In an isosceles trapezoid, the base angle is 45 degrees, the height is 10, and the larger base is 70. Find the smaller base.

We draw from the points of the smaller base B and C heights BH and CF. Since the toapecia is isosceles, the angles BAD and CDA are equal, and both are equal to 45, which follows from the problem statement. Both formed triangles are isosceles, since the sum of acute angles in a right-angled triangle is 90, therefore, both angles are 45. Then the legs in both right-angled triangles are = 10, which is also said in the problem statement. The larger base AD consists of two equal parts AH and FD and the segment HF, which is equal to the smaller base. We denote by x the smaller base, then:
70 = 10 + 10 + x
X = 50
Answer: 50 cm