A parallelogram is a fundamental shape in geometry, categorized as a special type of quadrilateral. What sets a parallelogram apart is its defining characteristic: both pairs of its opposite sides are parallel and equal in length.
Consider the parallelogram ABCD illustrated below. In this figure, side AB is parallel (II) to side CD, and side AD is parallel to side BC. This parallelism is not just a visual trait; it’s the core property that defines a parallelogram. Furthermore, the opposite sides are not only parallel but also equal in length: AD = BC and AB = CD.
However, it’s important to understand what is not a parallelogram. A trapezium (or trapezoid in US English) is a quadrilateral with only one pair of parallel sides. This single difference means a trapezium does not qualify as a parallelogram.
Key Properties of Parallelograms
Parallelograms possess several unique properties that distinguish them within the world of quadrilaterals. Understanding these properties is key to working with and recognizing parallelograms in geometry and real-life scenarios.
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Opposite Sides are Parallel: As mentioned earlier, this is the defining property. In parallelogram ABCD, AB || CD and AD || BC.
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Opposite Sides are Equal in Length: Not only are they parallel, but opposite sides are also congruent. Therefore, AB = CD and AD = BC.
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Opposite Angles are Equal: The angles opposite each other within a parallelogram are equal. In parallelogram ABCD, ∠A = ∠C and ∠B = ∠D.
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Sum of All Angles is 360°: Like all quadrilaterals, the interior angles of a parallelogram sum up to 360 degrees. ∠A + ∠B + ∠C + ∠D = 360°.
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Adjacent Angles are Supplementary: Adjacent angles, meaning angles that share a side, add up to 180°. For example, ∠A + ∠B = 180°, ∠B + ∠C = 180°, ∠C + ∠D = 180°, and ∠D + ∠A = 180°.
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Diagonals Bisect Each Other: The diagonals of a parallelogram, which are line segments connecting opposite vertices, intersect at a point that bisects each diagonal. If diagonals AC and BD intersect at point O, then OB = OD and OA = OC.
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Diagonals Form Congruent Triangles: Each diagonal divides the parallelogram into two congruent triangles. For instance, diagonal AC divides parallelogram ABCD into two congruent triangles, ΔABC and ΔCDA.
Types of Parallelograms
While all types of parallelograms share the core properties, there are specific classifications based on angles and side lengths. Three common types of parallelograms are:
- Rectangle: A rectangle is a parallelogram where all angles are right angles (90°). This also means that its diagonals are equal in length.
- Rhombus: A rhombus is a parallelogram where all sides are equal in length. Its diagonals bisect each other at right angles and also bisect the angles of the rhombus.
- Square: A square is a parallelogram that is both a rectangle and a rhombus. This means it has four right angles and four equal sides. The square is the most symmetrical type of parallelogram.
Real-Life Examples of Parallelograms
Parallelograms aren’t just abstract geometric shapes; they are prevalent in the world around us. Recognizing parallelograms in everyday objects helps solidify understanding and appreciate their practical applications.
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Buildings: Architectural designs often incorporate parallelograms for aesthetic and structural reasons. The Dockland Office Building in Hamburg, Germany, is a striking example of parallelogram shapes in modern architecture.
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Tiles: Many tiles, especially paving stones and wall tiles, are manufactured in parallelogram shapes, including rectangles and rhombuses, for efficient tiling patterns.
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Erasers: Many erasers, particularly rectangular prism erasers, have faces that are parallelograms.
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Paper and Books: While often rectangular, consider a stack of slightly shifted papers or a leaning book; they can visually represent parallelograms.
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Scissors Jacks: The mechanism of a car scissor jack utilizes parallelogram linkages to lift vehicles.
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Sliding Doors and Windows: Many sliding mechanisms are based on parallelogram principles for smooth, parallel motion.
Area of a Parallelogram
The area of a parallelogram is calculated using a straightforward formula: Area (A) = base (b) × height (h). Here, the ‘base’ is the length of one side, and the ‘height’ is the perpendicular distance between the base and its opposite side. It’s crucial to use the perpendicular height, not the length of an adjacent side, for accurate area calculation.
Perimeter of a Parallelogram
The perimeter of any shape is the total length of its boundary. For a parallelogram, the perimeter is the sum of all four sides. Since opposite sides of a parallelogram are equal, the formula simplifies to: Perimeter = 2 × (sum of adjacent sides) or Perimeter = 2 × (length + width), where ‘length’ and ‘width’ refer to the lengths of two adjacent sides.
Solved Examples
Let’s work through some examples to apply the properties and formulas of parallelograms.
Example 1:
In parallelogram ABCD, ∠DAB = 75° and ∠CBD = 60°. Calculate ∠BDC.
Solution:
Opposite angles of a parallelogram are equal, so ∠DCB = ∠DAB = 75°.
In triangle BCD, the sum of angles is 180°. Therefore, ∠BDC + ∠DCB + ∠CBD = 180°.
Substituting the known values: ∠BDC + 75° + 60° = 180°.
Simplifying: ∠BDC + 135° = 180°.
Solving for ∠BDC: ∠BDC = 180° – 135° = 45°.
Therefore, ∠BDC = 45°.
Example 2:
Find the area of a parallelogram with a base of 15 centimeters and a height of 6 centimeters.
Solution:
Area = base × height
Area = 15 cm × 6 cm
Area = 90 cm²
Example 3:
Two adjacent sides of a parallelogram are 5 cm and 3 cm. Calculate its perimeter.
Solution:
Perimeter = 2 × (sum of adjacent sides)
Perimeter = 2 × (5 cm + 3 cm)
Perimeter = 2 × (8 cm)
Perimeter = 16 cm
Practice Problems
Test your understanding of parallelograms with these practice questions.
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What is the perimeter of a parallelogram ABCD where the two adjacent sides are 12m and 8m respectively?
a) 40m b) 20m c) 4m d) 30m
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Which of the following is NOT a parallelogram?
a) Rectangle b) Rhombus c) Square d) Trapezium
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In a parallelogram ABCD, if ∠A = 60°, then ∠D is equal to:
a) 140° b) 130° c) 120° d) 110°
Frequently Asked Questions (FAQs)
Is a trapezium a parallelogram?
No, a trapezium is not a parallelogram. A parallelogram has two pairs of parallel sides, while a trapezium only has one pair of parallel sides.
Are all rectangles parallelograms?
Yes, all rectangles are parallelograms. Rectangles fulfill all the conditions of a parallelogram: they have two pairs of parallel sides and equal opposite sides. Additionally, rectangles have the specific property of having four right angles.
Are all parallelograms rectangles?
No, not all parallelograms are rectangles. While rectangles are special types of parallelograms, a parallelogram only needs to have opposite sides parallel and equal. It doesn’t necessarily need to have right angles. A parallelogram becomes a rectangle only when all its angles are 90°.
Do the diagonals of a parallelogram always bisect each other?
Yes, a key property of parallelograms is that their diagonals always bisect each other at their point of intersection.
Is it necessary for all sides to be equal in a quadrilateral to be a parallelogram?
No, it is not necessary for all sides of a quadrilateral to be equal for it to be a parallelogram. Only the opposite sides need to be equal in length. If all sides are equal in a parallelogram, it becomes a special type of parallelogram called a rhombus.
