ICSE Class 8 Maths Geometry Chapter 28 Quadrilaterals

Chapter 28: Quadrilaterals

Topics Covered:

Quadrilaterals

Types of Quadrilaterals

Properties of Quadrilaterals

Introduction

Four non-collinear co-planar points A, B, C, and D joined by four line segments form quadrilateral ABCD. The eight elements of a quadrilateral are its four sides and four angles.

In Figure 28.1, A, B, C, and D are the vertices of quadrilateral ABCD. ∠1, ∠2, ∠3, and ∠4 are its four interior angles. AB, BC, CD, and DA are its four sides. ∠5, ∠6, ∠7, and ∠8 are its four exterior angles. AC and BD, which connect the opposite vertices, are known as its diagonals.

Sum of Angles of a Quadrilateral

1. To prove: The sum of the interior angles of a quadrilateral is 360°.

Proof: Draw a quadrilateral ABCD and connect its opposite vertices B and D (Figure 28.2).

In △ABD: ∠1 + ∠2 + ∠3 = 180° (sum of interior angles)

In △BCD: ∠4 + ∠5 + ∠6 = 180° (sum of interior angles)

In quadrilateral ABCD, ∠1 + (∠2 + ∠5) + ∠4 + (∠3 + ∠6) = 180° + 180° (adding equations)

Sum of interior angles of quadrilateral ABCD = 360°. Q.E.D

2. To prove: The sum of the exterior angles of a quadrilateral is 360°.

Proof: Extend sides AB, BC, CD, and DA of quadrilateral ABCD to form exterior angles, ∠1, ∠2, ∠3, ∠4 respectively (Figure 28.3).

Now ∠1 + ∠5 = 180° (linear pair)

∠1 = 180° - ∠5

Similarly ∠2 = 180° - ∠6, ∠3 = 180° - ∠7 and ∠4 = 180° - ∠8

∠1 + ∠2 + ∠3 + ∠4 = 180° - ∠5 + 180° - ∠6 + 180° - ∠7 + 180° - ∠8 (adding the four equations)

∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8 = 720°

But ∠5 + ∠6 + ∠7 + ∠8 = 360° (sum of interior angles)

∠1 + ∠2 + ∠3 + ∠4 + 360° = 720°

or ∠1 + ∠2 + ∠3 + ∠4 = 360°

or Sum of exterior angles = 360°

Alternatively,

∠1 + ∠5 = 180°

∠2 + ∠6 = 180°

∠3 + ∠7 = 180°

∠4 + ∠8 = 180°

all linear pairs

Types of Quadrilaterals and Their Properties

Convex Quadrilaterals: All angles less than 180°

Two pairs of adjacent sides equal: This describes a Kite ABCD where AB = AD; BC = CD

Kite ABCD: AB = AD; BC = CD

One pair of opposite sides parallel but not equal: This describes a Trapezium ABCD where BC || AD and BC ≠ AD

Trapezium ABCD: BC || AD, BC ≠ AD

Isosceles trapezium ABCD: BC || AD; AB = CD, BC ≠ AD

Opposite sides parallel and equal: This describes a Parallelogram ABCD where AB || CD; AD || BC and AB = DC, AD = BC

Parallelogram ABCD: AB || CD; AD || BC, AB = DC, AD = BC

Rectangle ABCD: AB || CD; AD || BC, all angles = 90°, AB = DC, AD = BC

Opposite sides parallel and all sides equal: This describes a Rhombus ABCD where AB || CD; AD || BC, AB = BC = CD = DA

Rhombus ABCD: AB || CD; AD || BC, AB = BC = CD = DA

Square ABCD: AB || CD; AD || BC, AB = BC = CD = DA, all angles = 90°

Concave Quadrilaterals: One angle greater than 180°. Two pairs of adjacent sides equal.

Kite ABCD: AB = AD; BC = CD

Teacher's Note

Understanding quadrilaterals helps in real-world applications like designing floor tiles, windows, and furniture. Recognizing different types helps in calculating areas and determining structural properties of buildings.

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