COMPLEX NUMBERS , DEMOIVRE'S THEOREM & EULER'S FORM

       COMPLEX    NUMBERS

Any  number of the form  a+bi is known as  complex  number where a & b are real numbers & i=  √-1 , a is called real part & b is called imaginary part.

           A complex number is generally denoted by z. If a is 0 then 

      z is purely imaginary & if b is 0 then z is purely real.

NOTE:

  1) i  = √-1  

  2) i² = -1

  3) i³ = i² i =-1.i =-i

  4) i⁴ =i³.i =-i.i =-i² =-(-1) =1

  5) i⁵ = i⁴ .i =1.i =i

  6) i⁶ =i⁵.i = i.i = i² = -1

 7) i⁷ =i⁶ .i =-1. i =-i

 8 ) i⁸ =i⁷.i =-i.i =- i² = -(-1)=1

 9) i⁹ =  i⁸.i = 1.i =i

10)i¹º =i⁹.i =i.i =i² = -1

  &  so on .

Read blog on Line & it's Equation

                ALJEBRA OF COMPLEX NUMBERS

  1)  ADDITION : When two complex numbers 

   are added their sum is also a complex number.

   eg.

  if z₁=a₁+b₁i  & z₂=a₂+b₂i then z₁+ z₂ is given by

       z₁+ z₂ = (a₁+b₁i ) + (a₂+b₂i ) 

                  = (a₁+a₂) + (b₁+b₂)i

2)   SUBTRACTION : When two complex 

   numbers are subtracted their  difference is 

    also a complex number .

   eg.

    if z₁=a₁+b₁i  & z₂=a₂+b₂i then z₁-z₂ is given by

       z₁- z₂ = (a₁+b₁i ) - (a₂+b₂i ) 

                 = (a₁-a₂) + (b₁-b₂)i

2) MULTIPLICATION : When two complex numbers 

    are multiplied their multilication is also a complex number .

   eg.

 if z₁=a₁+b₁i  & z₂=a₂+b₂i then z₁. z₂ is given by

       z₁. z₂ = (a₁+b₁i ) . (a₂+b₂i ) 

                 = (a₁a₂-b₁b₂) + ( a₁b₂+b₂a₁)i

                 EQUALITY OF COMPLEX NUMBERS 

Two complex numbers are said to be equal if

1)Their real parts are equal

2)Their imaginary parts are also equal.

 eg. If z₁=a+bi  & z₂=2+3i 

             z₁=z₂ if a+bi  =2+3i i.e a=2 & b=3

         CONJUGATE OF A COMPLEX NUMBER

Two complex numbers which are differ only in the sign of their imaginary parts are said to be conjugates of each other.

If z is complex number then it's conjugate is denoted by z .ie. z bar.

if 2+3i is given complex number the it's conjugate is given by 2-3i.

          

                 z                              conjugate  z

          2+3i                                             2-3i.

               2-3i                               2+3i

             -2+3i                              -2-3i

             -2-3i                                -2+3i

              -3i                                     +3i

             +3i                                      -3i

NOTE : 1)SUM &product of two complex conjugates are real.

               2)difference of two complex conjugates are imaginary.

Read the blog on Pascal Triangle & FIBONACCI NUMBERS

                       MODULUS & ARGUMENTS

If z=a+bi is a complex number then modulus of z is given by r=√a²+b² .

Argument or amplitude of z is given by  𝛩＝tan⊣(y/x) 

i.e tan inverse of y/x

           POLAR FORM OF A COMPLEX NUMBER

IF z=a+bi is a complex number then it's 

   polar form is given by z=r(cos𝛩+ i sin 𝛩)

