Revision Notes on Quadrilaterals

Polygon: – It is made by Greek words Polus + Gonia, in which Polus means many and Gonia means Corner or angle. Thus, a plane close figure bounded by a finite straight-line segment in loop is called polygon.

Ex- Triangle, Quadrilateral, Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon etc.

Regular Polygon: – A polygon whose all sides and angles are equal is called regular polygon.

Ex – Equilateral triangle and square

Irregular polygon: –  A polygon whose all sides and angles are not equal is called regular polygon.

Ex – A rectangle has equal angles but not equal sides, and hence an irregular polygon.

Quadrilateral: – It is made by Latin words; Quardi + Latus. Quadri – means four and Latus means side.

A polygon figure bounded by four sides is called quadrilateral.

* A quadrilateral has 4 sides, 4 vertices and 4 angles.

* The sum of the interior angles of a quadrilateral is equal to 360 degrees.

* It is two dimensional (2-d) shapes.

* The quadrilateral is basically of 6 types such as:

Angle sum property of a polygon:

Note-   Sum of interior Angles of a polygon = (n – 2) × 180⁰

Where ‘n’ is the number of sides

Example:

1. Angle sum of a triangle = (3 – 2) × 180⁰ = 1 × 180⁰ = 180

2. Angle sum of a quadrilateral = (4 – 2) × 180⁰ = 2 × 180⁰ = 360⁰

3. Angle sum of a pentagon = (5 – 2) × 180⁰ = 3 × 180⁰ = 540⁰

Types of quadrilateral (6)
Rectangle आयत	Square वर्ग	Parallelogram समान्तर चतुर्भुज	Rhombus सम चतुर्भुज	Trapezium समलम्ब चतुर्भुज	Kite पतंग

Rectangles: –A quadrilateral whose opposite sides are equal and each angle is 90°  is called rectangle.

Properties of a Rectangle

* Opposite sides are parallel and congruent. (MN = OP and NO = PM)

* A rectangle is a special type of parallelogram whose angles 90°

* A rectangle has two lines of symmetry.

* The order of rotational symmetry is 2.

* The diagonals are congruent or equal and bisect each other.

* Diagonals bisect the angles.                        

* Each diagonal divides the rectangle into two congruent  right-angled triangles

* The adjacent angles are supplementary. That is

0^°

# Length of the diagonal of a rectangle = √(L² + B²)

# Area = L * B

# Perimeter = 2(L+B)

Squares: – A quadrilateral whose all sides are equal and each angle is 90° is called Square.

Properties of a square

* All sides and angles are congruent.

* Opposite sides are parallel to each other.

* A square is parallelogram whose all angles and sides are equal.

* A square has four lines of symmetry.

* The order of rotational symmetry is 4.

* The diagonals bisect each other at 90⁰.

* The diagonals are equal and congruent.

* Diagonals bisect the angles.

* Each diagonal divides the square into two congruent  isosceles right-angled triangles

* The adjacent angles are supplementary. That is

0°

* A parallelogram becomes a square when the diagonals are equal and right bisectors of each other.

Important formulas of Square

Length of the diagonal

# D = √2×side = √2× L

# Area = (side)² = L².

# Perimeter = 4×side = 4L

Parallelogram: – A quadrilateral whose opposite sides are parallel and equal is called parallelogram.

 

 

Properties of a parallelogram

* Opposite sides are parallel and congruent. (MN=OP and NO=PM)

* Opposite angles are congruent. ()

* Adjacent angles are supplementary.

 180⁰)

* Diagonals bisect each other.

* Diagonal divides the parallelogram into two congruent triangles.

* If one of the angles of a parallelogram is a right angle then all other angles are right and it becomes a rectangle.

 

Important formulas of parallelograms

# Area = Base ×Height = L × H

# Perimeter = 2(L+ B)

Rhombus: –  A quadrilateral whose all sides is equal and angles not equal to 90° is called rhombus.

 

 

 

 

Properties of a Rhombus

* All sides are congruent.

* Opposite angles are congruent.

* The diagonals are perpendicular to and bisect each other.

* Adjacent angles are supplementary (For ex- ∠A + ∠B = 180°).

* A rhombus is a parallelogram whose diagonals are perpendicular to each other.

