, 2022

Solution 1

Since $A$, $B$, $C$, $D$, $E$, and $F$ are equally spaced around the circle, moving from one point to the next corresponds to moving $\frac{1}{6}$ of the way around the circle. Therefore, moving from $A$ to $C$ corresponds to moving $\frac{2}{6}$ (or $\frac{1}{3}$) of the way around the circle. Since moving around the whole circle corresponds to moving through $360\degree$, then moving $\frac{1}{3}$ of the way around the circle corresponds to moving through $\$\frac{1}{3} \times 360\degree = 120\degree$$. Thus,$$\angle$ AOC = $120\degree$$. Solution 2 We join$A$,$B$,$C$,$D$,$E$, and$F$to$O$. [[IMAGE0]] Since$A$,$B$,$C$,$D$,$E$, and$F are equally spaced around the circle, the angles made at the centre by consecutive points are equal. That is,

$\angle AOB = \angle BOC = \angle COD = \angle DOE = \angle EOF = \angle FOA$ Since these 6 angles form a complete circle, the sum of their measures is

$360\degree$$. Therefore,$$\angle AOB = \angle BOC = \angle COD = \angle DOE = \angle EOF = \angle FOA = \tfrac{1}{6} \times 360\degree = 60\degree$$This means that$$\angle$ AOC = $\angle$ AOB + $\angle$ BOC = $60\degree + 60\degree = 120\degree\$.$