In triangle ABC BM is the median and BH is the height. It is known that AC = 216, HC = 54 and ∠ACB = 40 °. Find the angle AMB.

Since BM is the median, it means AM = MC = AC / 2 = 216/2 = 108
Consider the MBC triangle.
MH = MC-HC = 108-54 = 54, i.e. it turns out that MH = HC.
BH for this triangle is not only the height, but also the median. This is a property of an isosceles triangle.
By the property of an isosceles triangle: ∠BMC = ∠ACB = 40 °.
∠AMB = 180 ° -∠BMC = 180 ° -40 ° = 140 ° (as it is adjacent)
Answer: ∠AMB = 140 °

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