Quadrilaterals — Class 8 Maths | Preeti Kushwah Classes

📐 Introduction

Look around you — the screen you are reading this on, the top of your desk, the floor tiles, a kite soaring in the sky — all of these are examples of quadrilaterals. A quadrilateral is a closed figure made up of four line segments. In this chapter, we will explore the fascinating world of quadrilaterals, their properties, and the relationships between different types.

This chapter from the NCERT Ganita (2025-26) textbook builds upon what you learned about triangles and their properties. Just as the angle sum of a triangle is 180°, quadrilaterals have their own angle sum property — and much more to discover!

💡 Prerequisite Knowledge: Before studying this chapter, make sure you are comfortable with: (i) Properties of triangles (angle sum = 180°), (ii) Types of angles (acute, right, obtuse), (iii) Parallel lines and transversals, (iv) Basic properties of lines and angles.

📚 Chapter Overview: What You Will Learn

Topic	Key Concepts
Polygons	Classification, convex vs concave, regular polygons
Angle Sum Property	Sum of interior angles = 360°, diagonal method
Types of Quadrilaterals	Trapezium, kite, parallelogram, rectangle, rhombus, square
Properties	Sides, angles, diagonals of each type
Hierarchy	How types are related, special cases
Exterior Angles	Sum of exterior angles = 360°

⬟ Polygons & Their Classification

A polygon is a simple closed figure made up of only line segments. The word comes from the Greek words poly (many) and gonia (angle). Polygons are classified by the number of sides they have.

📜 Types of Polygons by Number of Sides

Number of Sides	Name	Sum of Interior Angles	Example
3	Triangle	180°	Traffic sign
4	Quadrilateral	360°	Book, screen
5	Pentagon	540°	Pentagon building
6	Hexagon	720°	Honeycomb cell
7	Heptagon	900°	Some coins
8	Octagon	1080°	Stop sign
n	n-gon	(n − 2) × 180°	—

Sum of Interior Angles of an n-sided polygon = (n − 2) × 180° This is derived by dividing the polygon into (n − 2) triangles using diagonals from one vertex.

📌 Convex vs Concave Polygons

✅ Convex Polygon

All interior angles are less than 180°. All diagonals lie completely inside the polygon. If you stand inside a convex polygon, you can see every wall from any point.

❌ Concave Polygon

At least one interior angle is greater than 180° (reflex angle). At least one diagonal passes outside the polygon. Think of an arrow-head shape.

Convex polygon (all angles < 180°) vs Concave polygon (one angle > 180°)

💫 Regular vs Irregular Polygons

⭐ Regular Polygon

All sides are equal in length and all angles are equal in measure. Examples: equilateral triangle, square, regular hexagon.

Each interior angle = (n − 2) × 180° / n

🔸 Irregular Polygon

Either the sides are not all equal, or the angles are not all equal, or both. Most real-world quadrilaterals (like a random piece of paper cut at an angle) are irregular.

💡 Memory Trick: To remember the angle sum formula, think: "Every time you add a side, you add another triangle (180°)." A quadrilateral = 2 triangles = 360°. A pentagon = 3 triangles = 540°, and so on!

📐 Angle Sum Property of Quadrilaterals

The Angle Sum Property of a quadrilateral states that the sum of all four interior angles of any quadrilateral is 360°.

∠A + ∠B + ∠C + ∠D = 360° This holds for ALL quadrilaterals — convex, concave, regular, or irregular.

🔐 Proof Using Diagonal

Consider a quadrilateral ABCD. Draw diagonal AC, which divides the quadrilateral into two triangles: △ABC and △ACD.

Diagonal AC divides quadrilateral ABCD into two triangles

Step 1: In △ABC, sum of angles = ∠BAC + ∠ABC + ∠BCA = 180° … (i)

Step 2: In △ACD, sum of angles = ∠DAC + ∠ACD + ∠ADC = 180° … (ii)

Step 3: Adding (i) and (ii):
(∠BAC + ∠DAC) + ∠ABC + (∠BCA + ∠ACD) + ∠ADC = 360°
∴ ∠A + ∠B + ∠C + ∠D = 360°

💡 Key Observation: ∠BAC + ∠DAC = ∠A (the full angle at vertex A), and ∠BCA + ∠ACD = ∠C (the full angle at vertex C). This is why the diagonal method works perfectly!

