Here is a fun one: The triangle ABC is isoseceles, with angle(ABC) = angle(ACB) = 80°. We draw now two segments inside the triangle. First, we choose the point D on segment AC such that angle(CBD) is 60°. Also, we choose the point E on AB, where angle(BCE) is 50°. We now extend DE until it intersects with the line on the BC segment in point F, as below:

 

The question is: what is the angle between the two lines on F? In other words, what is the value of angle(DFC)?