Properties of Parallelogram
A parallelogram is a quadrilateral whose opposite sides are parallel. It is one of the most important shapes in geometry, and many other quadrilaterals like rectangles, rhombuses, and squares are special cases of a parallelogram.
The properties of a parallelogram form the foundation for understanding all these related shapes. Once you know parallelogram properties, you can easily derive the properties of rectangles, rhombuses, and squares.
In a parallelogram ABCD, side AB is parallel to side CD, and side AD is parallel to side BC. This parallel nature of the sides gives rise to several important properties related to the sides, angles, and diagonals of the shape.
Parallelograms are found everywhere in daily life — from tiles on floors to the design of bridges, from the shape of a book cover viewed at an angle to the structure of mechanical linkages. Understanding their properties helps solve problems in geometry, engineering, and design.
In this topic, you will study the five fundamental properties of a parallelogram, learn how to prove that a quadrilateral is a parallelogram, work through solved examples involving sides, angles, and diagonals, and apply area and perimeter formulas.
What is Properties of Parallelogram?
Definition: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
AB ∥ CD and AD ∥ BC
In parallelogram ABCD:
- AB is parallel to CD — these are one pair of opposite sides.
- AD is parallel to BC — these are the other pair of opposite sides.
- The symbol for parallelogram is ∥gm or ▱.
- A parallelogram has 4 sides, 4 vertices, 4 angles, and 2 diagonals.
A parallelogram is denoted as ∥gm ABCD, where vertices are named in order (either clockwise or anti-clockwise).
Note: The word "parallelogram" comes from Greek words meaning "parallel lines." Any quadrilateral with two pairs of parallel sides automatically has all the properties listed below — equal opposite sides, equal opposite angles, supplementary consecutive angles, and bisecting diagonals.
Comparison with other quadrilaterals:
- A trapezium has only ONE pair of parallel sides — it is NOT a parallelogram.
- A rectangle has two pairs of parallel sides AND all angles are 90° — it IS a parallelogram.
- A rhombus has two pairs of parallel sides AND all sides are equal — it IS a parallelogram.
- A square has all sides equal AND all angles 90° — it IS a parallelogram (and also a rectangle and a rhombus).
Methods
Properties of a Parallelogram:
Property 1: Opposite sides are equal.
- AB = CD and AD = BC
- Each pair of opposite sides has the same length.
- Proof outline: Draw diagonal AC. In triangles △ABC and △CDA: ∠BAC = ∠DCA (alternate interior angles, AB ∥ CD), ∠BCA = ∠DAC (alternate interior angles, AD ∥ BC), and AC = AC (common). By ASA congruence, △ABC ≅ △CDA. Therefore AB = CD and BC = DA.
Property 2: Opposite angles are equal.
- ∠A = ∠C and ∠B = ∠D
- Angles at opposite vertices are equal in measure.
- Proof outline: Since △ABC ≅ △CDA (proved above), ∠B = ∠D. Similarly, drawing diagonal BD proves ∠A = ∠C.
Property 3: Consecutive angles are supplementary.
- ∠A + ∠B = 180°
- ∠B + ∠C = 180°
- ∠C + ∠D = 180°
- ∠D + ∠A = 180°
- Any two adjacent angles add up to 180° because they are co-interior angles (same-side interior angles) formed between parallel lines.
- Why? AB ∥ CD, and AD is a transversal. So ∠A + ∠D = 180° (co-interior angles on the same side of the transversal).
Property 4: Diagonals bisect each other.
- If diagonals AC and BD meet at point O, then:
- AO = OC and BO = OD
- The diagonals cut each other into two equal halves.
- Proof outline: In triangles △AOB and △COD: ∠OAB = ∠OCD (alternate interior angles), ∠OBA = ∠ODC (alternate interior angles), AB = CD (opposite sides). By ASA, △AOB ≅ △COD. Therefore AO = OC and BO = OD.
- Note: The diagonals of a parallelogram bisect each other but are generally NOT equal in length. They are equal only in a rectangle.
Property 5: Each diagonal divides the parallelogram into two congruent triangles.
- Diagonal AC divides ∥gm ABCD into △ABC ≅ △CDA
- Diagonal BD divides ∥gm ABCD into △ABD ≅ △CDB
- This means each triangle has the same area, which is half the area of the parallelogram.
Additional property — Mid-point theorem connection:
- The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half its length. This theorem is closely connected to parallelogram properties.
Solved Examples
Example 1: Example 1: Finding unknown angles
Problem: In a parallelogram ABCD, ∠A = 65°. Find ∠B, ∠C, and ∠D.
