In the parallelogram KMNP, the bisector of the angle MKP is drawn, which intersects the side MN at point E.

In the parallelogram KMNP, the bisector of the angle MKP is drawn, which intersects the side MN at point E. a) Prove that triangle KME is isosceles. b) Find the side of the KP if ME = 10 cm, and the parallelogram perimeter is 52 cm.

a) Since KE is the bisector of the angle, the angle MKE = EKP. The angle MEK = EKP as criss-crossing angles at the intersection of parallel lines KP and MN secant KE, then the angle MKE = MEK, and, accordingly, the triangle KME is isosceles, which was required to prove.

c) Since the triangle KME is isosceles, then ME = KM = PN = 10 cm.

Then the side KP = (P – KM – PN) / 2 = (52 – 10 – 10) / 2 = 16 cm.

Answer: Side KR = 16 cm.