CBSE Class 9 Answered

give me the solutions of the exercise 8.1

Asked by paddipawar | 02 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
      Double click to edit

CEB =

CBE        (angle opposite to equal sides are also equal)
     Now consider parallel lines AD and CE. AE is transversal line for them
     Double click to edit

A +

CEB = 180 (angles on the same side of transversal)
Double click to edit

A+

CBE = 180 (using the relation Double click to edit

CEB =

CBE) ... (1)
But Double click to edit

B +

CBE = 180    (linear pair angles)           ... (2)
     From equations (1) and (2), we have
      Double click to edit

A =

B
(ii) AB || CD
Double click to edit

A +

D = 180 (angles on the same side of transversal)
Also Double click to edit

C +

B = 180 (angles on the same side of transversal)

A +

D =

C +

But

A =

B [using the result obtained proved in (i)]

C =

D

(iii) In ABC and BAD
       AB = BA                (common side)
   BC = AD                (given)
        Double click to edit

B =

A (proved before)

ABC

BAD (SAS congruence rule)

(iv) Double click to edit

ABC

BAD

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) Double click to edit

ABC and

DEF.
        AB = DE                                (given)
        BC = EF                (given)
        AC = DF                                (ACFD is a parallelogram)

ABC

DEF (by SSS congruence rule)

10.

(i) In

APB and

CQD

APB =

CQD        (each 90^o)
     AB = CD            (opposite sides of parallelogram ABCD)
      Double click to edit

ABP =

CDQ (alternate interior angles for AB || CD)

APB

CQD (by AAS congruency)

(ii) By using the result obtained as above
Double click to edit

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 Double click to edit

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 Double click to edit

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.

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We have provided solutions of 9,10,11,12 of Ex.8.1 of NCERT.

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Answered by Vimala Ramamurthy | 03 Dec, 2014, 08:52: AM