✏️ Example 1: Finding a Missing Angle

Three angles of a quadrilateral are 75°, 90°, and 120°. Find the fourth angle.

Given: ∠A = 75°, ∠B = 90°, ∠C = 120°

Using angle sum property: ∠A + ∠B + ∠C + ∠D = 360°

Substituting: 75° + 90° + 120° + ∠D = 360°

Solving: 285° + ∠D = 360°
∠D = 360° − 285° = 75°

✏️ Example 2: Angles in Ratio

The angles of a quadrilateral are in the ratio 3 : 5 : 7 : 9. Find all the angles.

Let the angles be: 3x, 5x, 7x, and 9x

Using angle sum property: 3x + 5x + 7x + 9x = 360°

Solving: 24x = 360° ⇒ x = 15°

The angles are:
3 × 15° = 45°
5 × 15° = 75°
7 × 15° = 105°
9 × 15° = 135°
Verify: 45 + 75 + 105 + 135 = 360° ✅

🔬 Types of Quadrilaterals

Quadrilaterals are classified into several types based on the properties of their sides, angles, and diagonals. Let us explore each type in detail.

▲ Trapezium (Trapezoid)

A trapezium is a quadrilateral in which one pair of opposite sides is parallel. The parallel sides are called bases, and the non-parallel sides are called legs.

Trapezium ABCD: AB ∥ DC (one pair of parallel sides)

🔹 Isosceles Trapezium

A special trapezium where the non-parallel sides (legs) are equal. In an isosceles trapezium:

Base angles are equal
Diagonals are equal in length

📏 Area of Trapezium

Area = ½ × (sum of parallel sides) × height
= ½ × (a + b) × h
where a and b are the lengths of the two parallel sides.

🔸 Kite

A kite is a quadrilateral in which two pairs of adjacent sides are equal. That is, the sides come in two pairs of consecutive equal sides.

Kite ABCD: AB = AD (= a), CB = CD (= b), diagonals perpendicular

💫 Properties of a Kite

Two pairs of adjacent sides are equal (AB = AD and CB = CD)
Diagonals are perpendicular to each other (AC ⊥ BD)
The main diagonal (connecting the vertices between unequal sides) bisects the other diagonal
One pair of opposite angles is equal (∠B = ∠D)
The main diagonal bisects the vertex angles through which it passes
Area = ½ × d₁ × d₂ (product of diagonals divided by 2)

▭ Parallelogram & Its Properties

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. It is one of the most important types of quadrilaterals, and many other special quadrilaterals (rectangle, rhombus, square) are actually special cases of a parallelogram.

Parallelogram ABCD: AB ∥ DC and AD ∥ BC; diagonals bisect each other at O

🔐 Properties of a Parallelogram

1. Opposite Sides are Equal

In parallelogram ABCD:
AB = DC and AD = BC
Converse: If opposite sides of a quadrilateral are equal, it is a parallelogram.

2. Opposite Angles are Equal

∠A = ∠C and ∠B = ∠D
Converse: If opposite angles of a quadrilateral are equal, it is a parallelogram.

3. Diagonals Bisect Each Other

The diagonals AC and BD meet at point O such that:
AO = OC and BO = OD
Converse: If diagonals of a quadrilateral bisect each other, it is a parallelogram.

4. Consecutive Angles are Supplementary

∠A + ∠B = 180°
∠B + ∠C = 180°
(co-interior angles, since opposite sides are parallel)

💡 Important: Each of the above properties has a converse — meaning if you can prove any one of these conditions, you can conclude the quadrilateral is a parallelogram. There is also a fifth test: if one pair of opposite sides is both equal and parallel, the quadrilateral is a parallelogram.

✏️ Example 3: Using Parallelogram Properties

In a parallelogram PQRS, ∠P = (3x + 10)° and ∠R = (5x − 30)°. Find all angles.

Since PQRS is a parallelogram: Opposite angles are equal, so ∠P = ∠R

Setting up equation: 3x + 10 = 5x − 30

Solving: 10 + 30 = 5x − 3x ⇒ 40 = 2x ⇒ x = 20

∠P = ∠R = 3(20) + 10 = 70°

Consecutive angles are supplementary:
∠Q = 180° − 70° = 110°
∠S = ∠Q = 110° (opposite angles)
Verify: 70 + 110 + 70 + 110 = 360° ✅

✏️ Example 4: Diagonals Bisecting Each Other

The diagonals of a parallelogram ABCD intersect at O. If AO = (2x + 3) cm and OC = (x + 7) cm, find the length of diagonal AC.

