One of the corners of a rectangular trapezoid is 120 degrees, a large lateral side is 40cm, center line = 14cm. Find the base.

First, find the remaining angle of the trapezoid. The sum of all the angles of the trapezoid is 360 degrees, so the angle – let’s call it CDA – is 60 degrees, because: 360 – 90 – 90 – 120 = 60.
Let’s draw the height CH from point C to the larger base of the trapezoid – AD. We get a right-angled triangle CDH, one of the corners of which (CDA) is 60 degrees. Let’s find another angle in this triangle, since the sum of all the angles of the triangle is 180, then the angle we need is HCD = 180 – 60 – 90 = 30 degrees. We recall the theorem that reads: The leg, which lies opposite an angle of 30 degrees, is equal to half the hypotenuse. Accordingly, the segment HD = half of the side CD = 20 cm.
We see that the base AD consists of two segments = HD and AH. AH = BC.
And the middle line is equal to the half-sum of the bases. Let’s designate segments BC and AH as X and get the equation: x + x + 20/2 = 14
2 (x + 10) / 2 = 14
x + 10 = 14.
x = 4.
Find the bases: 4 and 24 cm