Introduction to Quadrilaterals

Types and properties of quadrilaterals.

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What Is a Quadrilateral?

The word quadrilateral comes from the Latin words quadri (four) and latus (side). A quadrilateral is a closed plane figure formed by joining four points (vertices), no three of which are collinear (lie on the same straight line), using four straight line segments (sides). In simple terms: any closed shape with exactly four straight sides is a quadrilateral.

You encounter quadrilaterals every day — the pages of your textbook, a football field, a door frame, floor tiles, and the faces of a brick are all quadrilaterals. Recognising them and understanding their properties is a key skill tested in the Class 9 board exams of CBSE, Telangana, and Andhra Pradesh.

4 Vertices 4 Sides 4 Angles 2 Diagonals Angle Sum = 360°

💡 Quick identification rule: Count the sides. If a closed figure has exactly 4 straight sides, it is a quadrilateral. A circle, triangle, or pentagon is not a quadrilateral.

Five of these six shapes are quadrilaterals. Only the circle (dashed) is not — it has no straight sides.

Parts of a Quadrilateral

Every quadrilateral ABCD has six important parts. Knowing their names and how to label them correctly is essential — board exam questions regularly ask you to identify sides, angles, and diagonals of a given quadrilateral.

Quadrilateral ABCD: vertices (blue), sides (green labels), diagonals AC and BD (orange dashed), and their intersection O.

Part	Count	In quadrilateral ABCD	What it means
Vertices	4	A, B, C, D	Corner points where sides meet
Sides	4	AB, BC, CD, DA	The four straight line segments forming the boundary
Angles	4	∠A, ∠B, ∠C, ∠D	Interior angles at each vertex
Diagonals	2	AC, BD	Line segments joining opposite vertices
Angle Sum	—	∠A + ∠B + ∠C + ∠D = 360°	Sum of all interior angles is always 360°

📐 Angle Sum Property: The sum of the four interior angles of any quadrilateral is always 360° (four right angles). This is true regardless of the shape — regular, irregular, convex, or concave. You can verify this by dividing any quadrilateral into two triangles using a diagonal: each triangle has an angle sum of 180°, so 2 × 180° = 360°.

∠A + ∠B + ∠C + ∠D = 360°

Convex and Concave Quadrilaterals

All quadrilaterals fall into one of two broad categories based on how their interior looks:

Left (Convex): any line drawn between two interior points stays inside the shape. Right (Concave): such a line can pass through the exterior.

Feature	Convex Quadrilateral	Concave Quadrilateral
Interior points line	Always stays inside	May pass outside
Interior angles	All less than 180°	At least one angle > 180° (reflex)
Diagonals	Both lie inside the shape	At least one lies outside
Common examples	Rectangle, square, trapezium	Arrow-head (dart) shape

💡 Simple test: Look at all four interior angles of the quadrilateral. If every angle is less than 180°, it is convex. If any one angle is greater than 180° (a reflex angle), it is concave. In your Class 9 syllabus, almost all quadrilaterals studied are convex.

Types of Quadrilaterals — All Five Explained

Based on how many sides are parallel and whether sides and angles are equal, quadrilaterals are classified into five main types. Each type is a special case of the one above it in the hierarchy. Let's understand each one with its definition, diagram, and all properties.

Trapezium ABCD

1. Trapezium

A quadrilateral with exactly one pair of opposite sides parallel.

iOne pair of opposite sides is parallel: AB ∥ DC
iiThe angles on the same side of a non-parallel leg are supplementary (add up to 180°): ∠A + ∠D = 180° and ∠B + ∠C = 180°
iiiThe other pair of opposite sides (the legs AD and BC) are not necessarily parallel or equal.

Parallelogram ABCD

2. Parallelogram

A quadrilateral with two pairs of opposite sides parallel.

iBoth pairs of opposite sides parallel: AB ∥ DC and AD ∥ BC
iiOpposite sides are equal: AB = DC, AD = BC
iiiOpposite angles are equal: ∠A = ∠C, ∠B = ∠D
ivAdjacent angles are supplementary: ∠A + ∠D = 180°, ∠A + ∠B = 180°, etc.
vDiagonals bisect each other (cut each other into equal halves): OA = OC, OB = OD

Rectangle ABCD (all 90° angles)

3. Rectangle

A parallelogram in which one angle (and therefore all angles) is 90°.

iAll properties of a parallelogram apply (parallel & equal opposite sides, bisecting diagonals, etc.)
iiEvery interior angle is a right angle: ∠A = ∠B = ∠C = ∠D = 90°
iiiOpposite sides are equal: AB = DC, AD = BC (but adjacent sides need not be equal)
ivExtra property: Diagonals are equal in length: AC = BD
vDiagonals bisect each other (but not at right angles): OA = OC, OB = OD

Rhombus (all sides equal; diagonals ⊥)

