In the parallelogram KMNP, the bisector of the angle MKP is drawn which intersects MN

In the parallelogram KMNP, the bisector of the angle MKP is drawn which intersects MN at the point E, prove that the triangle KME is isosceles.

Since KE is the bisector of the MKP angle, the MKE angle = EKP.

The angle EKP is equal to the angle MEK as the intersecting angles at the intersection of parallel straight lines KP and MN secant KE, then the angle MKE is equal to MEK.

Then in the triangle KME the angles at the base of KE are equal, and therefore, the triangle KME is isosceles, KM = ME, which was required to prove.