The angle M of an isosceles triangle KMF (KM = MF) is 120 degrees. The height МD is extended beyond

The angle M of an isosceles triangle KMF (KM = MF) is 120 degrees. The height МD is extended beyond the point D so that МD = DE. Find the degree measure of the MEK angle.

If you connect points K and E, and points E and F, then you get a rhombus KMFE, ME and KF are the diagonals of the KMFE rhombus, and D is the intersection point of the diagonals of the rhombus, which divides them in half. Also, angle KMF = angle KEF, since the opposite angles in a rhombus are equal.
It is known from the properties of the rhombus that the diagonals are the bisectors of the angles, then ME is the bisector of the angles КМF and КЕF. Since the angle KMF = angle KEF, then the angle KEF = 120 degrees. Then:
angle MEK = angle KEF / 2;
angle MEK = 120/2 = 60 (degrees).
Answer: MEK angle = 60 degrees.