Properties of 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 foursided 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.
1.  What are the Properties of Parallelogram? 
2.  Properties of Diagonal of Parallelogram 
3.  Theorems on Properties of Parallelogram 
4.  FAQs on Properties of Parallelogram 
What are the Properties of Parallelogram?
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 the angles and sides of a parallelogram. Also, the properties are helpful for calculations in problems relating to sides and angles of a parallelogram.
Properties of Parallelogram Related to Sides and Angles
The four important properties of a parallelogram are as follows.
 Opposites sides of a parallelogram are equal and parallel to each other.
 Opposite angles are equal. ∠A= ∠C, and ∠B = ∠D
 All the angles of a parallelogram add up to 360^{o}. ∠A + ∠B + ∠C + ∠D = 360^{o}.
 The consecutive angles of a parallelogram are supplementary. i.e.,
∠A + ∠B = 180^{o}
∠B + ∠C = 180^{o}
∠C + ∠D = 180^{o}
∠D + ∠A = 180^{o }
Properties of Diagonal of Parallelogram
First, we will recall the meaning of a diagonal. Diagonals are line segments that join the opposite vertices. In parallelogram ABCD, AC and BD are the diagonals. Let us assume that O is the intersecting point of the diagonals AB and BD. The properties of diagonals of a parallelogram are as follows:
 Diagonals of a parallelogram bisect each other.
i.e., OB =OD and OA = OB.  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., AB^{2}+BC^{2}+CD^{2}+DA^{2} = AC^{2}+BD^{2}.
Theorems on Properties of Parallelogram
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
 If one pair of opposite sides is equal and parallel in a quadrilateral then it is a parallelogram
Theorem 1 on Properties of Parallelogram
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 ABCB, compare triangles ABC and CDA. In these triangles:
 AC = CA (common sides)
 ∠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.
The 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 on Properties of Parallelogram
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=AC (common sides)
 ∠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 on Properties of Parallelogram
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. PE=ET and ER=EQ
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)
 ∠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 on Properties of Parallelogram
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 (opposite sides)
 ∠1 = ∠3 (alternate interior angles)
 ∠2 = ∠4 (alternate interior angles)
Thus, by SAS, 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 on Properties of Parallelogram
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 bisect each other
 one pair of opposite sides is equal and parallel
2. Note that any 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?
 Why is a kite not a parallelogram?
 Is an isosceles trapezoid a parallelogram?
Related Topics:
Examples on Properties of Parallelogram

Example 1: If one angle of a parallelogram is 90^{o}, show that all its angles will be equal to 90^{o}.
Solution:
Consider the parallelogram ABCD in the following figure, in which ∠A is a right angle:
We know that in any parallelogram, the opposite angles are equal. This implies ∠C=90^{o}.
Also, in any parallelogram, the adjacent angles are supplementary. This implies ∠B=180^{o}  ∠ A=180^{o}  90^{o} =90^{o}.
Similarly, ∠D=180^{o}  ∠C=180^{o}  90^{o} =90^{o}.
Hence, ∠A=∠B=∠C=∠D = 90^{o}.
Clearly, all the angles in this parallelogram (which is actually a rectangle) are equal to 90^{o}.
Answer: We have proved that when one angle of a parallelogram is 90^{0}, the parallelogram is a rectangle.

Example 2: In a quadrilateral ABCD, the diagonals AC and BD bisect each other at right angles. Show that the quadrilateral is a rhombus.
Solution:
Consider the following figure:
First of all, we note that since the diagonals bisect each other, we can conclude that ABCD is a parallelogram. So the opposite sides are equal
 AB = CD ... (1)
 AD = BC ... (2)
Now, let us compare ΔAEB and ΔAED:
 AE=AE
 BE =ED
 ∠AEB=∠AED= 90^{o}.
Thus, by the SAS criterion, the two triangles are congruent, which means that AB = AD.
Also, by (1) and (2), we can conclude that AB=BC=CD=AD. Clearly, ABCD is a rhombus.
Answer: We have proved that the quadrilateral in which the diagonals bisect each other at right angles is a rhombus.

Example 3: ABCD is a quadrilateral with AB = 8.5 units. The diagonals of ABCD bisect each other at right angles. Then find the perimeter of ABCD.
Solution:
From Example 2, if the diagonals of a quadrilateral bisect each other at right angles then it becomes a rhombus.
So ABCD is a rhombus and hence AB = BC = CD = DA.
Thus, the perimeter = 4(AB) = 4(8.5) = 34 units.
Answer:The perimeter of ABCD is 34 units.
FAQs on Properties of Parallelogram
What are the 7 Properties of Parallelogram?
The 7 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 angle of a parallelogram is supplementary.
 If one angle is a right angle, all the angles are right angles in a parallelogram.
 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 Parallelogram With Respect to Diagonals?
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.
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 rightangles.
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 examples of a parallelogram.
What are the Four Important Properties of Parallelogram?
The four important properties of a parallelogram are as follows.
 The opposite sides are equal.
 The opposite angles are equal.
 The adjacent angles are supplementary.
 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.
How Can We 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, 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.