EXPONENTIAL FORM OF A COMPLEX NUMBER

1)   eⁱӨ = e raise to iӨ =cos Ө+i sinӨ

2) e⁻ⁱө  =e raise to -iө  =cos Ө-i sinӨ

3) eⁱӨ +e⁻ⁱө =2cosө

4) eⁱӨ - e⁻ⁱө =2isinө

  DE MOIVRE'S THEOREM

For any rational number n the value or one of the value of 

(cos Ө+i sinӨ)ⁿ= (cos nӨ+i sin nӨ)

e.g. 1) if (cos Ө+i sinӨ)⁴ = (cos 4Ө+i sin4Ө)

       2) if (cos Ө-i sinӨ)⁴ = (cos 4Ө-i sin4Ө)

      3)  if (cos Ө+i sinӨ)⁻⁴ = (cos 4Ө-i sin4Ө)

      4)  if (cos Ө-i sinӨ)⁻⁴ = (cos 4Ө+i sin4Ө)

EULER'S FORM 

1) if z₁=a₁+b₁i  & z₂=a₂+b₂i then r=√a²+b² .

&     z₁

__________=  ( r₁/r₂) eⁱ(ө₁-ө₂)      

     z₂                

2) z₁ . z₂  = (r₁.r₂) eⁱ(ө₁+ө₂)  

i.e    (r₁.r₂). e raise to i.(ө₁+ө₂)  

 This is known as Euler's form

Line & It's Equations

                                    

          Line :- 

    A line is one dimensional figure which has                 length but no breath.

         

        

           FIG. RayOB & RayOA joined together.

                             

It is also defined as a combination of two rays 

joined together at a common vertex but in 

opposite direction.

Both the side it can be extended infinitely.

There are are different  types of lines like straight line, vertical line, thick line , thin line, horizontal line , zigzag line, curved line , spiral line ,diagonal line , skew line ,slant/oblique line etc.

Read the Blog on Pascal triangle

                       EQUATION OF LINE

There are two form of equation of  line ...

  1) ax+by=0  (Line passes through origin)

  2) ax+by+c=0 (Line does not pass through                                                                            origin)

TYPES OF EQUATION OF LINE : -

1)                 TWO POINT FORM : 

When Two points A(x₁,y₁) & B(x₂, y₂) 

of a line is known then the equation of line 

is given by ...

       y-y₁       y₁- y₂

       ------ =  --------

       x-x₁       x₁-x₂

2)             Slope Point form :

    When a point A(x₁,y₁) & slope (m) 

    of a line is known  then the equation 

    of a line is given by ..

        y-y₁=m(x-x₁)

3)   Double Intercept form  :

      When two intercept i.e 

       x-intercept= a

   & y-intercept= b 

       of a line is known then the equation 

       of a line is given by 

        x/a +y/b  =1

4)     SLOPE - INTERCEPT FORM :

    When slope (m) & Y-intercept-c 

       of a line is known then the   equation 

       of a line is given by 

                y=mx+c

EQUATIONS OF AXES

     1) Equation of x-axis is y=0

       2) Equation of y-axis is x=0

Note:

 

   1)If a line is perpendicular to x-axis then 

       it's equation is given by  x= ± a

2) If a line is perpendicular to y-axis then  

     it's equation is given by  y= ± b

                      

Read the blog on Angle & it's Types

TO FIND SLOPE  :

To find slope of a line there are three formulae

1) If Ө is the angle made by the line with the + direction of x-axis then the slope of the line is given by ...

                slope = TanӨ

 

2) When A(x₁,y₁) & B(x₂, y₂) two points of two line s are known then slope is given by

                        y₂-y₁

    slope =    -----------

                        x₂-x₁

3)When ax+by+c=0 is the equation of a given line then the slope is given by

      slope = -a / b

TO FIND SLOPE OF AXES

1) Slope of x-axis  = 0

2) Slope of y-axis  = not defined =∞

* * IF m₁ & m₂ slopes of two lines are then

 

1) If two lines are parallel then 

                  m₁ = m₂

2) If two lines are perpendicular then  

                 m₁.m₂= -1

3) If two lines are intersecting then angle between them is given by

  TanӨ= | m₁ - m₂/ 1 - m₁. m₂|

  

