NCERT Solutions for Class 9 Maths Chapter 6 - Lines And Angles

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Chapter 6 - Lines And Angles Exercise Ex. 6.1

Solution 1
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles
Solution 2

Let common ratio between a and b is x,  a = 2x and b = 3x.

XY is a straight line, OM and OP rays stands on it.

XOM + MOP + POY = 180    b + a + POY = 180

3x + 2x + 90 = 180

               5x  = 90

                 x = 18 

a = 2x

   = 2 * 18

   = 36

b = 3x

   = 3 * 18

   = 54

 

Now, MN is a straight line. OX ray stands on it. 

 Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Anglesb + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Anglesc = 180

54 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Anglesc = 180

Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Anglesc = 180 54   = 126 

          Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Anglesc = 126

Solution 3
In the given figure, ST is a straight line and QP ray stand on it.
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQS + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR = 180            (Linear Pair)
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR = 180 - Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQS             (1)
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRT + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRQ = 180            (Linear Pair)
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRQ = 180 - Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRT            (2)
    Given that Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRQ. Now, equating equations (1) and (2), we have
    180 - Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQS = 180  - Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRT
                      Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQS = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRT

Solution 4
We may observe that
    x + y + z + w = 360                (Complete angle)
    It is given that
x + y = z + w
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles x + y + x + y = 360

    2(x + y) = 360
    x + y = 180
    Since x and y form a linear pair, thus AOB is a line.
Solution 5
Given that OR Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQ  
 Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPOR = 90    
 
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPOS  + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSOR = 90
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesROS = 90 - Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPOS                ... (1)
 
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQOR = 90                     (As OR Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQ) 
 
   Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQOS - Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesROS = 90
 
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesROS = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQOS - 90             ... (2)
 
    On adding equations (1) and (2), we have
 
    2 Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesROS = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQOS - Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPOS
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles
Solution 6
                                         Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles
 
Given that line YQ bisects Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPYZ.
 Hence, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQYP = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesZYQ
 Now we may observe that PX is a line. YQ and YZ rays stand on it.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesXYZ + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesZYQ + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQYP = 180    
 
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles 64 + 2Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQYP = 180
 
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles 2Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQYP = 180 - 64 = 116
 
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQYP = 58
 
    Also, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesZYQ = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQYP = 58
 
    Reflex Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQYP = 360o - 58o = 302o
 
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesXYQ = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesXYZ + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesZYQ
 
         = 64o + 58o = 122o

Chapter 6 - Lines And Angles Exercise Ex. 6.2

Solution 1
We may observe that
50 + x = 180                   (Linear pair)
x = 130             ... (1)
Also, y = 130                    (vertically opposite angles)
As x and y are alternate interior angles for lines AB and CD and also measures of these angles are equal to each other, so line AB || CD
Solution 2
  Given that AB || CD and CD || EF
   Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles  AB || CD || EF    (Lines parallel to a same line are parallel to each other)
 
   Now we may observe that
   x = z             (alternate interior angles)    ... (1)
   Given that y: z = 3: 7
   Let common ratio between y and z be a
   Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles y = 3a and z = 7a

    Also x + y = 180     (co-interior angles on the same side of the transversal)
    z + y = 180             [Using equation (1)]
    7a + 3a = 180
    10a = 180
        a = 18
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Anglesx = 7 a = 7 Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles 18 = 126

Solution 3
 
It is given that
AB || CD                        
EF Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles   CD
GED = 126
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesGEF + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesFED = 126
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesGEF + 90 = 126    
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesGEF = 36
Now, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesAGE and Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesGED are alternate interior angles
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesAGE = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesGED = 126    
But Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesAGE +Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesFGE = 180      (linear pair)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles 126 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesFGE = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesFGE = 180 - 126 = 54
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesAGE = 126, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesGEF = 36, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesFGE = 54