 

Important formulas for a Rhombus

# Area = (d₁× d₂)

= (a× b)

# Perimeter = 4×side = 4L

Trapezium: – A quadrilateral whose one pair of opposite sides is parallel is called trapezium. 

  

Properties of a Trapezium

* The bases of the trapezium are parallel to each other (MN ⫽ OP).

* No sides, angles and diagonals are congruent.

Important Formulas for a Trapezium

# Area =  (Sum of parallel sides) × Height

      =  (a + b) × h

# Perimeter = Sum of all sides = L + L₁ + L₂ + L₃

Remark: A square, Rectangle and Rhombus are also a parallelogram.

Summary of properties

S.No.	Property	Paralle logram	Rec tangle	Rhom bus	Square
1	All sides are congruent	✕	✕	✓	✓
2	Opposite sides are parallel and congruent	✓	✓	✓	✓
3	All angles are congruent	✕	✓	✕	✓
4	Opposite angles are congruent	✓	✓	✓	✓
5	Diagonals are congruent	✕	✓	✕	✓
6	Diagonals are perpendicular	✕	✕	✓	✓
7	Diagonals bisect each other	✓	✓	✓	✓
8	Adjacent angles are supplementary	✓	✓	✓	✓

Angle Sum Property of a Quadrilateral

The sum of the four angles of a quadrilateral is 360°

If we draw a diagonal in the quadrilateral, it divides it into two triangles.  

And we know the angle sum property of a triangle i.e. the sum of all the three angles of a triangle is 180°.

The sum of angles of ∆ADC = 180°.

The sum of angles of ∆ABC = 180°.

By adding both we get ∠A + ∠B + ∠C + ∠D = 360°

Hence, the sum of the four angles of a quadrilateral is 360°.

Example

Find ∠A and ∠D, if BC∥ AD and ∠B = 52° and ∠C = 60° in the quadrilateral ABCD.

Solution:

Given BC ∥ AD, so ∠A and ∠B are consecutive interior angles.

So ∠A + ∠B = 180° (Sum of consecutive interior angles is 180°).

∠B = 52°

∠A = 180°- 52° = 128°

∠A + ∠B + ∠C + ∠D = 360° (Sum of the four angles of a quadrilateral is 360°).

∠C = 60°

128° + 52° + 60° + ∠D = 360°

∠D = 120°

∴ ∠A = 128° and ∠D = 120 °.

Remark: A square, Rectangle and Rhombus are also a parallelogram.

Properties of a Parallelogram

Theorem 1: When we divide a parallelogram into two parts diagonally then it divides it into two congruent triangles.

∆ABD ≅ ∆CDB

Theorem 2: In a parallelogram, opposite sides will always be equal.

Theorem 3: A quadrilateral will be a parallelogram if each pair of its opposite sides will be equal.

Here, AD = BC and AB = DC

Then ABCD is a parallelogram.

Theorem 4: In a parallelogram, opposite angles are equal.

In ABCD, ∠A = ∠C and ∠B = ∠D

Theorem 5: In a quadrilateral, if each pair of opposite angles is equal, then it is said to be a parallelogram. This is the reverse of Theorem 4.

Theorem 6: The diagonals of a parallelogram bisect each other.

Here, AC and BD are the diagonals of the parallelogram ABCD.

So the bisect each other at the centre.

DE = EB and AE = EC

Theorem 7: When the diagonals of the given quadrilateral bisect each other, then it is a parallelogram.

This is the reverse of the theorem 6.

The Mid-point Theorem

1. If a line segment joins the midpoints of the two sides of the triangle then it will be parallel to the third side of the triangle.

If AB = BC and CD = DE then BD ∥ AE.

2. If a line starts from the midpoint of one line and that line is parallel to the third line then it will intersect the midpoint of the third line. 

If D is the midpoint of AB and DE∥ BC then E is the midpoint of AC.

Example

Prove that C is the midpoint of BF if ABFE is a trapezium and AB ∥ EF.D is the midpoint of AE and EF∥ DC.

Solution:

Let BE cut DC at a point G.

Now in ∆AEB, D is the midpoint of AE and DG ∥ AB.

By midpoint theorem, G is the midpoint of EB.

Again in ∆BEF, G is the midpoint of BE and GC∥ EF.

So, by midpoint theorem C is the midpoint of BF.

Hence proved.