What are the Properties of a Parallelogram?

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 of a parallelogram, let us first know about parallelogram. It is a four-sided closed figure with opposite sides are equal and opposites angles are equal. The properties of a parallelogram mainly deal with its sides and angles. 

We all know that a parallelogram is a convex polygon with 4 edges and 4 vertices. The opposite sides are equal and parallel; the opposite angles are also equal. Let's learn more about the properties of parallelograms in detail in this lesson.

First, we will recall the meaning of a diagonal. Diagonals are line segments that join the opposite vertices. In parallelogram PQRS, PR and QS are the diagonals. The properties of diagonals of a parallelogram are as follows:

• Diagonals of a parallelogram bisect each other. OQ =OS  and OR = OP 
• Each diagonal divides the parallelogram into two congruent triangles, so, ΔRSP ≅ ΔPQR and ΔQPS ≅ ΔSRQ.
• Parallelogram Law: The sum of the squares of the sides is equal to the sum of the squares of the diagonals. PQ²+QR²+RS²+SP² = QS²+PR²

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 applies to solve various problems. Further, these theorems are also supportive to understand the concepts in other quadrilaterals. Four important theorems relating to the properties of a parallelogram are given below:

• Opposite sides of a parallelogram are equal
• Opposite angles of a parallelogram are equal
• Diagonals of a parallelogram bisect each other
• One pair of opposite sides is equal and parallel in a parallelogram

Theorem 1: In a Parallelogram the Opposite Sides Are Equal. This means, in a parallelogram, the opposite sides are equal.

Given: ABCD is a parallelogram.

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

Proof: In parallelogram ABCB, compare triangles ABC and CDA. In these triangles AC = CA (common sides). Also ∠BAC =∠DCA (alternate interior angles), and ∠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. If AB = CD and BC = AD in the given quadrilateral ABCD, then it is a parallelogram.

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

To Prove: ABCD is a parallelogram.

Proof: In 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), and 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. 

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

To Prove: ∠A =∠C and ∠B=∠D

Proof: Let us assume that ABCD is a parallelogram. Now compare triangles ABC, and CDA. Here we have  AC=AC (common sides),  ∠1=∠4 (alternate interior angles), and ∠2=∠3 (alternate interior angles). Thus, 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.

Given: ∠A=∠C and ∠B=∠D in the quadrilateral ABCD. To Prove: ABCD is a parallelogram.

Proof: 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. That means, in a parallelogram, the diagonals bisect each other. 

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

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

Proof: First, let us assume that PQTR is a parallelogram. Compare triangles RET, and triangle PEQ. We have PQ = RT (opposite sides of the parallelogram), ∠QRT = ∠PQR (alternate interior angles), and ∠PTR = ∠QPT  (alternate interior angles). By the ASA criterion, the two triangles are congruent, which means that PE = ET, and RE = EQ. Thus, the two diagonals PT, and RQ bisect each other, and PE=ET and ER=EQ

The 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: One Pair of Opposite Sides Is Equal and Parallel in a Parallelogram.

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

To Prove: ABCD is a parallelogram.

Proof: Let us compare the triangle AEB, and triangle DEC. We have AB = CD (opposite sides), ∠1 = ∠3  (alternate interior angles), and ∠2  = ∠4 (alternate interior angles). Thus, the two triangles are congruent. Hence we can conclude that  AE=EC, BE=ED. Therefore, the diagonals AC and BD bisect each other, and this further means that ABCD is a parallelogram.

Important Notes

1.  A quadrilateral is a parallelogram when:

• the opposite sides of a quadrilateral are equal
• the opposite angles of a quadrilateral are equal
• the diagonals of a quadrilateral bisect each other
• one pair of opposite sides is equal and parallel

2.  Note that the relation between two lines intersected by a transversal, when the angles on the same side of the transversal are supplementary, are parallel to each other.

Do you know?

1. Why is a kite not a parallelogram?
2. Is an isosceles trapezoid a parallelogram?

What are the 7 Properties of a Parallelogram?

The 7 properties of a parallelogram are as follows:

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

What are the Properties of the 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 equal. The opposite sides and opposite angles of a parallelogram are equal. And these opposite sides and angles make up for two congruent triangles, with the two diagonals being the sides of these two congruent triangles. Hence the diagonals of the parallelogram are equal.

What is a Parallelogram?

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

What are the Examples of a Parallelogram?

The square and a rectangle are the two simple examples of a parallelogram. Hence the flat surfaces of the furniture such as a table, a cot, a plain sheet of A4 paper can all be counted as an example of a parallelogram.

What are the Four Important Properties of a Parallelogram?

The four important properties of a parallelogram are as follows.

• The opposite sides are equal.
• The opposite angles are equal.
• The diagonals are equal.
• The opposite angles are equal.

Can a Rectangle be called a Parallelogram?

A rectangle satisfies all the properties of a parallelogram. The opposite sides of a rectangle are equal and each angle of a rectangle is a right angle. Hence with these features, a rectangle satisfies all the properties of a parallelogram and it can be called a parallelogram.

What is the Difference Between a Parallelogram and a Quadrilateral?

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