The bisectors AD and BE of triangle ABC meet at point O. Find the angle C of the triangle if the angle AOE = 50 degrees.

The AOE angle given by condition is the outer angle for the AOB triangle.
Let’s use the triangle outer angle theorem and write:
∠ AOE = ∠ ABO + ∠ BAO = 1/2 ∠ ABC + 1/2 ∠ BAC = 50 °.
We get that in triangle ABC the sum of two angles is equal to:
∠ ABC + ∠ BAC = 100 °.
We find the degree measure of the angle C of the triangle ABC:
∠ С = 180 ° – (∠ ABC + ∠ BAC) = 180 ° – 100 ° = 80 °.
Answer: the angle C of the triangle ABC is 80 °.