$2$. It's clear that we can have $2$ angles satisfying given conditions. Now we assume that we have $3$ angles satisfying given conditions. Let $A,B$ and $C$ be vertices of the angles and define $l_A,r_A$ be rays of angle $A$ such that directed angle between $l_A$ and $r_A$ is $+60^{\circ}$ (here "l" satnds for "left", "r" for "right")(here counterclockwise is positive) . Define $l_B,r_B,l_C,r_C$ similarly. It's not hard to check that angles $A$ and $B$ have $4$ intersection points if and only if directed angle between $l_B$ and $l_A$ is larger than $+60^{\circ}$. So among the angles, every pair of two angles have $4$ intersection points implies directed angle between $(l_B,l_A) (l_C,l_B) (l_A,l_C) $ are all larger than $+60^{\circ}$ , which contradicts fact that this angles up to $+180^{\circ}$