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Properties of a parallelogram help us identify whether a figure is a parallelogram. A parallelogram is a four-sided quadrilateral whose opposite sides and angles are equal. It is a two-dimensional shape (2D shape). Parallelogram has four angles in its vertices.
- Properties of a parallelogram are very useful in identifying the figures, whether it is quadrilateral, square, rectangle or rhombus.
- The sum of angles of the parallelogram formed on the opposite sides is 1800 .
- The total sum of angles is 3600.
- It is based on the Euclidean parallel postulate.
- Properties of a parallelogram are also helpful in solving the geometrical problems in mathematics.
- The trapezium is a non-example of a parallelogram.
- It forms two congruent triangle on bisection with the help of diagonals.
Read More: Surface Area and Volumes
Key Terms: Parallelogram, Angles, Quadrilaterals, Diagonals, Area, Perimeter, Properties of a Parallelogram, Diagonals, Square, Rectangle, Rhombus
Parallelogram
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Parallelogram is a quadrilateral that has the same opposite sides and opposite angles. The pair of opposite sides are equal and parallel. In a parallelogram, diagonals bisect each other.
- The quadrilateral formed by joining the midpoints of the sides is a parallelogram.
- Diagonals of the figure bisect each other at 90 degrees.
- The area and perimeter measure the distance along the boundary.
- The three-dimensional figure of the figure is called parallelepiped.
Parallelogram diagram
From the above diagram, it is clear that opposite sides are equal such as
- AB=CD
- AD=BC
- Also, opposite angles are equal.
- ∠A=∠C
- ∠B=∠D
- Moreover, diagonals of a parallelogram are also equal such as,
- DO=OB
- AO=OC
- The sum of the angle of vertices is 1800. Then,
- ∠A+∠B=1800
- ∠A+∠D=1800
- ∠B+∠C=1800
- ∠C+∠D=1800
Read More: Triangle Theorems
Solved Example of a ParallelogramExample: The area of a playground which is in the shape of a parallelogram is 3500 in2, with one side measuring 350 in. Find the corresponding altitude. Solution: Area of the playground = 3500 in2 Side of a playground = 350 in By using the area of the parallelogram formula, Area = Side × Altitude 3500 = 350 × Altitude Altitude = 3500/350 = 10 in |
Read More: Area of Parallelogram
Properties of a Parallelogram
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Some of the important properties of a parallelogram are mentioned below:
- Opposite sides of a parallelogram are equal as well as congruent (i.e.having the same size and shape).
- In this figure, opposite angles are equal.
- Consecutive angles of the three-dimensional figure are supplementary.
- Diagonals bisect each other at 90 degrees.
- In a parallelogram, if any one angle is a right angle, then all the angles will be at the right angle.
- The sum of the squares of all the sides is equal to the sum of the squares of the diagonals.
- Two congruent triangles are formed when each diagonal bisects in a parallelogram.
Properties of a Parallelogram
Read More: Perimeter of a Parallelogram
Theorems on Properties of a Parallelogram
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On the basis of properties of a parallelogram, there are four most important theorems related to the parallelogram which are as follows:
Theorem 1
Statement: In a Parallelogram, a Diagonal Divides it into two Congruent Triangles.
Proof: Let us consider a parallelogram ABCD with diagonal AC which divides the parallelogram into two triangles as ΔABC and ΔCDA.
To prove that ΔABC and ΔCDA are congruent triangles.
Parallelogram with two congruent triangles
Here, in ΔABC and ΔCDA,
- BC || AD (BC is parallel to AD)
- Also, AC is a transversal
- So, pair of alternate angles such as
- ∠BCA =∠DCA
- Also, AB || DC (AB is parallel to DC) and AC is a transversal.
- So, pair of alternate angles such as
- ∠BAC =∠DCA
- And AC = CA (common)
- According to ASA rule,
- ΔABC ≅ ΔCDA
Hence, diagonal AC divides parallelogram ABCD into two congruent triangles ABC and CDA.
Theorem 2
Statement: The Opposite Sides are Equal in a Parallelogram.
