Prove that the segments joining the midpoint of consecutive sides of an isosceles trapezoid form a rhombus. find the slope of de and fg

Let abcd be a trapezoid and points k,l,m and n be the mid-points of ab, bc, cd and da, respectively. as trapezoid abcd is isosceles, then ak=kb=cm=md and ∠a=∠d, ∠b=∠c. also, bl=lc (because l is mid-point of bc) and an=nd (point n is mid-point of ad). then you have to pairs of equal triangles δakn=δdmn, δkbl=δmcl. this implies that kn=mn and kl=lm.  in quadrilateral klmn line km is a middle line of trapezoid abcd, so km||bc||ad and ln is the height of trapezoid abcd, so ln⊥bc and ln⊥ad.  let consider two triangles δklo and δkon. they are right with common side ko and equal legs lo and on (because middle line of trapezoid divides height into equal parts). from sas theorem δklo=δkon. thus, kl=kn.from  kn=nmkl=lmkl=knyou can conclude that kl=lm=mn=kn and klmn is rhombus. the angle between km and ln is right.