The bisector of angles A and D of parallelogram ABCD meets at point M lying on side BC.

The bisector of angles A and D of parallelogram ABCD meets at point M lying on side BC. Find the sides of the parallelogram if its P = 36 cm.

Let’s designate the intersection point as M and consider the triangle CDM, the angle CDM will be equal to 1/2 of the angle CDA, as DM is the bisector. The sum of the angles BAD and CDA is 180 (since ABCD is a parallelogram), and the angle BAD is equal to BCD, then:
CMD = 180- (MCD + CDM) = 180-180 + CDA-1 / 2CDA = 1 / 2CDA = CDM
From the obtained equality it follows that the triangle MCD is isosceles, then:
CD = CM
Similarly, we need to consider the triangle ABM, then we get BC = 1 / 2CD.
Take the side ABC, we get the equation:
x + 2x + x + 2x = 36
6x = 36
x = 6
2nd side: 6 * 2 = 12