๐ Quadrilaterals
Ganita Prakash โข Grade 8 โข Chapter 4
โ๏ธ Handwritten Notes
๐จ Colourful
๐ฑ Mobile Friendly
@edugrown
๐ Table of Contents
๐ท
โ Back to top
What is a Quadrilateral?
The word quadrilateral comes from Latin โ quadri meaning four and latus meaning sides. So a quadrilateral is simply any four-sided closed figure!
๐ Definition
A quadrilateral is a polygon with exactly 4 sides, 4 angles, and 4 vertices.
The angles are measured between its adjacent sides.
@edugrown
4๏ธโฃ
4 Sides
Must have exactly four straight sides
๐
4 Angles
One angle at each vertex between two sides
๐ต
4 Vertices
Four corner points where sides meet
๐
Closed Figure
All sides connected, no gaps
๐ฅ
๐ฅ Rectangle
๐ Definition (Simple!)
A rectangle is a quadrilateral in which all the angles are 90ยฐ.
@edugrown
Properties of a Rectangle โ remember these! ๐
| # | Property | Key Word |
|---|---|---|
| 1 | All angles = 90ยฐ each | Right angles โ |
| 2 | Opposite sides are equal | AB = CD, BC = AD โ |
| 3 | Opposite sides are parallel | AB โฅ CD, BC โฅ AD โ |
| 4 | Diagonals are equal length & bisect each other | AC = BD, OA = OC โ |
๐ก Fun Fact โ Carpenter's Trick!
To make a rectangular frame, join two equal wooden strips at their midpoints.
The endpoints will always form a rectangle โ no matter the angle between the strips! ๐ช
๐ Two Equivalent Definitions of Rectangle
Def 1: A quadrilateral where all angles = 90ยฐ
Def 2: A quadrilateral whose diagonals are equal and bisect each other
โ Back to top
Def 2: A quadrilateral whose diagonals are equal and bisect each other
๐จ
๐จ Square โ A Special Rectangle!
๐ Definition
A square is a quadrilateral where all angles = 90ยฐ AND all sides are equal.
Every square is a rectangle ๐จโ๐ฅ
But NOT every rectangle is a square!
But NOT every rectangle is a square!
@edugrown
| # | Property | Unique to Square? |
|---|---|---|
| 1 | All 4 sides equal | โญ Yes! |
| 2 | Opposite sides parallel | No (rect has it too) |
| 3 | All angles = 90ยฐ | No (rect has it too) |
| 4 | Diagonals equal & bisect at 90ยฐ | โญ Yes! (rect: equal, bisect but NOT at 90ยฐ) |
| 5 | Diagonals bisect the angles (each = 45ยฐ) | โญ Yes! |
โ๏ธ Key Difference: Rectangle vs Square
Rectangle โ Opposite sides equal | Square โ ALL sides equal
Rectangle diagonal โฅ? โ Not necessarily | Square diagonal โฅ? โ Always 90ยฐ!
โ Back to top
Rectangle diagonal โฅ? โ Not necessarily | Square diagonal โฅ? โ Always 90ยฐ!
๐
๐ Angles in a Quadrilateral
๐ Sum of ALL angles in ANY quadrilateral = 360ยฐ
Why? The Proof Idea ๐ค
โ
Take any quadrilateral ABCD. Draw a diagonal โ this cuts it into 2 triangles.
โก
Sum of angles in Triangle 1 = 180ยฐ
โข
Sum of angles in Triangle 2 = 180ยฐ
โฃ
Total = 180ยฐ + 180ยฐ = 360ยฐ โ
@edugrown
๐ก Useful Trick!
If you know 3 angles of a quadrilateral โ 4th angle = 360ยฐ โ (sum of other 3)
Example: angles are 90ยฐ, 90ยฐ, 90ยฐ โ 4th = 360ยฐ โ 270ยฐ = 90ยฐ (that's a rectangle!)
โ Back to top
Example: angles are 90ยฐ, 90ยฐ, 90ยฐ โ 4th = 360ยฐ โ 270ยฐ = 90ยฐ (that's a rectangle!)
๐ต
๐ต Parallelogram
๐ Definition
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
@edugrown
| # | Property | Remember As |
|---|---|---|
| 1 | Opposite sides are equal | AB = DC, AD = BC |
| 2 | Opposite sides are parallel | AB โฅ DC, AD โฅ BC |
| 3 | Adjacent angles add up to 180ยฐ | โ A + โ B = 180ยฐ |
| 4 | Opposite angles are equal | โ A = โ C, โ B = โ D |
| 5 | Diagonals bisect each other | OA = OC, OB = OD |
โ ๏ธ What Parallelograms DON'T have
โ Diagonals are NOT necessarily equal (unlike rectangle)
โ Diagonals are NOT necessarily perpendicular (unlike rhombus)
โ Diagonals are NOT necessarily perpendicular (unlike rhombus)
๐ต Relationship Diagram:
@edugrown
โ Back to top
๐
๐ Rhombus
๐ Definition
A rhombus is a quadrilateral in which all four sides are of equal length.