Solution:
Given:
- ∥gm ABCD with ∠A = 65°
Using properties:
- ∠C = ∠A = 65° (opposite angles are equal)
- ∠A + ∠B = 180° (consecutive angles are supplementary)
- 65° + ∠B = 180°
- ∠B = 180° − 65° = 115°
- ∠D = ∠B = 115° (opposite angles are equal)
Answer: ∠B = 115°, ∠C = 65°, ∠D = 115°.
Example 2: Example 2: Finding side lengths
Problem: In ∥gm PQRS, PQ = 12 cm and QR = 8 cm. Find RS and PS.
Solution:
Given:
- PQ = 12 cm, QR = 8 cm
Using property (opposite sides are equal):
- RS = PQ = 12 cm (opposite sides)
- PS = QR = 8 cm (opposite sides)
Answer: RS = 12 cm, PS = 8 cm.
Example 3: Example 3: Diagonals bisecting each other
Problem: The diagonals of parallelogram ABCD meet at O. If AC = 16 cm and BD = 10 cm, find AO, OC, BO, and OD.
Solution:
Given:
- AC = 16 cm, BD = 10 cm
Using property (diagonals bisect each other):
- AO = OC = AC/2 = 16/2 = 8 cm
- BO = OD = BD/2 = 10/2 = 5 cm
Answer: AO = OC = 8 cm, BO = OD = 5 cm.
Example 4: Example 4: Finding perimeter
Problem: The two adjacent sides of a parallelogram are 9 cm and 6 cm. Find its perimeter.
Solution:
Given:
- a = 9 cm, b = 6 cm
Using formula:
- Perimeter = 2(a + b)
- Perimeter = 2(9 + 6)
- Perimeter = 2 × 15 = 30 cm
Answer: Perimeter = 30 cm.
Example 5: Example 5: Finding area
Problem: A parallelogram has base 14 cm and height 9 cm. Find its area.
Solution:
Given:
- Base = 14 cm, Height = 9 cm
Using formula:
- Area = base × height
- Area = 14 × 9 = 126 cm²
Answer: Area = 126 cm².
Example 6: Example 6: Using algebra with sides
Problem: In ∥gm ABCD, AB = (3x + 5) cm and CD = (5x − 7) cm. Find x and the length of AB.
Solution:
Given:
- AB = (3x + 5) cm, CD = (5x − 7) cm
Since opposite sides are equal:
- AB = CD
- 3x + 5 = 5x − 7
- 5 + 7 = 5x − 3x
- 12 = 2x
- x = 6
Finding AB:
- AB = 3(6) + 5 = 18 + 5 = 23 cm
Answer: x = 6, AB = 23 cm.
Example 7: Example 7: Using algebra with angles
Problem: In ∥gm ABCD, ∠A = (2x + 30)° and ∠B = (3x − 10)°. Find x and all angles.
Solution:
Given:
- ∠A = (2x + 30)°, ∠B = (3x − 10)°
Since consecutive angles are supplementary:
- ∠A + ∠B = 180°
- (2x + 30) + (3x − 10) = 180
- 5x + 20 = 180
- 5x = 160
- x = 32
Finding angles:
- ∠A = 2(32) + 30 = 94°
- ∠B = 3(32) − 10 = 86°
- ∠C = ∠A = 94°
- ∠D = ∠B = 86°
Answer: x = 32, ∠A = 94°, ∠B = 86°, ∠C = 94°, ∠D = 86°.
Example 8: Example 8: Finding height from area
Problem: The area of a parallelogram is 270 cm² and its base is 18 cm. Find the height.
Solution:
Given:
- Area = 270 cm², Base = 18 cm
Using formula:
- Area = base × height
- 270 = 18 × height
- height = 270/18 = 15 cm
Answer: Height = 15 cm.
Example 9: Example 9: Proving a quadrilateral is a parallelogram
Problem: In quadrilateral ABCD, AB = CD = 7 cm and AB ∥ CD. Is ABCD a parallelogram?
Solution:
Given:
- AB = CD (one pair of opposite sides is equal)
- AB ∥ CD (the same pair is parallel)
Conclusion:
- If one pair of opposite sides is both equal and parallel, the quadrilateral is a parallelogram.
- Therefore, ABCD is a parallelogram.
Answer: Yes, ABCD is a parallelogram.
Example 10: Example 10: Diagonal intersection
Problem: In ∥gm ABCD, diagonals meet at O. If AO = (2x + 3) cm and OC = (4x − 7) cm, find x and AC.
