In triangle ABC, bisector AK is drawn, find angle B if angle C = 30 degrees, and angle AKC = 110 degrees

Consider the triangle AKS and find the angle CAK in it, knowing the other two angles.
∠САK = 180 ° – (∠ С + ∠ АКС) = 180 ° – (30 ° + 110 °) 180 ° – 140 ° = 40 °.
We find the angle A of the triangle ABC:
∠ А = 2 * ∠САK = 80 ° (AK – bisector by condition).
Two angles of the triangle ABC are known, we find the angle B:
∠ B = 180 ° – (∠ A + ∠ C) = 180 ° – (80 ° + 30 °) = 70 °.
Answer: The degree measure of angle B is 70 °.