(Think: a "pushed-over" square!)
@edugrown
| # | Property | Key Point |
|---|---|---|
| 1 | All 4 sides are equal | โญ Defining property |
| 2 | Opposite sides parallel | It's a parallelogram! |
| 3 | Adjacent angles sum = 180ยฐ, opposite = equal | Like all parallelograms |
| 4 | Diagonals bisect each other | Like all parallelograms |
| 5 | Diagonals bisect the angles | โญ Unique to rhombus! |
| 6 | Diagonals are perpendicular (90ยฐ) | โญ Unique to rhombus! |
๐ Rhombus = Pushed Square
A square has all sides equal AND all angles 90ยฐ.
A rhombus has all sides equal but angles are NOT necessarily 90ยฐ.
โด Every square is a rhombus, but every rhombus is NOT a square!
โ Back to top
A rhombus has all sides equal but angles are NOT necessarily 90ยฐ.
โด Every square is a rhombus, but every rhombus is NOT a square!
๐ช
๐ช Kite
๐ Definition
A kite is a quadrilateral with two pairs of adjacent sides equal.
i.e., AB = BC and CD = DA (the equal pairs share a vertex)
@edugrown
| # | Property of Kite |
|---|---|
| 1 | Diagonal BD bisects angles โ ABC and โ ADC |
| 2 | Diagonal BD is perpendicular to diagonal AC (BD โฅ AC) |
| 3 | Diagonal BD bisects AC โ AO = OC |
| 4 | One pair of opposite angles are equal (โ A = โ C) |
๐ช Real Life Example!
The actual flying kite ๐ช is shaped like a geometric kite โ
two pairs of adjacent sides equal with a longer diagonal as the main spine!
โ Back to top
๐ฉ
๐ฉ Trapezium
๐ Definition
A trapezium is a quadrilateral with at least one pair of parallel opposite sides.
The parallel sides are called bases.
@edugrown
| Key Property of Trapezium | |
|---|---|
| 1 | Co-interior angles (same side) sum = 180ยฐ i.e., โ S + โ P = 180ยฐ and โ R + โ Q = 180ยฐ |
๐ Isosceles Trapezium
When the non-parallel sides are equal in length, the trapezium is called an
isosceles trapezium.
Extra property: Base angles are equal โ โ U = โ V
๐ง Is a Parallelogram a Trapezium?
YES! โ
A parallelogram has TWO pairs of parallel sides, so it satisfies
"at least one pair" โ it IS a trapezium (but a special one).
โ Back to top
๐
๐ Quick Revision Summary
๐บ๏ธ All Quadrilaterals at a Glance!
| Shape | All sides = | Opp. sides โฅ | All โ = 90ยฐ | Diagonals = | Diag. โฅ | Diag. bisect |
|---|---|---|---|---|---|---|
| ๐ฅ Rectangle | โ | โ | โ | โ | โ | โ |
| ๐จ Square | โ | โ | โ | โ | โ | โ |
| ๐ต Parallelogram | โ | โ | โ | โ | โ | โ |
| ๐ Rhombus | โ | โ | โ | โ | โ | โ |
| ๐ช Kite | โ | โ | โ | โ | โ | one |
| ๐ฉ Trapezium | โ | one pair | โ | โ | โ | โ |
๐๏ธ The Big Family Tree of Quadrilaterals:
@edugrown
๐ฎ Key Formulas to Remember:
โ A + โ B + โ C + โ D = 360ยฐ
Rect: diagโ = diagโ
Para: adj โ s = 180ยฐ
Rhombus: diag โฅ diag
Trap: co-int โ s = 180ยฐ
๐ Golden Rules to Never Forget!
1๏ธโฃ Sum of angles of any quadrilateral = 360ยฐ
2๏ธโฃ Rectangle diagonals are equal (not โฅ) | Square diagonals are equal AND โฅ
3๏ธโฃ Rhombus diagonals are โฅ but NOT equal (unless it's a square!)
4๏ธโฃ Every Square is a Rectangle, Rhombus, AND Parallelogram โ
5๏ธโฃ Every Rectangle and Rhombus is a Parallelogram โ
โ Back to top
2๏ธโฃ Rectangle diagonals are equal (not โฅ) | Square diagonals are equal AND โฅ
3๏ธโฃ Rhombus diagonals are โฅ but NOT equal (unless it's a square!)
4๏ธโฃ Every Square is a Rectangle, Rhombus, AND Parallelogram โ
5๏ธโฃ Every Rectangle and Rhombus is a Parallelogram โ
โ๏ธ Notes by @edugrown โข Ganita Prakash Grade 8