question_answer

D and E are the mid-points of the sides AB and AC, respectively of \[\Delta ABC.DE\]is produced to F. To prove that CF is equal and parallel to DA, we need an additional information which is

A) \[\angle DAE=\angle EFC\]         

B)         AE = EF         

C)         DE = EF                     

D)        \[\angle ADE=\angle ECF\]         

Correct Answer: C

Solution :

We have produced DE to F such that DE = EF                                           ?(i) In\[\Delta \Alpha DE\]and \[\Delta CFE,\] AE = CE           [\[\therefore \]E is the mid-point of AC] \[\angle AED=\angle CEF\][Vertically opposite angles] DF = FE                                    [By (i)] \[\therefore \]\[\Delta ADE\cong \Delta CFE\]           [By SAS congruency] \[\therefore \]\[AD=CF\]and \[\angle ADE=\angle CFE\]   [By CPCT] This show that alternate interior angles are equal. Thus, \[AD||CF\] Therefore, the additional information which we need is DE = EF