give me the solutions of the exercise 8.1

Asked by paddipawar | 2nd Dec, 2014, 09:22: PM

Expert Answer:

12.
Extend AB. Draw a line through C, which is parallel to AD, intersecting AE at point E. 
Now, AECD is a parallelogram.

(i)   AD = CE                  (opposite sides of parallelogram AECD)  
      But AD = BC             (given)
      So, BC = CE 
     CEB = CBE         (angle opposite to equal sides are also equal) 
     Now consider parallel lines AD and CE. AE is transversal line for them 
    A + CEB = 180         (angles on the same side of transversal)
    A+ CBE = 180         (using the relationCEB = CBE)    ... (1)  
     But B + CBE = 180     (linear pair angles)            ... (2) 
     From equations (1) and (2), we have
     A = B  
(ii)  AB || CD
      A + D = 180            (angles on the same side of transversal) 
      Also C + B = 180          (angles on the same side of transversal)  
       A + D = C + B

      But A = B             [using the result obtained proved in (i)]
       C = 

(iii)  In ABC and BAD
       AB = BA                 (common side) 
       BC = AD                 (given)
       B = A                 (proved before) 
       ABC  BAD            (SAS congruence rule) 

(iv)   ABCBAD  
       AC = BD                 (by CPCT) 
 
11.
 
(i)   Here AB = DE and AB || DE.     
      Now, if two opposite sides of a quadrilateral are equal and parallel to each other, it will be
      a parallelogram. 
      Therefore, quadrilateral ABED is a parallelogram. 

(ii)  Again BC = EF and BC || EF. 
      Therefore, quadrilateral BEFC is a parallelogram.
 
(iii)  Here ABED and BEFC are parallelograms. 
       AD = BE, and AD || BE          
       (Opposite sides of parallelogram are equal and parallel) 
       And BE = CF, and BE || CF         
       (Opposite sides of parallelogram are equal and parallel)
         AD = CF, and AD || CF 

(iv)   Here one pair of opposite sides (AD and CF) of quadrilateral ACFD are equal and
        parallel to each other, 
        so, it is a parallelogram. 
(v)    As ACFD is a parallelogram, so, pair of opposite sides will be equal and parallel to each
        other. 
         AC || DF and AC = DF 

(vi)   ABC and DEF.
        AB = DE                                (given)
        BC = EF                                (given) 
        AC = DF                                (ACFD is a parallelogram)  
        ABCDEF                (by SSS congruence rule)
 
10.
(i)  In APB and CQD
     APB = CQD         (each 90o
     AB = CD             (opposite sides of parallelogram ABCD) 
     ABP = CDQ         (alternate interior angles for AB || CD) 
      APB CQD        (by AAS congruency) 

(ii) By using the result obtained as above 
     APB CQD, we have 
     AP = CQ             (by CPCT) 
9.
 
(i)  In APD and CQB
     ADP = CBQ                   (alternate interior angles for BC || AD) 
     AD = CB                       (opposite sides of parallelogram ABCD) 
     DP = BQ                       (given) 
     APD CQB         (using SAS congruence rule) 

(ii)  As we had observed that APD  CQB 
       AP = CQ                     (CPCT)
 
(iii)  In AQB and CPD
       ABQ = CDP          (alternate interior angles for AB || CD)
       AB = CD                     (opposite sides of parallelogram ABCD) 
       BQ = DP                     (given) 
       AQB  CPD               (using SAS congruence rule) 

(iv)  As we had observed that AQB CPD 
        AQ = CP             (CPCT)

(v)   From the result obtained in (ii) and (iv), we have
              AQ = CP and  AP = CQ  
       Since opposite sides in quadrilateral APCQ are equal to each other. So, APCQ is a
       parallelogram.
You have not mentioned the book for which you want to get the solutions
We have provided solutions of 9,10,11,12 of Ex.8.1 of NCERT.
If you need all the solutions, please enroll for textbook solutions from topper's site.
  

Answered by Vimala Ramamurthy | 3rd Dec, 2014, 08:52: AM

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