Property: Diagonals of a parallelogram bisect each other, so AO = OC

Setting up equation: 2x + 3 = x + 7

Solving: 2x − x = 7 − 3 ⇒ x = 4

AO = OC = 2(4) + 3 = 11 cm

AC = AO + OC = 11 + 11 = 22 cm

🔐 Proving a Quadrilateral is a Parallelogram

You can prove that a quadrilateral is a parallelogram by showing any one of the following:

Both pairs of opposite sides are parallel (definition)
Both pairs of opposite sides are equal
Both pairs of opposite angles are equal
Diagonals bisect each other
One pair of opposite sides is both equal and parallel

⬜ Rectangle, Rhombus & Square

Rectangle, rhombus, and square are all special types of parallelogram. Each one inherits all the properties of a parallelogram plus has additional properties of its own.

🔷 Rectangle

A rectangle is a parallelogram in which each angle is a right angle (90°).

Rectangle ABCD: All angles = 90°, diagonals are equal (d₁ = d₂)

💫 Properties of a Rectangle

All properties of a parallelogram (opposite sides equal & parallel, diagonals bisect each other)
All four angles are 90°
Diagonals are equal in length (AC = BD)
Diagonals bisect each other (but NOT necessarily at 90°)
Area = length × breadth
Perimeter = 2(length + breadth)

🔶 Rhombus

A rhombus is a parallelogram in which all four sides are equal. It looks like a "tilted square."

Rhombus ABCD: All sides = a, diagonals bisect each other at 90°

💫 Properties of a Rhombus

All properties of a parallelogram
All four sides are equal
Diagonals bisect each other at right angles (90°)
Diagonals bisect the vertex angles
Opposite angles are equal (but NOT necessarily 90°)
Area = ½ × d₁ × d₂ (half the product of diagonals)
Perimeter = 4 × side

⬛ Square

A square is a parallelogram that is both a rectangle and a rhombus. It has all angles equal (90°) AND all sides equal.

Square ABCD: All sides = a, all angles = 90°, diagonals equal & perpendicular

💫 Properties of a Square

All properties of a parallelogram, rectangle, AND rhombus
All four sides are equal
All four angles are 90°
Diagonals are equal in length
Diagonals bisect each other at right angles (90°)
Diagonals bisect the vertex angles (each diagonal makes 45° with each side)
Area = side² = ½ × diagonal²
Diagonal = side × √2
Perimeter = 4 × side

✏️ Example 5: Rectangle Diagonal

The diagonals of a rectangle ABCD meet at O. If ∠AOB = 118°, find ∠ABO.

Key property: In a rectangle, diagonals are equal and bisect each other. So AO = BO (half-diagonals are equal).

In △AOB: Since AO = BO, triangle AOB is isosceles.

∠OAB = ∠OBA (base angles of isosceles triangle)

Sum of angles in △AOB: ∠AOB + ∠OAB + ∠OBA = 180°

Solving: 118° + 2 × ∠OBA = 180°
2 × ∠OBA = 62°
∠ABO = 31°

✏️ Example 6: Rhombus Diagonals

The diagonals of a rhombus are 16 cm and 12 cm. Find the side of the rhombus.

Property: Diagonals of a rhombus bisect each other at right angles.

Half-diagonals: d₁/2 = 16/2 = 8 cm, d₂/2 = 12/2 = 6 cm

Using Pythagoras theorem in the right triangle formed:
side² = 8² + 6² = 64 + 36 = 100

side = √100 = 10 cm

Area = ½ × 16 × 12 = 96 cm²

📊 Comparison Table

Property	Parallelogram	Rectangle	Rhombus	Square
Opposite sides parallel	✅	✅	✅	✅
Opposite sides equal	✅	✅	✅ (all equal)	✅ (all equal)
All sides equal	❌	❌	✅	✅
Opposite angles equal	✅	✅ (all 90°)	✅	✅ (all 90°)
All angles 90°	❌	✅	❌	✅
Diagonals bisect each other	✅	✅	✅	✅
Diagonals equal	❌	✅	❌	✅
Diagonals perpendicular	❌	❌	✅	✅
Diagonals bisect vertex angles	❌	❌	✅	✅

💡 Remember: A square is the most "special" quadrilateral — it has ALL the properties of a parallelogram, rectangle, AND rhombus combined. Think of it as: Square = Rectangle + Rhombus.

