Brilliant's logo made a whole lot complex

Draw a circle circumscribing the regular 18-gon. Draw the segment from the centre to the points 1 & 4.

Since, $\angle 124$ & $\angle 134$ inscribe the same arc, $\angle 124 = \angle 134$ .

The angle formed by arc joining points 4,11,1 is 300 degrees. Since, angle formed at the circumference is half the inscribed angle, $\angle 124 = \angle 134 = 150$
Their sum is 300.

The triangle 234 has angles at the three vertices 2: 10 degrees 3: 160 degrees 4: 10 degrees angle 124 = 160 degrees - angle of vertex 2 in triangle 234 so the angle 124 = 150 degrees its the same for the other angle