Question

abcd is a square and ef parallel bd prove be = df

Answer 1

draw a diagonal, join BC
now you have two triangles CAB and CDB
proove these triangles congruent

so in CAB and CDB

CD = CD (common side)
Angle A = Angle Angle D (90 degree)
AB = CD (sides of square)

so triangle CAB is congruent to CDB (RHS)

therefore, it is a square