In the parallelogram EKFM, the diagonals intersect at point O, with the KOF angle being 138 degrees

In the parallelogram EKFM, the diagonals intersect at point O, with the KOF angle being 138 degrees and the FEM angle being 34 degrees. Find the angle K of the parallelogram if KM is 2 times MF

The diagonals of the parallelogram, at the point of their intersection, are divided in half, then OK = ОМ = КМ / 2. By condition, КМ = 2 * FM, then ОМ = FM, and the triangle ОМF is isosceles. Angle FOM and FOK are adjacent angles, then angle FOM = 180 – 138 = 42. Since triangle FOM is isosceles, angle FOM = OFM = 42. In triangle EFM, define the angle EMF.

Angle EMF = 180 – FEM – EFM = 180 – 34 – 42 = 104. Since the opposite angles of the parallelogram are equal, the angle EKF = EMF = 104.

Answer: Angle EKF is 104