Find the smaller base of the isosceles trapezoid if its larger base is 32 cm, the side is 20 cm, and one of the corners is 60 degrees.

Let’s call this isosceles trapezoid ABCD. Draw the height BE from the top B. Consider a right-angled triangle ABE, whose angle BAE = 60 degrees by condition, and the angle BEA = 90 degrees by construction. Then the angle ABE = 30 degrees. By the property that the leg lying opposite an angle of 30 degrees is equal to half of the hypotenosis. So AE = 20: 2 = 10 (see) Let’s draw the height CK from the vertex C. Triangle ABE = triangle CKD. therefore AE = KD = 10 (see), then EK = BC = 32 -10 -10 = 12 (see)
Answer: 12 cm.