* If a point is on x- axis then it's 

    coordinate is given by 

                (± a , 0)  

* If a point is on y-axis then it's 

    coordinate is given by 

                ( 0, ± b)

*If u=a₁x+b₁y+c₁=0 & 

      v=a₂x+b₂y+c₂=0 

  are two lines then equation of a 

  line passing through the 

  intersection of these two line is given by

                            u+kv=0

*Joint equation of  two line is given by

                        u.v=0

* Joint Equation / Homogenous equation 

    of pair of line passing through origin is 

    given by 

                     ax²+2hxy+by²=0

* General equation of pair of line is given by

  ax²+2hxy+by²+2gx+2fy+c=0

* If the two line are perpendicular then

                a+b=0

*If two line are parallel then 

                h²-ab=0  

* If the two lines are intersecting then

the angle between them is given by

TanӨ= | 2√ h²-ab/a+b |    

**Shortest Distance between a Point  P(x₁,y₁)  

    & a point M on the line ax+by+c=0 is 

    given by 

      PM = | ax₁+by₁+c₁/√ a²+b²|        

   Read the Blog on Story "MASOOM DIL"   

***

In 3-Dimension  :

If a line makes an angle 𝝰 ,𝞫 &𝜸 

with the + direction of x-axis,y-axis

 &z-axis respectively.

Then cos 𝝰 ,cos 𝞫 , & cos𝜸 are 

known as Direction cosine of the line.

It is also known as dc's.

 also cos𝝰=l  ,cos𝞫=m & cos𝜸=n

**cos²𝝰 +cos²𝞫 +cos²𝜸 = 1

i.e.   l²+m²+n² =1

* If  a/cos𝝰=b/cos𝞫=c/cos𝜸 then a, b, c 

are known as direction ratios. 

It's is denoted by dr's.

*If A(x₁,y₁,z₁) & B(x₂, y₂,z₂) are two point

then it's direction ratio is given by 

a=x₂-x₁ , b=y₂-y₁,c =z₂-z₁

*The Vector equation of the line passing through A(ā)& parallel to Б is given by 

  r=ā+⋋b

* Cartesian equations of the line passing through A(x₁,y₁,z₁) & having direction 

ratio a, b & c are 

          x-x₁/a   = y-y₁/b    =z-z₁/c

PASCAL TRIANGLE & FIBONACCI NUMBERS

 

                                PASCAL  TRIANGLE

                                                   1  

                                           1              1

                                   1              2             1

                            1              3              3                1

                    1             4               6             4                   1

          1               5             10            10                5                  1

1                6             15             20             15                 6                     1

  PASCAL TRIANGLE GIVES THE BINOMIAL 

COEFFICIENT IN THE EXPANSION OF BINOMIAL THEOREM.

READ THE BLOG ON "INDIAN RAILWAY "

FOR Example 

 1)  In the expansion of  (a+b)¹ = a + b = 1a+ 1b

   The coefficient of a & b is  1  &  1 .

  The Coefficient of 1st term & 2 nd term is 1 & 1 i.e.  1       1

This is shown in the second row of Pascal's Triangle .

2) In the  expansion of (a+b)² = a² +2ab+b² =1a² +2ab+1b² 

  The Coefficient of 1st term is 1 , Coefficient of 2nd term is 2 & Coefficient of 3rd term is 1 i.e.  1      2      1

This is shown in the 3rd row of Pascal's Triangle.

3)In the expansion of  (a+b)³ = a³ +3a²b+3ab²+b³

                                                  = 1a³ +3a²b+3ab²+1b³

The Coefficient of 1st term is 1, coefficient of 2nd term is 3, coefficient of 3rd term is 3 & Coefficient of last term is 1. i.e.

The Coefficient are   1    3    3    1

This is shown in the 4th row of Pascal's Triangle.