Proof: Let us consider a parallelogram ABCD.
To prove that opposite sides are equal such as AD=BC and AB=CD.
Parallelogram with equal opposite sides
Now, compare the triangles ΔABC and ΔCDA, we get,
- AC = CA (common)
- And pair of alternate angles such as
- ∠BCA =∠DCA
- ∠BAC =∠DCA
- As per the ASA rule, both triangles are congruent. If both the triangles are congruent then opposite sides are also equal.
- Hence, AD=BC ,and AB=CD
- Therefore, it is proved that opposite sides of parallelogram are equal. If this theorem is verified then its converse is also true.
Read More: Area of Rectangle
Theorem 3
Statement: The Opposite Angles are Equal in a Parallelogram.
Proof: Let us consider a parallelogram ABCD with angles at each vertex. Let us assume ∠1= ∠A, ∠2= ∠B, ∠3= ∠C, ∠4= ∠D.
Parallelogram with angle at each vertices
We all know, AC is transversal in a parallelogram ABCD and it divides the parallelogram into two congruent triangles like ΔABC and ΔCDA.
- Now, compare the triangles ΔABC and ΔCDA, we get,
- AC = CA (common)
- Also, the alternate interior angles
- ∠1= ∠4 (i.e. ∠A = ∠C)
- ∠2=∠3 (i.e. ∠B = ∠D)
- According to the ASA criterion, both the triangles are congruent. If both the triangles are congruent then opposite sides are also equal. It means that the angels ∠A = ∠C.
- Similarly, ∠B = ∠D.
Hence, it is proved that opposite angles in any parallelogram are equal. If this theorem is verified then its converse is also true.
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Theorem 4
Statement: In a Parallelogram, Diagonals Bisect Each Other.
Proof: Let us consider a parallelogram ABCD and AC and BD are the diagonals. Also, ‘O’ is the midpoint.
To prove that AC and BD bisect each other(i.e. AO=OC, BO=OD).
Parallelogram bisects each other
Now, compare the triangles AOB and COD, we get,
- AB = CD (opposite sides of parallelogram)
- Also, alternate interior angles are
- ∠ABC=∠ADC
- ∠ABD=∠BCD
- According to ASA criterion, two triangles are congruent. Then, as per CPCTC (corresponding parts of congruent triangles are congruent)
- I.e. AO=OC, and BO=OD
Hence, two diagonals AC and BD bisect each other, and AO=OC, BO=OD.
Read More: Quadrilateral Formulas
Important Formulas Related to Parallelogram
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There are two most important formulas related to the properties of a parallelogram which are as follows:
Area of a Parallelogram
Area of a parallelogram is equal to the total space occupied with the four sides of the parallelogram. The formula to calculate the area of a parallelogram is given as:
A = Base * Height
Solved Example of Area of a ParallelogramExample: What will be the area of a sheet in the shape of a parallelogram, if its base is 30 cm, and its height is 8 cm? Solution: Given that
Using the Area of Parallelogram Formula Area of Parallelogram = b x h Area of Sheet = b × h = (30) × (8) = 240 cm2 Thus, the area of the sheet is 240 in2. |
Read More: Hexagon
Perimeter of Parallelogram
Perimeter of a parallelogram is defined as the sum of the sides of the figure. The formula to calculate the perimeter of a parallelogram is given as:
P= 2(l+m) units
- Where, l and m are the lengths of the sides of parallelogram
Solved Example of Perimeter of a ParallelogramExample: Find the perimeter of a parallelogram whose adjacent sides are 19 units and 9 units. Solution: The adjacent sides of the given parallelogram are, a = 19 units and b = 9 units. Then its perimeter (P) is, P = 2 (a + b) P = 2 (19 + 9) = 2 (28) = 56 units. |
Read More:
Things to Remember
- Properties of a parallelogram help us select which figure is a parallelogram.
- Parallelogram, rhombus, square and rectangle are all quadrilaterals.
- In a quadrilateral, the sum of the angles is 3600.
- When a diagonal bisects a parallelogram then congruent triangles are formed.