📈 Hierarchy of Quadrilaterals

Quadrilaterals form a beautiful hierarchy where each type is a special case of a more general type. Understanding this hierarchy helps you apply properties correctly: every property of a general type automatically applies to its special types.

Hierarchy of Quadrilaterals: each lower type is a special case of the type above it

🔄 Reading the Hierarchy

Every square IS a rectangle, rhombus, parallelogram, and quadrilateral
Every rectangle IS a parallelogram, but NOT necessarily a rhombus
Every rhombus IS a parallelogram, but NOT necessarily a rectangle
A trapezium is NOT a parallelogram (only 1 pair of parallel sides)

💡 True or False?

"Every square is a rhombus" — TRUE
"Every rhombus is a square" — FALSE
"Every rectangle is a square" — FALSE
"Every square is a rectangle" — TRUE
"Every parallelogram is a trapezium" — TRUE (has at least 1 pair of parallel sides)

🔄 Exterior Angles of Quadrilaterals

An exterior angle of a polygon is the angle formed between one side of the polygon and the extension of an adjacent side. At each vertex, the interior angle and exterior angle form a linear pair (they add up to 180°).

Sum of Exterior Angles of ANY Convex Polygon = 360° This is true regardless of the number of sides! (Taking one exterior angle at each vertex)

e₁ + e₂ + e₃ + e₄ = 360° (sum of exterior angles)

🔐 Why Is the Sum 360°?

At each vertex: Interior angle + Exterior angle = 180°

For a quadrilateral with 4 vertices:

Sum of all (interior + exterior) = 4 × 180° = 720°

We know sum of interior angles = 360°

So, sum of exterior angles = 720° − 360° = 360°

✏️ Example 7: Finding Exterior Angles

The exterior angles of a quadrilateral are (x + 5)°, (2x + 3)°, (3x − 7)°, and (4x + 9)°. Find all the exterior angles.

Sum of exterior angles = 360°

Setting up equation: (x + 5) + (2x + 3) + (3x − 7) + (4x + 9) = 360

Simplifying: 10x + 10 = 360 ⇒ 10x = 350 ⇒ x = 35

The exterior angles are:
35 + 5 = 40°
2(35) + 3 = 73°
3(35) − 7 = 98°
4(35) + 9 = 149°
Verify: 40 + 73 + 98 + 149 = 360° ✅

💡 Regular Polygon Shortcut: For a regular polygon with n sides, each exterior angle = 360° / n. So for a regular quadrilateral (square): each exterior angle = 360° / 4 = 90°. For a regular hexagon: 360° / 6 = 60°.

✏️ More Worked Examples

✏️ Example 8: Angles of a Parallelogram

Two adjacent angles of a parallelogram are (2x + 15)° and (3x − 25)°. Find all four angles.

Property: Adjacent angles of a parallelogram are supplementary (sum = 180°)

Setting up equation: (2x + 15) + (3x − 25) = 180

Simplifying: 5x − 10 = 180 ⇒ 5x = 190 ⇒ x = 38

The angles are:
2(38) + 15 = 91°
3(38) − 25 = 89°
Opposite to 91° ⇒ 91°
Opposite to 89° ⇒ 89°
Verify: 91 + 89 + 91 + 89 = 360° ✅

✏️ Example 9: Square Diagonal Calculation

The side of a square is 7 cm. Find the length of its diagonal.

Formula: Diagonal of a square = side × √2

Substituting: d = 7 × √2 = 7√2 cm

Approximate value: d = 7 × 1.414 ≈ 9.9 cm

Alternatively, using Pythagoras:
d² = 7² + 7² = 49 + 49 = 98
d = √98 = √(49 × 2) = 7√2 cm ✅

✏️ Example 10: Identifying the Quadrilateral

A quadrilateral has all four sides equal and diagonals that are equal. What type of quadrilateral is it?

All four sides equal: This means it is at least a rhombus.

Diagonals are equal: In a rhombus, diagonals are generally NOT equal. Diagonals are equal in a rectangle.

Combining both: A quadrilateral that is both a rhombus (all sides equal) and a rectangle (all angles 90°, diagonals equal) is a SQUARE.