4. Rhombus

A parallelogram in which all four sides are equal (equivalently, two adjacent sides of a parallelogram are equal).

iAll properties of a parallelogram apply.
iiAll four sides are equal: AB = BC = CD = DA
iiiOpposite angles are equal: ∠A = ∠C, ∠B = ∠D
ivAdjacent angles are supplementary: ∠A + ∠B = 180°, etc.
vExtra property: Diagonals bisect each other at right angles: ∠AOB = 90°

Square (all sides equal; all angles 90°; diagonals ⊥ and equal)

5. Square

A rectangle with all sides equal, or equivalently, a rhombus with all angles equal to 90°. The square is the most special quadrilateral — it satisfies all conditions of both rectangle and rhombus.

iAll sides are equal: AB = BC = CD = DA
iiEvery angle is 90°: ∠A = ∠B = ∠C = ∠D = 90°
iiiOpposite sides are parallel: AB ∥ DC, AD ∥ BC
ivDiagonals are equal in length: AC = BD
vDiagonals bisect each other at right angles: OA = OC, OB = OD, ∠AOB = 90°

Hierarchy of Quadrilaterals — How They Are Related

Each special type of quadrilateral is a more restricted version of the type above it. Understanding this hierarchy helps you instantly know which properties apply — a square inherits all properties of a rectangle and a rhombus and a parallelogram.

📌 How to read the hierarchy: Every shape lower in the diagram has all the properties of every shape above it, plus extra ones. A square is both a rectangle (all right angles) and a rhombus (all sides equal), making it the most special of all. A rectangle is a special parallelogram; a parallelogram is a special trapezium.

Master Comparison Table — All Properties at a Glance

Property	Trapezium	Parallelogram	Rectangle	Rhombus	Square
1 pair of ∥ sides	✔	✔	✔	✔	✔
2 pairs of ∥ sides	✘	✔	✔	✔	✔
Opposite sides equal	✘	✔	✔	✔	✔
All sides equal	✘	✘	✘	✔	✔
All angles = 90°	✘	✘	✔	✘	✔
Opposite angles equal	✘	✔	✔	✔	✔
Diagonals bisect each other	✘	✔	✔	✔	✔
Diagonals are equal	✘	✘	✔	✘	✔
Diagonals ⊥ (at right angles)	✘	✘	✘	✔	✔

Common Mistakes to Avoid

Confusing rhombus and square: A rhombus has all sides equal but angles are NOT necessarily 90°. A square has all sides equal AND all angles 90°. Every square is a rhombus, but not every rhombus is a square.
Forgetting the angle sum: The sum of angles in a quadrilateral is always 360°, not 180° (which is for triangles). In an exam, if three angles are given, always find the fourth as 360° minus the sum of the other three.
Thinking a rectangle's diagonals meet at right angles: In a rectangle, the diagonals are equal in length and bisect each other, but they do NOT cross at 90° unless it is also a square.
Thinking a rhombus has equal diagonals: In a rhombus, the diagonals bisect each other at 90°, but they are NOT equal in length unless it is also a square.
Calling any quadrilateral a parallelogram: A parallelogram requires BOTH pairs of opposite sides to be parallel. A trapezium has only ONE such pair and is NOT a parallelogram.

⛔ Most common board exam error: Saying "diagonals of a rectangle bisect each other at right angles." This is wrong — perpendicular diagonals are a property of the rhombus and square, not the rectangle (unless the rectangle is also a square).

What This Introduction Prepares You For

The properties you learned here are used directly in the theorems and exercises of Chapter 8. The next topics cover proofs related to the mid-point theorem and properties of parallelograms, where you will use the fact that opposite sides are equal or diagonals bisect each other as reasons in formal geometry proofs.

In Chapter 7 (Triangles), congruence conditions are applied to triangles formed inside quadrilaterals. Understanding that a diagonal divides a parallelogram into two congruent triangles is a key connecting idea. In Class 10, these concepts reappear in Coordinate Geometry (proving shapes are parallelograms or rectangles using coordinates) and in Similar Triangles.

For the Telangana and Andhra Pradesh board exams, Chapter 8 typically contributes 4–6 marks through fill-in-the-blank property questions, 2-mark "state and use a property" problems, and 4-mark proof questions. Memorising the comparison table above will help you score full marks on property-based questions.

✅ Chapter 8 Roadmap: Introduction (this page) → Exercise 8.1 — Properties of Parallelograms → Exercise 8.2 — The Mid-Point Theorem

Introduction to Quadrilaterals

Quadrilaterals — Introduction

What Is a Quadrilateral?

Parts of a Quadrilateral

Convex and Concave Quadrilaterals

Types of Quadrilaterals — All Five Explained

Hierarchy of Quadrilaterals — How They Are Related

Master Comparison Table — All Properties at a Glance

Common Mistakes to Avoid

What This Introduction Prepares You For