AK bisector of triangle ABC, angle A = 40 °, angle AKB = 120 °. Find angle C.

The bisector AK divides the angle A, which is 40 ° by condition, into two equal angles:

∠ СAK = ∠ ВAK = ∠ A / 2 = 40 ° / 2 = 20 °.

The angles AKB and AKС are adjacent, and since the sum of adjacent angles is 180 °, then:

∠ AKВ+ ∠ AKС = 180 °.

Hence, ∠ AKC = 180 ° – ∠ AKВ = 180 ° – 120 ° = 60 °.

In the triangle СAK ∠ СAK = 20 °, ∠ AKC = 60 °, and since the sum of the angles of any triangle is 180 °, then ∠ С = 180 ° – ∠ СAK – ∠ AKC = 180 ° – 20 ° – 60 ° = 100 °.