Answer: The quadrilateral is a square.

✏️ Example 11: Interior Angles of a Regular Polygon

Find the number of sides of a regular polygon whose each interior angle is 135°.

Each interior angle of a regular n-gon: (n − 2) × 180° / n

Setting up equation: (n − 2) × 180 / n = 135

Solving: 180n − 360 = 135n

45n = 360 ⇒ n = 8

Answer: The polygon has 8 sides (it is a regular octagon).

📊 Summary & Key Formulas

Sum of Interior Angles of a Quadrilateral = 360°

Sum of Exterior Angles of ANY Convex Polygon = 360°

Sum of Interior Angles of n-sided Polygon = (n − 2) × 180°

📋 Quick Reference: Properties at a Glance

⬟ Trapezium

One pair of parallel sides
Area = ½(a+b)h

🔸 Kite

2 pairs of adjacent sides equal
Diagonals perpendicular
One pair of opposite angles equal
Area = ½ d₁d₂

▭ Parallelogram

Both pairs of opposite sides parallel & equal
Opposite angles equal
Diagonals bisect each other
Consecutive angles supplementary

🔷 Rectangle

Parallelogram + all angles 90°
Diagonals equal & bisect each other
Area = l × b

🔶 Rhombus

Parallelogram + all sides equal
Diagonals perpendicular & bisect vertex angles
Area = ½ d₁d₂

⬛ Square

Rectangle + Rhombus
All sides equal, all angles 90°
Diagonals equal, perpendicular, bisect vertex angles
Diagonal = side × √2

💡 Super Memory Trick — "PRRSS": Remember the hierarchy as Parallelogram → Rectangle / Rhombus → Square. Each step adds a special property. Rectangle adds "all angles 90°". Rhombus adds "all sides equal". Square adds BOTH.

🧠 MCQ Quiz — Test Your Understanding

Click on an option to check your answer. Try to solve each question before clicking!

Q1. The sum of interior angles of a quadrilateral is:
- 180°
- 270°
- 360°
- 540°
✅ Correct! The sum of interior angles of a quadrilateral is always 360° (since it can be divided into 2 triangles: 2 × 180° = 360°).
Q2. A quadrilateral with only one pair of parallel sides is called a:
- Parallelogram
- Trapezium
- Rhombus
- Kite
✅ Correct! A trapezium has exactly one pair of parallel sides (called bases).
Q3. In a parallelogram, opposite angles are:
- Supplementary
- Complementary
- Equal
- None of these
✅ Correct! In a parallelogram, opposite angles are always equal. Adjacent (consecutive) angles are supplementary.
Q4. The diagonals of a rectangle:
- Are perpendicular to each other
- Are equal and bisect each other
- Bisect the vertex angles
- Are unequal
✅ Correct! In a rectangle, diagonals are equal in length and bisect each other. They are NOT necessarily perpendicular (that is a property of a rhombus).
Q5. Which quadrilateral has diagonals that bisect each other at right angles?
- Rectangle
- Trapezium
- Rhombus
- Isosceles trapezium
✅ Correct! In a rhombus (and also in a square), the diagonals bisect each other at right angles (90°).
Q6. If the angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4, the smallest angle is:
- 36°
- 40°
- 60°
- 72°
✅ Correct! Let the angles be x, 2x, 3x, 4x. Then x + 2x + 3x + 4x = 360° ⇒ 10x = 360° ⇒ x = 36°.
Q7. Every square is a:
- Trapezium but not a parallelogram
- Rhombus but not a rectangle
- Rectangle but not a rhombus
- Rectangle, rhombus, and parallelogram
✅ Correct! A square is the most special quadrilateral — it is simultaneously a rectangle (all angles 90°), a rhombus (all sides equal), and a parallelogram (opposite sides parallel).
Q8. The diagonals of a rhombus are 24 cm and 10 cm. The side of the rhombus is:
- 12 cm
- 13 cm
- 14 cm
- 15 cm
✅ Correct! Half-diagonals = 12 cm and 5 cm. By Pythagoras: side = √(12² + 5²) = √(144 + 25) = √169 = 13 cm.
Q9. The sum of exterior angles of a convex polygon with 20 sides is:
- 3240°
- 3600°
- 720°
- 360°
✅ Correct! The sum of exterior angles of ANY convex polygon is always 360°, regardless of the number of sides.
Q10. A quadrilateral in which two pairs of adjacent sides are equal is called a:
- Parallelogram
- Trapezium
- Kite
- Rectangle
✅ Correct! A kite is defined as a quadrilateral with two pairs of equal adjacent (consecutive) sides.

