You are given a regular hexagon ABCDEF. Prove that the angle ACB = 30 degrees.

Since the hexagon is regular, a circle can be described around it.

Let’s build the radii ОА, ОВ and ОВ.

Central angles AOB = BOC = 360/6 = 60.

ОА = ОВ = ОС = Р, angle AOB = BOС = 60, then triangles AOB and BOС are equilateral, and then all their internal angles are equal to 60.

Then the angle ABC = (ABO + СВO) = 60 + 60 = 120.

The ABC triangle is isosceles, then the angle BAC = BCA = (180 – 120) / 2 = 30, which was required to prove.