Bisectors AD and BE of triangle ABC intersect at point O. Angle AOB = 140 °. Find the angle C of triangle ABC.

The angles of a triangle add up to 180 degrees.
In the AOB triangle, the AOB angle is 140 degrees, which means the sum of the ABO and AAO angles is equal to: ABO angle + BAO angle = 180-140 = 40 degrees.
Since BE and AD are the bisectors of angles A and B, then angle B = 2 * angle ABO, angle A = 2 * angle BAO. Obviously, the sum of the angles A and B is twice the sum of the angles ABO and BAO: angle A + angle B = 2 * angle BAO + 2 * angle ABO = 2 * (angle BAO + angle ABO) = 2 * 40 = 80 degrees.
In triangle ABC: angle C = 180-angle A-angle B = 180- (angle A + angle B) = 180-80 = 100 degrees.