that PQTR is a parallelogram. Compare triangles TER and triangle PEQ. We have,

• PQ = RT (opposite sides of the parallelogram PQTR)
• ∠QRT = ∠PQR (alternate interior angles)
• ∠PTR = ∠QPT (alternate interior angles).

By the ASA criterion, the two triangles are congruent, which means by CPCTC, PE = ET, and RE = EQ. Thus, the two diagonals PT and RQ bisect each other, and PE = ET and ER = EQ.

Converse of Theorem 3: If the diagonals in a quadrilateral bisect each other, then it is a parallelogram. In the quadrilateral PQTR, if PE=ET and ER=EQ, then it is a parallelogram.

Given: The diagonals PT and QR bisect each other.

To Prove: PQRT is a parallelogram.

Proof: Suppose that the diagonals PT and QR bisect each other. Compare triangle RET, and triangle PEQ once again. We have:

• RE = EQ
• ET = PE (Diagonals bisect each other)
• ∠RET =∠PEQ (vertically opposite angles).

Hence by the SAS criterion, the two triangles are congruent. This means that ∠QRT = ∠PQR, and ∠PRT = ∠QPT. Hence, PQ || RT, and RT || QT. Thus PQRT is a parallelogram.

Theorem 4: If One Pair of Opposite Sides is Equal and Parallel in a Quadrilateral then it is a Parallelogram.

Proof:

Given: It is given that AB = CD and AB || CD.

To Prove: ABCD is a parallelogram.

Let us compare the triangle AEB and triangle CED. We have,

• AB = CD (given)
• ∠1 = ∠3 (alternate interior angles)
• ∠2 = ∠4 (alternate interior angles)

Thus, by ASA criterion, the two triangles are congruent. Hence we can conclude that by CPCTC, AE = EC, and BE = ED. Therefore, the diagonals AC and BD bisect each other, and this further means that ABCD is a parallelogram.

Important Notes:

A quadrilateral is a parallelogram when:

• The opposite sides of a quadrilateral are equal and parallel.
• The opposite angles of a quadrilateral are equal.
• The diagonals bisect each other.
• One pair of opposite sides is equal and parallel.

• Parallelogram Formula
• Area of Parallelogram
• Properties of Rectangle
• Difference Between Rectangle and Parallelogram

What are the 7 Properties of Parallelogram?

The seven properties of a parallelogram are as follows:

• The opposite sides of a parallelogram are equal.
• The opposite angles of a parallelogram are equal.
• The consecutive angles of a parallelogram are supplementary.
• If one angle of a parallelogram is a right angle, then all the angles are right angles.
• The diagonals of a parallelogram bisect each other.
• Each diagonal of a parallelogram bisects it into two congruent triangles.
• If one pair of opposite sides of a quadrilateral is equal and parallel, then the quadrilateral is a parallelogram.

What are the Properties of Diagonals of a Parallelogram?

There are two important properties of the diagonals of a parallelogram. The diagonal of a parallelogram divides the parallelogram into two congruent triangles. And the diagonals of a parallelogram bisect each other.

Are the Diagonals of a Parallelogram Equal?

The diagonals of a parallelogram are NOT equal. The opposite sides and opposite angles of a parallelogram are equal. The diagonals of a square and a rectangle are equal which are special types of parallelograms.

Write the Properties of a Parallelogram.

A parallelogram is a quadrilateral with opposite sides equal and parallel. The opposite angles of a parallelogram are also equal. In short, a parallelogram can be considered a twisted rectangle. It is more of a rectangle, but the angles at the vertices need not be right angles. The four important properties of a parallelogram are as follows:

• The opposite sides of a parallelogram are equal.
• The opposite angles of a parallelogram are equal.
• The adjacent angles of a parallelogram are supplementary.
• The diagonals of a parallelogram bisect each other.

Can a Rectangle be called a Parallelogram?

The opposite sides of a rectangle are equal and parallel. So a rectangle satisfies all the properties of a parallelogram and hence a rectangle can be called a parallelogram.

What are the Unique Properties of Parallelograms?

The unique characteristics of parallelograms that make it different from other quadrilaterals are given below:

• The opposite sides of every parallelogram are equal and parallel.
• The opposite angles are always equal.
• The sum of consecutive interior angles is always equal to 180°.

What are the Different Properties of Each Special Parallelogram?

There are three special types of parallelograms - square, rectangle, and rhombus. A square is a four-sided polygon in which all sides and angles are equal. A rectangle has opposite sides equal and parallel. All the angles of squares and rectangles are equal and measure 90° each. A rhombus is a parallelogram with four equal sides, but its angles need not be right angles.

How to Differentiate a Parallelogram and a Quadrilateral by Properties of Parallelogram?

Every parallelogram can be called a quadrilateral, but every quadrilateral cannot be called a parallelogram. A trapezium and a kite can be called quadrilaterals, but they do not fully satisfy the properties of a parallelogram and hence cannot be called a parallelogram.

What do the Opposite Angles in a Parallelogram add up to?

The opposite angles in a parallelogram are always equal. However, it is to be noted that the consecutive interior angles of a parallelogram always add up to 180°.

What are the Rules of a Parallelogram?

The rules of a parallelogram are the characteristics of a parallelogram that make it distinct from the other polygons. In other words, they are all the properties of a parallelogram that help us identify it. The basic characteristics of a parallelogram are given below:

• The opposite sides of a parallelogram are equal.
• The opposite angles of a parallelogram are equal.
• The consecutive angles of a parallelogram are supplementary.
• The diagonals of a parallelogram bisect each other.
• Each diagonal of a parallelogram bisects it into two congruent triangles.