✍️ Short Answer Questions (NCERT-style)

Q1. What is a polygon? Give two examples.

A polygon is a simple closed figure made up of only line segments. Examples: triangle (3 sides), quadrilateral (4 sides), pentagon (5 sides).
Q2. State the angle sum property of a quadrilateral.

The sum of all four interior angles of a quadrilateral is 360°. This can be proven by drawing a diagonal, which divides the quadrilateral into two triangles (each with angle sum 180°).
Q3. What is the difference between a convex and a concave polygon?

In a convex polygon, all interior angles are less than 180° and all diagonals lie inside the polygon. In a concave polygon, at least one interior angle is greater than 180° (reflex) and at least one diagonal lies outside.
Q4. What are the special properties of a kite?

A kite has: (i) two pairs of adjacent equal sides, (ii) diagonals are perpendicular, (iii) one diagonal bisects the other, (iv) one pair of opposite angles is equal.
Q5. Name 5 conditions that prove a quadrilateral is a parallelogram.

(i) Both pairs of opposite sides are parallel. (ii) Both pairs of opposite sides are equal. (iii) Both pairs of opposite angles are equal. (iv) Diagonals bisect each other. (v) One pair of opposite sides is both equal and parallel.
Q6. Find the number of diagonals in a hexagon.

Number of diagonals in an n-sided polygon = n(n − 3) / 2. For a hexagon (n = 6): 6(6 − 3)/2 = 6 × 3 / 2 = 9 diagonals.
Q7. Is every rectangle a square? Justify.

No. A rectangle has all angles 90° but its adjacent sides need not be equal. A square requires all sides to be equal AND all angles to be 90°. So every square is a rectangle, but not every rectangle is a square.
Q8. The angles of a quadrilateral are x°, (x + 10)°, (x + 20)°, and (x + 30)°. Find x.

x + (x + 10) + (x + 20) + (x + 30) = 360°
4x + 60 = 360 ⇒ 4x = 300 ⇒ x = 75°
The angles are: 75°, 85°, 95°, 105°.

📖 Long Answer Questions (NCERT-style)

Q1. Prove that the sum of interior angles of a quadrilateral is 360°.

Proof: Let ABCD be a quadrilateral. Draw diagonal AC.

This divides ABCD into two triangles: △ABC and △ACD.

In △ABC: ∠BAC + ∠ABC + ∠BCA = 180° …(i)
In △ACD: ∠DAC + ∠ACD + ∠ADC = 180° …(ii)

Adding (i) and (ii):
(∠BAC + ∠DAC) + ∠ABC + (∠BCA + ∠ACD) + ∠ADC = 360°

Now, ∠BAC + ∠DAC = ∠BAD = ∠A, and ∠BCA + ∠ACD = ∠BCD = ∠C.

∴ ∠A + ∠B + ∠C + ∠D = 360°. Hence proved.

Q2. Prove that the opposite angles of a parallelogram are equal.

Given: ABCD is a parallelogram (AB ∥ DC and AD ∥ BC).
To prove: ∠A = ∠C and ∠B = ∠D.

Proof: Since AB ∥ DC and AD is a transversal:
∠A + ∠D = 180° (co-interior angles) …(i)

Since AD ∥ BC and DC is a transversal:
∠D + ∠C = 180° (co-interior angles) …(ii)

From (i) and (ii):
∠A + ∠D = ∠D + ∠C
∴ ∠A = ∠C

Similarly, we can prove ∠B = ∠D. Hence proved.

Q3. Prove that the diagonals of a parallelogram bisect each other.

Given: ABCD is a parallelogram. Diagonals AC and BD intersect at O.
To prove: AO = OC and BO = OD.

Proof: In △AOB and △COD:
(i) AB = CD (opposite sides of parallelogram)
(ii) ∠OAB = ∠OCD (alternate interior angles, AB ∥ DC, transversal AC)
(iii) ∠OBA = ∠ODC (alternate interior angles, AB ∥ DC, transversal BD)

By ASA congruence: △AOB ≅ △COD

By CPCT: AO = OC and BO = OD. Hence proved.

