Notes
angles in two hexagons solution

Angles in Two Hexagons

Solution by Transformations

With the points labelled as above, a rotation of $60^\circ$ clockwise about $O$ takes point $A$ to point $B$ and point $C$ to point $D$. It therefore takes the line segment $A C$ to $B D$ and so the angle between these line segments is $60^\circ$. This is angle $B \hat{E} A$, so angle $A \hat{E} D$ is $120^\circ$. Since angles in a triangle add up to $120^\circ$, angles $D \hat{A} E$ and $E \hat{D} A$ therefore add up to $60^\circ$ and so the sum of the two marked angles is $180^\circ - 60^\circ = 120^\circ$.

Solution by Invariance Principle

In the case where the two hexagons the same size, the two angles individually are seen to be $60^\circ$ and their sum is therefore $120^\circ$.