In an isosceles triangle, the angle opposite to the base is 120, and the bisector to the base is 8. Find the side.

Triangle ABC: AB = BC – lateral sides, AC – base, ВK = 8 – bisector of angle B, angle B = 120 degrees.
1. In an isosceles triangle, the bisector, median and height drawn to the base are equal. Then the angle ВKA = angle ВKС = 90 degrees, AK = CК = AC / 2, angle ABK = angle СВK = angle B / 2 = 120/2 = 60 degrees. Thus, triangles AKB and СKB are equal.
2. Consider the triangle AKВ: angle ВKA = 90 degrees, angle ABK = 60 degrees, ВK = 8 and AK – legs, AB – hypotenuse. By the theorem on the sum of the angles of a triangle:
angle ВKA + angle AВK + angle ВAK = 180 degrees;
90 + 60 + ВAK angle = 180;
angle ВAK = 180 – 150;
angle ВАK = 30 degrees.
3. From the properties of a right-angled triangle: opposite an angle of 30 degrees, there is a leg, which is 2 times less than the hypotenuse, then:
ВK = AB / 2;
AB / 2 = 8;
AB = 2 * 8;
AB = 16.
AB = BC = 16.
Answer: 16.