The bisectors of angles A and B at the lateral side AB of the trapezoid ABCD intersect at point F. Find AB if AF = 16, BF = 12

∠GAE = ∠BEA (as they are cross-lying)
∠GAE = ∠BEA = ∠BAE (because AE is a bisector).
It turns out that the triangle ABE is isosceles.
BF is a bisector, and by the property of an isosceles triangle, it is also the median and height.
Thus, it turns out that the triangle ABF is rectangular.
By the Pythagorean theorem:
AB2 = AF2 + BF2
AB ^ 2 = 16 ^ 2 + 12 ^ 2
AB ^ 2 = 256 + 144 = 400
AB = 20
Answer: AB = 20

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