4) Same way expansion of of (a+b)⁴

  (a+b)⁴= a⁴ +a³b¹+a²b²+a¹b³+b⁴=

             = 1a⁴+4a³b¹+6a²b²+4a¹b³+1b⁴

The Coefficients are    1    4     6    4    1 

This is shown in the 5th row of Pascal's Triangle . 

This way Next row in Pascal triangle shows 

the Coefficients of (a+b)⁵ i.e. 1    5   10    10    5    1  & so on .

NOTE : 1) IN THE THIRD ROW OF PASCAL TRIANGLE 2

IS OBTAIN BY  ADDING NUMBER IN PREVIOS ROW i.e. 1+1

2)Fourth row is obtain from 3rd row. i.e. 3 is obtain from 1+2 & next 3 is obtain from 2+1.

3)simillarly fifth row is obtain from 4th row i.e. 4 is obtain from1+3 , 6is obtain from 3+3 , next 4is obtain from 3+1  & so on.

THE FIBONACCI SEQUENCE IS ALSO OBTAIN FROM

PASCAL'S TRIANGLE.

The Sequence is  1      1      2       3        5        8        13       21        34      55        89        144       233      377       610       987        1497  .......  so on

READ THE BLOG ON "BINOMIAL THEOREM"

             

In Fibonacci Sequence  1st & 2nd term is 1 i.e.   sequence is  1     1

3rd term is  1+1 =2  i.e now the sequence is   1    1    2

4th term is  1+2=3 i.e now the sequence is      1    1    2     3 

5th term is  2+3=5 i.e now the sequence is  1   1    2    3    5 

6th term is 3+5=8 i.e. now the sequence is 1    1    2     3      5    8

7thet erm is 5+8=13 i.e. now the sequence is 1  1   2   3    5     8    13

8the term is 8+13=21  , 9th term 13+21=34  ,  10 th term  21+34=55 &  so on .

BASIC ALJEBRA FORMUAE

 

BASIC  ALJEBRA FORMUAE

1) (a+b)¹  =a +b 

2) (a+b)² =a² +2ab +b²

3) (a-b)² =a² -2ab +b²

4) (a+b)³ =a³ +3a²b+3ab²+b³

5)(a-b)³ =a³-3a²b+3ab²-b³

6)(a+b)⁴=a⁴+4a³b¹ +6a²b² +4a¹b³ +b⁴ 

7)(a+b)⁵=a⁵+5a⁴b¹+10a³b²+10a²b³+5a¹b⁴+b⁵   

8) (a²-b²) =(a+b) (a-b)

9) (a+b)(a-b)  =a²-b²

10) (a³+b³)  =(a+b) (a²-ab+b² )

11) (a³-b³)   = (a-b) (a²+ab+b² )

12)(a+b+c)² =a²+b²+c²+2ab+2bc+2ca

13) (a+1/a)² = a²+2+1/a²

  .i.e a²+1/a² = (a+1/a)²-2

14) (a-1/a)² =a²-2+1/a²

i.e. a²+1/a²  = (a-1/a)² +2

15) (a+1/a)³ = a³+3a+3.1/a+1/a³

16) (a+b)²-2ab =a²+b²

17) (a+b)² - 4ab = (a-b)² 

18) (a-b)² +4ab= (a+b)²

19) (x +a) (x+b) =x (x+b) +a (x+b)

                            = x² +x.b +a.x+a.b

20) (x +a) (x-b)  = x (x-b) +a (x-b)

                            =x² - x.b + a.x - a.b

21) (x-a) (x+b)   = x (x+b) - a(x+b)

                           =x² +x.b - a.x - a.b

22) (x-a) (x-b)   = x (x-b) -a (x-b)

                          = x² - x.b - a.x + a.b

READ THE BLOG ON ANGLE & IT'S TYPE

      Laws of Indices  :

1) (a.b)ⁿ=aⁿ.bⁿ  (Power is common base is different)

2) aⁿ. aⁱ= aⁿ⁺ⁱ   (Base is same power is different)

