In triangle ABC, angles A and C are 30 ° and 50 °, respectively. Find the angle between the BH height and the BD bisector

By the theorem on the sum of the angles of a triangle: 180 ° = ∠A + ∠B + ∠C, hence ∠B = 180 ° -∠A-∠C = 180 ° -30 ° -50 ° = 100 °.
∠ABD = ∠B / 2 = 50 ° (because BD is a bisector).
Consider the triangle BHC, by the theorem on the sum of the angles of the triangle we get 180 ° = 50 ° + 90 ° + ∠CBH => ∠CBH = 40 °.
Then the desired angle is ∠DBH = ∠B-∠ABD-∠CBH = 100 ° -50 ° -40 ° = 10 °.
Answer: ∠DBH = 10 °