Q4. ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C.

Given: ABCD is a rhombus, so AB = BC = CD = DA.
To prove: ∠DAC = ∠BAC and ∠DCA = ∠BCA.

Proof: In △ABC:
AB = BC (sides of rhombus), so △ABC is isosceles.
∴ ∠BAC = ∠BCA …(i)

In △ADC:
AD = DC (sides of rhombus), so △ADC is isosceles.
∴ ∠DAC = ∠DCA …(ii)

Since ABCD is a parallelogram:
∠BAC = ∠DCA (alternate interior angles) …(iii)

From (i), (ii) and (iii):
∠BAC = ∠DAC (so AC bisects ∠A)
∠BCA = ∠DCA (so AC bisects ∠C). Hence proved.

Q5. Explain the hierarchy of quadrilaterals with suitable examples.

Quadrilaterals form a hierarchy where each type is a special case of a more general type:

1. Quadrilateral (most general): Any closed figure with 4 sides.
2. Trapezium: A quadrilateral with at least one pair of parallel sides.
3. Parallelogram: A quadrilateral with both pairs of opposite sides parallel (special case of trapezium).
4. Rectangle: A parallelogram with all angles = 90°.
5. Rhombus: A parallelogram with all sides equal.
6. Square: A parallelogram that is BOTH a rectangle and a rhombus (all sides equal AND all angles 90°).

Key insight: Every property of a general type is inherited by its special types. For example, since diagonals of a parallelogram bisect each other, this is also true for rectangles, rhombi, and squares.

🌟 Fun Facts & Did You Know?

🏠 Architecture Everywhere

Almost every building in the world is based on rectangular quadrilaterals. Windows, doors, walls, bricks, screens — all rectangles! Architects use quadrilateral properties daily.

💨 Kites That Fly

The mathematical kite is named after the flying kite! The diamond shape of a traditional kite is exactly a mathematical kite (two pairs of adjacent equal sides). The perpendicular diagonals help it fly stably.

🔮 Honeycomb Magic

Bees build their honeycombs using regular hexagons (6-sided polygons), not quadrilaterals. Hexagons tessellate perfectly and use the least amount of wax per unit area — nature's optimal design!

🇮🇳 Ancient Indian Geometry

The Sulbasutras (800 BCE) contain rules for constructing precise rectangles and squares for fire altars. Baudhayana described how to convert a rectangle into a square of equal area — over 2500 years ago!

📈 Baseball Diamond

A baseball "diamond" is actually a square with 90-foot sides! The bases form a perfect square, and the diagonal from home plate to second base is 90√2 ≈ 127.3 feet.

💻 Computer Screens

Every pixel on your screen is a tiny rectangle. A Full HD screen has 1920 × 1080 = over 2 million tiny rectangular pixels, each displaying a colour to create the images you see!

⬜ Quadrilaterals

✅ Convex Polygon

❌ Concave Polygon

⭐ Regular Polygon

🔸 Irregular Polygon

✏️ Example 1: Finding a Missing Angle

✏️ Example 2: Angles in Ratio

🔹 Isosceles Trapezium

📏 Area of Trapezium

💫 Properties of a Kite

1. Opposite Sides are Equal

2. Opposite Angles are Equal

3. Diagonals Bisect Each Other

4. Consecutive Angles are Supplementary

✏️ Example 3: Using Parallelogram Properties

✏️ Example 4: Diagonals Bisecting Each Other

💫 Properties of a Rectangle

💫 Properties of a Rhombus

💫 Properties of a Square

✏️ Example 5: Rectangle Diagonal

✏️ Example 6: Rhombus Diagonals

🔄 Reading the Hierarchy

💡 True or False?

✏️ Example 7: Finding Exterior Angles

✏️ Example 8: Angles of a Parallelogram

✏️ Example 9: Square Diagonal Calculation

✏️ Example 10: Identifying the Quadrilateral

✏️ Example 11: Interior Angles of a Regular Polygon

⬟ Trapezium

🔸 Kite

▭ Parallelogram

🔷 Rectangle

🔶 Rhombus

⬛ Square

🏠 Architecture Everywhere

💨 Kites That Fly

🔮 Honeycomb Magic

🇮🇳 Ancient Indian Geometry

📈 Baseball Diamond

💻 Computer Screens

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