3) 1/aⁱ  = a⁻ⁱ

3) aⁿ/aⁱ  =aⁿ➗ aⁱ = aª⁻ⁱ

4) (aⁿ)ⁱ = aⁿⁱ

5)( (aⁿ)ⁱ )ˡ = (a)ⁿⁱˡ

6) aº  = 1

7) a¹ = a 

8) a⁻ⁿ = 1/ aⁿ

9) (a)¹/ⁿ = n√a  e.g. ∜a = (a)¹/4

10) (a)ⁿ/ˡ  =1 / (a)ˡ/n

11) (a/b)ⁿ  = aⁿ/bⁿ

READ THE BLOG ON " INDIAN RAILWAY "

Note :

1)  (-) × (-)  = +

2) (-) × (+)  = -

3) (+) × (-)  = -

4) (+) × (+) = +

                          Binomial Theorem

`    Factorial Notation  :For a natural number n , the factorial 

   of n is written as n! or ∟n & read as " n factorial " is the 

   productof n natural numbers from 1 to n .

            i.e. n! =n (n-1)(n-2)(n-3)(n-4) ........4 . 3 . 2 . 1

                       =n (n-1)!

                       =n.(n-1).(n-2)!

          e.g.

               0!=1

               1!=1 

               2!=2✕1=2.1=2

               3!=3✕2✕1=3.2.1=6

               4!=4✕3✕2✕1=4.3.2.1=24

               5! =5✕4✕3✕2✕1  =5.4.3.2.1=120

               6!=6✕5✕4✕3✕2✕1=6.5.4.3.2.1=720

               7! =7✕6✕5✕4✕3✕2✕1=7.6.5.4.3.2.1=5040

                        ......... so on.                    

Read the Blog on ANGLE & IT'S TYPE

COMBINATION : A Combination of a set of n object taken r 

        at a time without repetition is known as Combination in                        Mathematics

        i.e.   Out of n object if r object is selected at time , 

           where order is not important. It is denoted by 

            nCr or C(n ,r) or nCr .

                        nCr =        n!        =    n!/r!(n-r)!

                                   r! (n-r)!

                e.g.    10C4 =10!/ 4! (10-4)! 

                                    =    10.9.8.7.6!    

                                           4.3.2.1.  6!

                                             =  10.9.8.7       = 210

                                           4.3.2.1

           The  Above calculation is according to the defination

             of Combination.

             The Calculation can also be done like this .

                  10C4 =    10.9.8.7        

                             4.3.2.1 

                20C6 =  20.19.18.17.16.15   

                               6.5.4.3.2.1

  i.e. If the value of r is 4 then just write 

Product of n to (n-3) narural number in the numerator 

and Product of 4 natural number from1to 4 in denominator 

     as in 1st example.

                 Note : nC0=nCn=1 & nC1=n

                 Bionomial Theorem  :

                             (a+b)¹ = a + b

                             (a+b)²=  a² +2ab+ b²    

                   (a+b)³= a³ + 3a²b + 3ab² + b³

                   (a+b)⁴= ?

           When the higher powers of  (a+b) comes then it 

       becomes difficult for us to remember. If someone 

           asks us what is 29th power of (a+b) =?

     Then it becomes difficult for us to answer . 

     At that time we can use Binomial Theorem

        to Find Higher Powers of (a+b) or any variable.

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     Binomial Theorem : For a,b∊R and n∊N

(a+b)ⁿ =nC0aⁿbº +nC1aⁿ⁻¹b¹ +nC2aⁿ⁻²b²+nC3aⁿ⁻³b³ +........+nCnaºbⁿ

  For Example :

 (a+b)⁴ = 4C0 a⁴ b⁰+4C1a³b¹ +4C2 a²b² +4C3 a¹b³+4C4 a⁰b⁴

            = a⁴ b⁰+4C1a³b¹+4C2 a²b² +4C3 a¹b³ +b⁴    (4C0=4C1=1)