Solution 4
                  Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles
Let us draw a line XY parallel to ST and passing through point R.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQRX = 180     (co-interior angles on the same side of transversal QR)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles 110 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQRX = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQRX = 70
Now,
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesRST +Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSRY = 180    (co-interior angles on the same side of transversal SR)
130 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSRY = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSRY = 50
XY is a straight line. RQ and RS stand on it.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQRX + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQRS + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSRY = 180     
70 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQRS + 50 = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQRS = 180 - 120 = 60

Solution 5
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesAPR = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRD                 (alternate interior angles)
50 + y = 127
           y = 127 - 50
           y = 77
Also Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesAPQ = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR         (alternate interior angles)
             50 = x
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles x = 50 and y = 77

Solution 6
                                  Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles
 
Let us draw BM  Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles PQ and CN Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles RS.
 As PQ || RS
So, BM || CN

 Thus, BM and CN are two parallel lines and a transversal line BC cuts them at B and C respectively.
  Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesNcert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles2 = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles3                               (alternate interior angles)
 
But Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles1 = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles2 and Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles3 = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles4      (By laws of reflection)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles1 = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles2 = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles3 = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles4
Now, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles1 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles2 = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles3 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles4
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesABC = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesDCB
 
But, these are alternate interior angles
 Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles AB || CD

Chapter 6 - Lines And Angles Exercise Ex. 6.3

Solution 1
Given that
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSPR = 135 and Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQT = 110
    Now, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSPR + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQPR = 180             (linear pair angles)
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles 135 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQPR = 180
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQPR = 45                         
    Also, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQT + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR = 180             (linear pair angles)
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles 110 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR = 180
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesNcert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR = 70     
    As we know that sum of all interior angles of a triangle is 180, so, for Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR  
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQPR + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRQ = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles 45 + 70 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRQ = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesNcert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRQ = 180 - 115
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRQ = 65

Solution 2
As we know that sum of all interior angles of a triangle is 180, so for Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesXYZ
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesX + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesXYZ + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesXZY = 180    
62 + 54 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesXZY = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesXZY = 180 - 116
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesXZY = 64
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesOZY =  Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And Angles = 32         (OZ is angle bisector of Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesXZY)
Similarly, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesOYZ =  = 27
Using angle sum property for Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesOYZ, we have
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesOYZ + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesYOZ + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesOZY = 180º
27 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesYOZ + 32 = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesYOZ = 180 - 59
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesYOZ = 121

Solution 3
AB || DE and AE is a transversal                    
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesBAC =Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesCED                 (alternate interior angle)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesNcert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesCED = 35
In Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesCDE,
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesCDE + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesCED + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesDCE = 180         (angle sum properly of a triangle)
53 + 35 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesDCE = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesDCE = 180 - 88
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesDCE = 92

Solution 4
Using angle sum property for Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRT, we have
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRT + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesRPT + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPTR = 180
40 + 95 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPTR = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPTR = 180 - 135
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPTR = 45
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSTQ = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPTR = 45             (vertically opposite angles)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSTQ = 45
By using angle sum property for Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSTQ, we have
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSTQ + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSQT + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQST = 180
45 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSQT + 75 = 180
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSQT = 180 - 120
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSQT = 60  

Solution 5
Given that PQ || SR and QR is a transversal line
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQRT             (alternate interior angles)
x + 28 = 65
x = 65 - 28
x = 37
By using angle sum property for Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSPQ, we have
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesSPQ + x + y = 180
90 + 37 + y = 180
y = 180 - 127
y = 53
 x = 37 and y = 53.

Solution 6
In Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQTR, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesTRS is an exterior angle.
 Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesNcert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQTR + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesTQR = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesTRS
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQTR = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesTRS - Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesTQR        (1)
For Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR, Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRS is external angle
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesNcert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQPR + Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPQR = Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesPRS
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQPR + 2Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesTQR = 2Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesTRS    (As QT and RT are angle bisectors)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQPR = 2(Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesTRS - Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesTQR)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQPR = 2Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQTR            [By using equation (1)]
Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQTR =  Ncert Solutions Cbse Class 9 Mathematics Chapter - Lines And AnglesQPR