Question

E and F are respectively the mid-points of the non-parallel sides AD and BC of a trapezium ABCD.Prove that EF is parallel to AB and EF=1/2(AB+CD)

Answer 1

 

ABCD is a trapezium in which AB || DC and E, F are mid points of AD, BC respectively.

Join CE and produce it to meet BA produced at G.

In ΔEDC and ΔEAG,

ED = EA    ( E is mid point of AD)

∠CED = ∠GEC ( Vertically opposite angles)

∠ECD = ∠EGA ( alternate angles) ( DC||AB, DC||GB and CG transversal)

∴ ΔEDC ≅ ΔEAG

CD  = GA and EC = EG 

In ΔCGB,

E is mid point of CG ( EC = EG proved)

F is a mid point of BC  (given)

∴ By mid point theorem EF ||AB and EF = (1/2)GB.

But GB = GA + AB = CD + AB

Hence EF||AB and EF = (1/2)( AB + CD).