(a+b)⁴= a⁴ +4a³b¹ +6a²b² +4a¹b³ +b⁴   

  Simillarly we can calculate          

 1)    (a+b)⁵ 

= a⁵+5a⁴b¹ +10 a³b² +10a²b³ +5a¹b⁴+b⁵ 

2) (a+b)⁶ 

 =a⁶+6a⁵b¹+15a⁴b²+20a³b³+15a²b⁴+6a¹b⁵+b⁶

3) (a+b)⁷

=a⁷+7a⁶b¹+21a⁵b²+35a⁴b³+35a³b⁴+21a²b⁵+7a¹b⁶+b⁷

 This way we can calculate any power of (a+b) by using Binomial Expansion.

     Note :

       1) In the expansion of (a+b)ⁿ contains (n+1) terms .

   For Example (a+b)² contain 3 terms , (a+b)³ contains 4 terms & so on .

    Expansion contain one more term  than the power of expansion.

        2) First term is alway aⁿ & last term is bⁿ  .

            For Example in the expansion of  (a+b)² first term is a² 

            and last term is b².

     In Example of (a+b)³ , First term is a³ and last term is b³ so on .

      3) In each term , the sum of  power of each term

            i.e a &b is alway n

     For example in the expansion of (a+b)², power of 1st term is 2 , 

      in second term sum of power of a & b is 2 and 

      Power of last term is also 2.

      simillarly in (A+b)³& so on

       4) In the successive term in the expansion we can

    see that power of a decreases by 1 and Power of b increases by one.

Binomial Theorem can used for negative term also .

          For example 

    (a-b)⁴= a⁴-4a³b¹+6 a²b² - 4a¹b³ +b⁴   

   (a-b)⁵= a⁵-5a⁴b¹+10 a³b²-10a²b³ +5a¹b⁴- b⁵ 

    Here we can note that First term is +ve , 

     second term is negative & so on. i.e. 

        Alternate terms are +ve & -ve .

   Binomial can also used for fractional & negative powers also .

        1/1+x = (1+x)⁻¹  =1-x+x²-x³+x⁴-x⁵+.......

        1/1-x =(1-x)⁻¹  = 1+x+x²+x³+x⁴+x⁵+.......

  1/(1+x)²  = (1+x)⁻²  =1-2x+3x²-4x³+........

  1/(1-x)²  =(1-x)⁻²  =1+2x+3x²+4x³+......

1/√(1+x)  =(1+x)⁻¹/²  = 1+x/2-x²/8+x³/16+........

  1/√(1-x) =(1-x)⁻¹/² =  1-x/2-x²/8-x³/16-.........

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11

ANGLE & IT'S TYPES

                

                        ANGLE   &  IT'S  TYPES 

ANGLE : An angle is the figure formed by two rays joined together at a common point called vertex .

         Two rays acts as sides of angle.

If OA is initial arm &OB is final arm as in fig1.The rotation of initial arm is in anticlockwise direction .

Then the angle formed is known as positive angle .

If OB is initial arm & OA is final arm as in fig2.The rotation of initial arm is in clockwise direction. Then the angle formed is known as Negative angle.

          Fig.1                                 Fig.2

   Angle formed by two ray generally lie in the same plane.

  Intersection of two planes also formed the angle,which is known as Dihedral angles. 

The angles are measured in 

1)Degree (Sexagesimal system)

2) Radian (Circular system).

There are different types of angles in geometry.

1)Right angle  :  An angle who measure is 90 degree is known as right angle.

                  

2)Acute angle  : An angle who measure is less than 90 degree is known as acute angle.

3)Obtuse angle  : An angle who measure is more than 90 degree is known as obtuse angle.

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4)Straight angle  :When two rays join together in opposite direction at a common point they form a straight angle.

  A  straight angle is also called as flat angle .

       It's measure is 180 degree .

5)Zero angle  : An angle whose measure is zero degree is called a zero angle.

  It is formed when both the rays which formed the side of angle lie in the same line or they overlap each other or they coincide with each other .

