Class 8 Mathematics Chapter 3: Understanding Quadrilaterals 

These notes of Understanding Quadrilaterals cover the entire NCERT chapter, including all theorems, properties, and typical CBSE exam questions. Use diagrams while revising for better understanding. Good luck!

1. Introduction to Quadrilaterals: Understanding Quadrilaterals

• A quadrilateral is a closed polygon with exactly four sides, four vertices and four angles.
• The word “quadrilateral” comes from Latin: “quad” = four, “latus” = side.
• Sum of interior angles of any quadrilateral = 360° (Proof: A quadrilateral can be divided into two triangles by drawing one diagonal. Sum of angles in a triangle = 180° × 2 = 360°)

2. Elements of a Quadrilateral

• Sides: Four line segments (AB, BC, CD, DA)
• Vertices: Four points (A, B, C, D)
• Angles: ∠DAB, ∠ABC, ∠BCD, ∠CDA
• Diagonals: Line segments joining non-adjacent vertices (AC and BD)
• Adjacent sides: Sides with a common vertex (AB and BC)
• Opposite sides: Sides not sharing a vertex (AB and CD, AD and BC)

3. Types of Quadrilaterals (Based on Sides and Angles)

A. Trapezium

• Exactly one pair of parallel sides.
• The parallel sides are called bases, non-parallel sides are legs.
• In an isosceles trapezium: – Non-parallel sides (legs) are equal. – Base angles are equal (∠A = ∠D, ∠B = ∠C). – Diagonals are equal (AC = BD).

B. Kite

• Two pairs of adjacent equal sides (AB = AD and CB = CD).
• Diagonals are perpendicular.
• One diagonal is the axis of symmetry (usually the diagonal between equal sides).
• One pair of opposite angles are equal (angles between unequal sides).

C. Parallelogram

• Both pairs of opposite sides are parallel and equal.
• Opposite angles are equal.
• Consecutive angles are supplementary (add up to 180°).
• Diagonals bisect each other (intersect at midpoint).

D. Rectangle

• A parallelogram with all angles = 90°.
• Opposite sides equal and parallel.
• Diagonals are equal and bisect each other.
• Has four right angles.

E. Rhombus

• A parallelogram with all sides equal.
• Opposite angles equal, consecutive angles supplementary.
• Diagonals bisect each other at 90° and bisect vertex angles.
• Every rhombus is a kite, but every kite is not a rhombus.

F. Square

• A rectangle with all sides equal, or a rhombus with right angles.
• All properties of parallelogram, rectangle, and rhombus.
• Diagonals are equal, perpendicular, and bisect vertex angles.
• Most symmetrical quadrilateral (4 lines of symmetry, rotational symmetry of 90°).

4. Angle Sum Property

• Interior angles of a quadrilateral = 360°.
• Exterior angle = 180° – Interior angle at that vertex.
• Sum of all exterior angles (one at each vertex) = 360°.

5. Special Properties Summary Table

6. Diagonals in Quadrilaterals

• Trapezium: Generally unequal (equal only in isosceles).
• Kite: Perpendicular, one bisected by the other.
• Parallelogram: Bisect each other.
• Rectangle: Equal and bisect each other.
• Rhombus: Perpendicular bisectors of each other.
• Square: Equal, perpendicular, bisect angles.

7. Important Theorems & Constructions

Theorem 3.1

The diagonal of a parallelogram divides it into two congruent triangles. (Proof using ASA/SAS congruence)

Theorem 3.2

In a parallelogram: (a) Opposite sides equal and parallel (b) Opposite angles equal (c) Consecutive angles supplementary (d) Diagonals bisect each other (Converse is also true)

Theorem 3.3

A quadrilateral is a parallelogram if:

1. Opposite sides are equal and parallel, OR
2. Opposite angles are equal, OR
3. Diagonals bisect each other, OR
4. One pair of opposite sides is equal and parallel.

Mid-point Theorem (for triangles, used in parallelograms)

The line segment joining mid-points of two sides of a triangle is parallel to the third side and half as long.

8. Construction-Related Questions

• Construct a parallelogram when two sides and one angle are given.
• Construct a rhombus when side and one angle are given.
• Construct a square when side is given (using perpendiculars).

9. Key Formulas

• Perimeter of quadrilateral = Sum of all sides
• Area of trapezium = (1/2) × (sum of parallel sides) × height
• Area of rhombus = (d₁ × d₂)/2 or base × height
• Area of parallelogram = base × height
• Area of rectangle = length × breadth
• Area of square = side² or (diagonal²)/2

10. Common Numerical Problems

1. Find the fourth angle if three angles are given (subtract sum from 360°).
2. In parallelogram ABCD, if ∠A = 70°, find all other angles (∠B = 110°, ∠C = 70°, ∠D = 110°).
3. Diagonal AC of rhombus ABCD = 24 cm, BD = 18 cm → Area = (24 × 18)/2 = 216 cm².
4. Trapezium with parallel sides 10 cm and 20 cm, height 8 cm → Area = (1/2) × (10 + 20) × 8 = 120 cm².

11. Important Points for Exams

• Always draw neat labelled diagrams.
• Remember the hierarchy: Square → Rhombus → Parallelogram Square → Rectangle → Parallelogram
• A square is a special case of all the above except trapezium and kite.
• If a parallelogram has one right angle → it is a rectangle.
• If a parallelogram has equal diagonals → it is a rectangle.
• If a rhombus has one right angle → it is a square.

12. Quick Revision One-liners

• Total angle sum = 360°
• Parallelogram → opposite everything equal, diagonals bisect
• Rectangle → all angles 90°, diagonals equal
• Rhombus → all sides equal, diagonals perpendicular
• Square → all sides equal + all angles 90°
• Kite → two pairs adjacent equal sides, diagonals ⊥
• Trapezium → exactly one pair parallel

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Chapter 4: Practical Geometry

 

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