Quadrilaterals: Properties of Parallelograms, Rectangles, Squares, Rhombus, and Trapeziums
This module delves into the fundamental properties of common quadrilaterals, essential for mastering geometry in competitive exams like the CAT. Understanding these shapes and their unique characteristics will equip you to solve a wide range of quantitative aptitude problems.
Parallelograms: The Foundation
A parallelogram is a quadrilateral with two pairs of parallel sides. This simple definition leads to several key properties that are crucial for problem-solving.
Opposite sides and angles of a parallelogram are equal, and diagonals bisect each other.
In any parallelogram, opposite sides are equal in length, and opposite angles are equal in measure. Furthermore, the diagonals of a parallelogram intersect at their midpoints.
Let ABCD be a parallelogram. Then, AB || DC and AD || BC. This implies that AB = DC, AD = BC, ∠A = ∠C, and ∠B = ∠D. If the diagonals AC and BD intersect at point O, then AO = OC and BO = OD. Consecutive angles are supplementary (e.g., ∠A + ∠B = 180°).
Opposite sides are equal in length, and opposite angles are equal in measure.
Rectangles: The Right-Angled Parallelogram
A rectangle is a special type of parallelogram where all four interior angles are right angles (90°). This additional property gives it unique characteristics.
Rectangles have all properties of parallelograms plus equal diagonals.
A rectangle inherits all properties of a parallelogram. Additionally, its diagonals are equal in length and bisect each other.
If ABCD is a rectangle, then it is also a parallelogram. Therefore, AB = DC, AD = BC, ∠A = ∠B = ∠C = ∠D = 90°, and diagonals AC and BD bisect each other. The key distinguishing feature is that AC = BD.
All four interior angles are right angles (90°), and its diagonals are equal in length.
Squares: The Pinnacle of Regularity
A square is a quadrilateral that is both a rectangle and a rhombus. This means it possesses all the properties of both, making it a highly symmetrical shape.
Squares have equal sides, right angles, and perpendicular, equal diagonals that bisect each other.
A square has all the properties of a rectangle and a rhombus. This includes equal sides, all right angles, equal diagonals that bisect each other, and diagonals that are perpendicular bisectors of each other.
For a square ABCD, AB = BC = CD = DA. All angles are 90°. Diagonals AC and BD are equal (AC = BD), bisect each other, and are perpendicular (AC ⊥ BD). Each diagonal also bisects the angles at the vertices it connects (e.g., AC bisects ∠A and ∠C).
Visualizing the properties of a square: Imagine a perfect square. All sides are the same length. All corners are perfect right angles. If you draw the diagonals, they are the same length, they cross exactly in the middle, and they meet at a perfect right angle. Each diagonal also cuts the corner angles exactly in half.
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Equal sides, perpendicular diagonals, and diagonals that bisect the vertex angles.
Rhombus: The Diamond Shape
A rhombus is a parallelogram with all four sides equal in length. While its angles are not necessarily right angles, its diagonals have special properties.
Rhombus has equal sides and perpendicular diagonals that bisect each other.
A rhombus is a parallelogram with all sides equal. Its diagonals are perpendicular bisectors of each other and bisect the angles of the rhombus.
In a rhombus ABCD, AB = BC = CD = DA. Opposite angles are equal (∠A = ∠C, ∠B = ∠D), and consecutive angles are supplementary. The diagonals AC and BD are perpendicular (AC ⊥ BD) and bisect each other. Unlike a square, the diagonals are not necessarily equal in length, and the angles are not necessarily 90°.
They are perpendicular bisectors of each other.
Trapezium (Trapezoid): At Least One Pair of Parallel Sides
A trapezium (or trapezoid in American English) is a quadrilateral with at least one pair of parallel sides. The non-parallel sides and angles have varying properties depending on the type of trapezium.
Trapeziums have one pair of parallel sides; consecutive angles between parallel sides are supplementary.
In a trapezium, the sides that are parallel are called bases, and the other two sides are called legs. Consecutive angles between the parallel sides are supplementary.
Let ABCD be a trapezium with AB || DC. Then, ∠A + ∠D = 180° and ∠B + ∠C = 180°. If the trapezium is isosceles (AD = BC), then the base angles are equal (∠A = ∠B and ∠D = ∠C), and the diagonals are equal (AC = BD).
| Property | Parallelogram | Rectangle | Square | Rhombus | Trapezium |
|---|---|---|---|---|---|
| Opposite Sides Parallel | Yes | Yes | Yes | Yes | No (at least one pair) |
| Opposite Sides Equal | Yes | Yes | Yes | Yes | No |
| All Angles 90° | No | Yes | Yes | No | No |
| All Sides Equal | No | No | Yes | Yes | No |
| Diagonals Bisect Each Other | Yes | Yes | Yes | Yes | No |
| Diagonals Equal | No | Yes | Yes | No | No (only isosceles) |
| Diagonals Perpendicular | No | No | Yes | Yes | No |
Remember that a square is a special case of a rectangle, a rhombus, and a parallelogram. Similarly, a rectangle and a rhombus are special cases of a parallelogram.
Learning Resources
Provides a clear, visual overview of different types of quadrilaterals and their properties, including parallelograms, rectangles, squares, rhombuses, and trapezoids.
A comprehensive video lesson explaining the definitions and key properties of various quadrilaterals, with examples.
A collection of essential geometry formulas, including those for quadrilaterals, which are vital for quick recall during exams.
Chapter from the NCERT Class 9 textbook focusing on quadrilaterals, offering foundational concepts and proofs of properties.
A blog post tailored for CAT aspirants, detailing strategies and important concepts related to quadrilaterals.
A focused tutorial on the properties of parallelograms, often including practice questions and explanations.
Interactive learning resources and explanations for geometric shapes, including quadrilaterals, suitable for building intuition.
Detailed explanations and diagrams of the properties of various quadrilaterals, with a focus on clarity for learners.
Study material specifically curated for CAT preparation, covering quadrilaterals and polygons with relevant examples.
The foundational geometric proof regarding parallel lines and transversals, which underpins many quadrilateral properties (specifically, the supplementary angles between parallel lines).