- In a quadrilateral, if each pair of opposite sides is equal and parallel then it is a parallelogram.
Read More: Areas of Parallelograms and Triangle
Sample Questions
Ques. What is the angle sum property of a quadrilateral? Explain it. [5 marks]
Ans. Angle sum property of a quadrilateral is defined as the sum of the angles of the quadrilateral is 3600. Let us consider an example to understand it more clearly.
Let us consider a quadrilateral ABCD and AC is a diagonal. The diagonal AC divides the quadrilateral ABCD into two triangles ABC and ADC.
Quadrilateral with diagonal
We know that, In Δ ADC,
∠ DAC + ∠ ACD + ∠ D = 180° —--------- (1)
Similarly, in Δ ABC,
∠ACB + ∠CAB +∠B = 1800 —--------------(2)
Add both the above equations(1) &(2), we get,
∠ DAC + ∠ ACD + ∠ D +∠ACB + ∠CAB +∠B = 1800 + 1800 —-----(3)
Also, ∠ DAC + ∠CAB = ∠A and ∠ ACD +∠ACB = ∠C
By substituting the above values in equation(3), we get,
∠A+ ∠D+ ∠B + ∠C = 3600
So, ∠A+∠B + ∠C + ∠D = 3600
Hence, the sum of the angles of a quadrilateral is 3600.
Ques. Why is a kite not a parallelogram? [2 marks]
Ans. Any quadrilateral is a parallelogram if its opposite sides are equal and parallel to each other. But, in the case of a kite, only two pairs of adjacent sides are equal. So, a kite is not parallelogram.
Ques. What are the conditions to be a parallelogram in a quadrilateral? [2 marks]
Ans. A quadrilateral is a parallelogram if it satisfies the four main conditions which are written below:
- Opposite sides are equal
- Opposite angles are equal
- Each and every pair of opposite sides is equal and parallel
- Diagonals bisect each other in a parallelogram
Ques. Show that the diagonals of a rhombus are perpendicular to each other. [4 marks]
Ans. First, we need to consider a rhombus ABCD.
Rhombus diagram
We know that, AB=BC=CD=DA
Now, In Δ AOD and Δ COD, diagonals of a parallelogram bisect each other
I.e. OA=OC
Also,
OD=OD (Common)
AD=CD
According to SSS congruence rule,
Δ AOD ≅ Δ COD
Also,
∠ AOD = ∠ COD ( a per CPCT criterion)
But, ∠ AOD + ∠ COD= 1800 (linear pair)
So, 2 ∠ AOD = 1800
∠ AOD = 1800/2
∠ AOD = 900 .,
Hence, it is clear that the diagonals of a rhombus are perpendicular to each other.
Ques.What is the difference between rhombus and parallelogram? [4 marks]
Ans. Rhombus and parallelogram are similar in many ways but there are some things which make them separate.
| S.No. | Rhombus | Parallelogram |
|---|---|---|
| 1. | In a rhombus, all four sides are equal in length. | In a parallelogram, only opposite sides are equal. |
| 2. | In rhombus, all opposite sides are parallel. | In parallelogram, only opposite sides are parallel. |
| 3. | Every rhombus is a parallelogram | Every parallelogram is not a rhombus |
| 4. | When diagonals bisect each other, scalene triangles are formed. | When diagonals bisect each other then congruent triangles are formed |
| 5. | In rhombus, perimeter, P= 4a where ‘a’ is the side | In parallelogram, perimeter, P= 2(a+b), where ‘a’ is the side and ‘b’ is the base. |
| 6. | In rhombus, Area, A=d1d2/2, where d1 and d2 are the diagonals. | In a parallelogram, Area, A=b x h, where ’b’ is base and ‘h’ is height. |
Ques. What do you mean by a parallelogram? What are its types? [2 marks]
Ans. Parallelogram is a two-dimensional shape which has opposite sides that are equal and parallel. Also, in parallelogram opposite angles are equal. When a diagonal bisects a parallelogram then two congruent triangles are formed. Moreover, all the properties of a parallelogram are true for the figures which are considered parallelograms. There are many shapes/figures which are parallelogram shown below:
- Rectangle
- Rhombus
- Quadrilateral
- Square etc.