Ray OA & RayOB Coincide with each other .

6)Reflex angle : A reflex angle is an angle which is more than 180 degree but less than 360 degree.

   If one angle is more than 180 degree than other angle formed in a plane can be either acute or obtuse 

i.e opposite of reflex angle is either acute or             obtuse .

7)Directed angle : The ordered pair of rays (OA,OB) together with the ROTATION OF THE RAY OA to the position of the ray OB is called the directed angle AOB.

  If the rotation of the initial ray is anticlockwise then the measure of directed angle is cosidered positive and if the rotation is clockwise then the measure of the angle is considered as negative.

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8)One rotation angle :After one complete rotation  if the initial ray OA coincide with the final ray i.e terminal ray OB then the angle so formed is known as one rotation angle. For example 360 Degree.

9)Standard angle : If the vertex of the angle is at the origin and the initial ray is along positive direction of x -axis of the coordinate system , such angle is known as  standard angle.

Angle POX , AngleXOQ , Angle XOR are          standard angles .

10)Quadrantal Angle : A directed angle in standard position whose terminal ray lie along X-axix or Y-axis is called a quadrantal angle.

Angle POQ , Angle POR , Angle POS , Angle POQ (Clockwise) are quadrantal angles.

11)Co- terminal angles : Directed angles of different amount of rotation having the same position of initial and final rays are known as Co-terminal angles.

Angle AOB (30 degree) , Angle AOB (-330 degree) , Angle AOB (390 degree) are Co-terminal angles. 

12)Supplementary angle : When sum of measure of two angle is 180 Degree then those angles are known as supplementary angle. Eg. If one angle is 60 Degree then it's supplementary angle is 120 Degree.

13) Complementary Angle : When sum of measure of two angle is 90 Degree then those angles are known as Complementary angles.

14) Adjacent angles : Two angles are said to be adjacent if 1)They have common vertex 2)they have one side common & 3) both the angles are on opposite side of common side.

Angle  CBD & angle DBE are adjacent angle .

Angle ABC & Angle CBD are adjacent angle . 

But Angle ABC & angle DBE are not adjacent angle .

15)Vertically opposite angles : Two angles are said to be vertically opposite if they are formed when two lines intersect at a common point & they are formed in opposite direction.      

                  Angle AOC & Angle DOB

                 Angle  AOD & Angle COB are                                        vertically opposite angles.

Their measure is same.

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16 ) FOR TWO PARALLEL LINE :

 If the two line are parallel (i.e. two line never intersect each other) & a transeverse intersect the two parallel lines at two different points  then following angles are formed .

1)Alternate angles 

2)Corresponding angles

3)interior angle 

4)Exterior angle

5)Vertcally opposite angle

 

          from fig.

1)Alternate angles are of equal measures 

            & they are : 

            Angle c & angle e

            Angle b & angle h

2)Corrosponding angles are ...

            Angle d & angle h

            angle c & angle g

            angle a & angle e

            angle b & angle f

            

3)Interior angles are :

     Angle c , angle h , angle b , angle e

     Aso   angle c+ angle h =180degree

    &  angle b+ angle e = 180 degree 

4) Exterior angles are :

      Angle d & angle g

       Angle a & angle f

5) Vertically opposite angles are :

           angle d & angle b

           angle a & angle c

           angle h & angle f

           angle e & angle g

   

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17) Plane Angle  :     An Angle between two Intersecting lines is                            known as plane angle in the same plane .  

                  It is measured in Radian. 

                   It is defined in two dimension .

In case of circle it is define as angle subtended by arc at the centre of the circle .

18) Solid Angle  : It is 3 Dimensional angular volume .

              It is measured in steradians. If the surface covers the                         entire sphere then the solid angle subtended at the center is 4 pi radian . Solid angle concept is generally used in physics.

19) Space Angle :  

It is a angle between a line and a Plane in the space . 

It is also angle between two plane in the space. 

It is also an angle between two line in the space.

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