Solution:
Given:
- AO = (2x + 3) cm, OC = (4x − 7) cm
Since diagonals bisect each other:
- AO = OC
- 2x + 3 = 4x − 7
- 3 + 7 = 4x − 2x
- 10 = 2x
- x = 5
Finding AC:
- AO = 2(5) + 3 = 13 cm
- AC = AO + OC = 13 + 13 = 26 cm
Answer: x = 5, AC = 26 cm.
Real-World Applications
Real-world applications of parallelogram properties:
- Architecture and construction: Parallelogram-shaped windows, tiles, and roof trusses use the property of equal opposite sides for structural balance.
- Mechanical linkages: Parallelogram linkages in machines ensure one side stays horizontal while the structure moves — used in drafting tables and bus doors.
- Map reading: Grid-based maps use parallelogram properties to measure areas of land plots.
- Engineering: Bridges and trusses use parallelogram-shaped supports because the diagonal bisecting property distributes forces evenly.
- Art and design: Tessellations (repeating patterns) use parallelograms because they tile a plane without gaps.
- Physics: Parallelogram law of vector addition uses the shape to find the resultant of two forces.
- Everyday objects: Erasers, chocolate bars, book covers, and tables often have parallelogram cross-sections.
Key Points to Remember
- A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
- Opposite sides of a parallelogram are equal in length.
- Opposite angles of a parallelogram are equal.
- Consecutive (adjacent) angles are supplementary (add up to 180°).
- Diagonals bisect each other — they cut each other into two equal parts.
- Each diagonal divides the parallelogram into two congruent triangles.
- Area = base × height and Perimeter = 2(a + b).
- Rectangle, rhombus, and square are special types of parallelograms.
- A quadrilateral is a parallelogram if one pair of opposite sides is both equal and parallel.
- The diagonals of a parallelogram are generally not equal (unlike a rectangle).
Practice Problems
- In ∥gm ABCD, ∠A = 72°. Find the remaining three angles.
- The adjacent sides of a parallelogram are 11 cm and 7 cm. Find its perimeter.
- Diagonals of a parallelogram are 20 cm and 14 cm. Find the length of each half-diagonal.
- In ∥gm PQRS, PQ = (4x − 3) cm and RS = (2x + 9) cm. Find x and the length of PQ.
- Find the area of a parallelogram with base 16 cm and height 11 cm.
- In ∥gm ABCD, ∠A = (3x + 15)° and ∠C = (5x − 25)°. Find x and ∠A.
- The area of a parallelogram is 360 cm² and its height is 15 cm. Find the base.
- Prove that the diagonals of parallelogram ABCD bisect each other, given AB ∥ CD and AD ∥ BC.
Frequently Asked Questions
Q1. What is a parallelogram?
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Examples include rectangles, rhombuses, and squares.
Q2. What are the 5 main properties of a parallelogram?
(1) Opposite sides are equal. (2) Opposite angles are equal. (3) Consecutive angles are supplementary. (4) Diagonals bisect each other. (5) Each diagonal divides it into two congruent triangles.
Q3. Is a rectangle a parallelogram?
Yes. A rectangle is a special parallelogram where all four angles are 90°. It has all parallelogram properties plus the additional property that diagonals are equal.
Q4. Is a rhombus a parallelogram?
Yes. A rhombus is a special parallelogram where all four sides are equal. It has all parallelogram properties plus the additional property that diagonals bisect each other at right angles.
Q5. Do the diagonals of a parallelogram bisect the angles?
Not in a general parallelogram. Diagonals bisect the vertex angles only in a rhombus (or square). In a general parallelogram, diagonals only bisect each other.
Q6. Are diagonals of a parallelogram equal?
Not in a general parallelogram. Diagonals are equal only in a rectangle (or square). In a general parallelogram, diagonals bisect each other but are usually of different lengths.
Q7. How do you find the area of a parallelogram?
Area = base × height, where the height is the perpendicular distance between the base and the opposite side. Do not use the slant side as the height.
Q8. What is the angle sum of a parallelogram?
The sum of all interior angles of a parallelogram is 360° (same as any quadrilateral). Since opposite angles are equal and consecutive angles are supplementary, this follows automatically.
Q9. How do you prove a quadrilateral is a parallelogram?
Show any ONE of: (a) both pairs of opposite sides are parallel, (b) both pairs of opposite sides are equal, (c) both pairs of opposite angles are equal, (d) diagonals bisect each other, or (e) one pair of opposite sides is both equal and parallel.
Q10. Is a trapezium a parallelogram?
No. A trapezium has only ONE pair of parallel sides. A parallelogram requires BOTH pairs of opposite sides to be parallel.