Ques. What is the area of a parallelogram if the base is 7cm and height 12 cm. [3 marks]
Ans. Given,
Base of a parallelogram,b= 7 cm
Height of a parallelogram, h= 12 cm.
Now, the area of a parallelogram,A=base(b) x height(h)
A= 7 X 12
A= 84 Sq.cm
Hence, the area of a parallelogram is 84 Sq.cm.
Ques. In a parallelogram ABCD, P and Q are mid points of opposite sides AB and CD(see in the given diagram).Also, if AQ intersects DP at S and BQ intersects CP at R, show that:
(i) APCQ is a parallelogram
(ii)DPBQ is a parallelogram
(iii)PSQR is a parallelogram [4 marks]
Ans. Given,
Parallelogram ABCD
(i) In a quadrilateral APCQ,
AP || QC ( since AB ||CD)....(1)
And given,
AP= ½ AB, CQ=½ CD
Also, AB = CD (opposite sides are equal)
AP=CQ ….(2)
From equations (1) and (2), it is clear that opposite sides are equal and parallel.
Therefore, APCQ is a parallelogram.
(ii) Similarly, quadrilateral DPBQ is a parallelogram, because
DQ || PB ( DQ is parallel to PB)
And DQ = PB
(iii) In quadrilateral PSQR,
SP || QR (SP is a part of DP and QR is a part of QB)
Similarly, , SQ || PR
Hence, PSQR is a parallelogram.
Ques. What will be the area of a parallelogram if its base and height are 14 cm and 8 cm respectively? (3 Marks)
Ans. Given that,
- Base of Parallelogram = 14 cm
- Height of Parallelogram = 8 cm
Using the Area of Parallelogram Formula
Area of Parallelogram = b x h
Area of Parallelogram = b × h = 14 × 8 = 112 cm2
Thus, the area of the given parallelogram is 112 cm2.
Ques: What is the perimeter of parallelogram when sides are 33 m and 17 m. (3 marks)
Ans: Given: Sides of parallelogram ‘a’ =23 m & ’b’ = 17 m
We also know that the formula for perimeter of parallelogram;
= 2 (23+ 17)
= 2 (40)
= 2 (40)
= 80 m
Therefore, the perimeter of the parallelogram is 80 m.
Ques. Find the area of the parallelogram if the length of its adjacent sides is 10 cm and 25 cm and the measure of the angle between the adjacent sides is 34°. (3 Marks)
Ans. Given that,
- Length of First Side = 10 cm
- Length of Second Side = 25 cm
- Angle between Both Sides = 34o
Using the Area of Parallelogram Formula,
Area of Parallelogram = ab sin(x)
Area of Parallelogram = 10 x 25 x sin(34°)
= 250 x 0.55919
= 139.795 cm2
Thus, the area of the given parallelogram is 139.795 cm2.
Ques. Find the height and base of a parallelogram when the area of a parallelogram is 800 sq. cm and its height is twice its base. (3 Marks)
Ans. Given that,
- Area of Parallelogram = 800 cm2
- Height, h = 2 x Base = 2b
Using the Area of Parallelogram Formula
Area of Parallelogram = b x h
800 = b x 2b
2b2 = 800
b2 = 400
b = 20 cm
Height, h = 2 x b = 40 cm.
Therefore, the height of the parallelogram is 40 cm.
Ques. Find the base and height of a parallelogram when the base is thrice its height and the area is 195 cm2. (3 Marks)
Ans. Given that,
- Area of Parallelogram = 195 cm2
- Base, b = 3 x Height = 3h
Using the Area of Parallelogram Formula
Area of Parallelogram = b x h
195 = 3h x h
3 x h2 = 195
h2 = 65 = 8.06 cm
Since height is 8 cm, therefore base = 3 x h = 3 x 8.06 = 24.18 cm.
Thus, the base and height of the parallelogram are 24 cm and 8 cm respectively.
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