🔶 Quadrilaterals - Class 9

Complete notes with properties, theorems, and formula sheet

1. Introduction to Quadrilaterals

A quadrilateral is a closed figure with four sides, four vertices, and four angles. The word "quadrilateral" comes from Latin: "quadri" means four and "lateral" means side.

📖 What is a Quadrilateral?

A quadrilateral is a polygon with four sides, four angles, and four vertices.

Sum of all four angles = 360°

Notation: Quadrilateral ABCD has vertices A, B, C, D and sides AB, BC, CD, DA.

1.1 Elements of a Quadrilateral

Sides: Four line segments AB, BC, CD, DA.
Vertices: Four corner points A, B, C, D.
Angles: Four interior angles ∠A, ∠B, ∠C, ∠D.
Diagonals: Line segments joining opposite vertices (AC and BD).
Adjacent sides: Sides that share a common vertex.
Opposite sides: Sides that don't share a common vertex.

⚠️ Angle Sum Property of Quadrilateral

The sum of all four interior angles of a quadrilateral is 360°.

∠A + ∠B + ∠C + ∠D = 360°

This can be proved by dividing the quadrilateral into two triangles using a diagonal.

2. Types of Quadrilaterals

2.1 Parallelogram

📖 Parallelogram

A quadrilateral with both pairs of opposite sides parallel.

Opposite sides are equal: AB = CD and BC = AD
Opposite angles are equal: ∠A = ∠C and ∠B = ∠D
Adjacent angles are supplementary: ∠A + ∠B = 180°
Diagonals bisect each other: AC and BD bisect at their midpoint
Each diagonal divides it into two congruent triangles

2.2 Rectangle

📖 Rectangle

A parallelogram with all four angles equal to 90°.

All properties of parallelogram apply
All angles are 90°: ∠A = ∠B = ∠C = ∠D = 90°
Diagonals are equal: AC = BD
Diagonals bisect each other
Opposite sides are equal and parallel

2.3 Square

📖 Square

A rectangle with all four sides equal.

All properties of rectangle and parallelogram apply
All sides are equal: AB = BC = CD = DA
All angles are 90°
Diagonals are equal and bisect each other at 90°
Diagonals bisect the angles (each diagonal makes 45° with sides)

2.4 Rhombus

📖 Rhombus

A parallelogram with all four sides equal.

All properties of parallelogram apply
All sides are equal: AB = BC = CD = DA
Opposite angles are equal
Diagonals bisect each other at 90°
Diagonals bisect the angles

2.5 Trapezium

📖 Trapezium

A quadrilateral with exactly one pair of parallel sides.

One pair of opposite sides is parallel (called bases)
Other pair is non-parallel (called legs)
Isosceles Trapezium: Non-parallel sides are equal
In isosceles trapezium, base angles are equal
In isosceles trapezium, diagonals are equal

2.6 Kite

📖 Kite

A quadrilateral with two pairs of adjacent sides equal.

Two pairs of adjacent sides are equal: AB = AD and CB = CD
One diagonal is the perpendicular bisector of the other
One pair of opposite angles are equal (the angles between unequal sides)

3. Important Theorems

⚠️ Theorem 1: Diagonal of Parallelogram

A diagonal of a parallelogram divides it into two congruent triangles.

⚠️ Theorem 2: Opposite Sides of Parallelogram

In a parallelogram, opposite sides are equal.

Converse: If opposite sides of a quadrilateral are equal, it's a parallelogram.

⚠️ Theorem 3: Diagonals of Parallelogram

The diagonals of a parallelogram bisect each other.

Converse: If diagonals of a quadrilateral bisect each other, it's a parallelogram.

⚠️ Theorem 4: Mid-point Theorem

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

📝 Example: Parallelogram Properties

Q: In parallelogram ABCD, ∠A = 70°. Find all other angles.

Solution:

Given: ∠A = 70°

∠C = ∠A = 70° (opposite angles equal)

∠A + ∠B = 180° (adjacent angles supplementary)

70° + ∠B = 180°

∠B = 110°

∠D = ∠B = 110° (opposite angles equal)

Angles: 70°, 110°, 70°, 110°

Quadrilateral	Sides	Angles	Diagonals
Parallelogram	Opposite equal	Opposite equal	Bisect each other
Rectangle	Opposite equal	All 90°	Equal, bisect each other
Square	All equal	All 90°	Equal, ⊥ bisect
Rhombus	All equal	Opposite equal	⊥ bisect each other
Trapezium	One pair parallel	No special property	Not necessarily equal

📚 Quick Formula Sheet - Quadrilaterals

Basic Property

Sum of all angles = 360°

∠A + ∠B + ∠C + ∠D = 360°

Universal for all quadrilaterals

Parallelogram

Opposite sides equal

Opposite angles equal

Adjacent angles = 180°

Diagonals bisect each other

Rectangle

All angles = 90°

Opposite sides equal

Diagonals equal

Diagonals bisect each other

Square

All sides equal

All angles = 90°

Diagonals equal

Diagonals bisect at 90°

Rhombus

All sides equal

Opposite angles equal

Diagonals bisect at 90°

Diagonals bisect angles

Trapezium

One pair parallel sides

Isosceles: legs equal

Isosceles: base angles equal

Isosceles: diagonals equal

Tests for Parallelogram

Opposite sides equal → Parallelogram

Opposite angles equal → Parallelogram

Diagonals bisect → Parallelogram

One pair parallel & equal → Parallelogram

Mid-point Theorem

Line joining mid-points

∥ to third side

= ½ × third side

Important for triangles

💡 Study Tips for Quadrilaterals

• Draw clear diagrams and mark equal sides/angles with same symbols.

• Understand the hierarchy: Square → Rectangle → Parallelogram.

• Square → Rhombus → Parallelogram.

• Every square is a rectangle, rhombus, and parallelogram!

• Focus on properties of diagonals - they're crucial for identification.

• Practice converting "if-then" theorems to their converses.

• Memorize the angle sum property (360°) - use it frequently!

🔑 Common Mistakes to Avoid

Don't confuse parallelogram with trapezium (parallelogram has BOTH pairs parallel)
Rectangle ≠ Square (rectangle doesn't need all sides equal)
Rhombus angles are NOT necessarily 90° (that's a square)
Remember: In rhombus, diagonals bisect at 90°, but aren't necessarily equal
Trapezium has only ONE pair of parallel sides (not both)
Mark all given information on your diagram before solving

📝 Practice Problems

Q1: Diagonals of a quadrilateral bisect each other. What is it?

Answer: Parallelogram (by converse theorem)

Q2: All sides equal + all angles 90°. What is it?

Answer: Square

Q3: In a quadrilateral, three angles are 80°, 95°, 105°. Find the fourth.

Answer: 360° - (80° + 95° + 105°) = 360° - 280° = 80°