Notes
three regular hexagons iv solution

Solution to the Three Regular Hexagons IV Puzzle

Solution by Angles in Parallel Lines, Angles in a Regular Hexagon, Isosceles Triangles, and Congruent Triangles

In the above diagram, line segment $D E$ is parallel to line segment $B A$ and of the same length, so that $A B D E$ is a parallelogram.

Triangles $D E F$ and $B C F$ are then congruent and are related by a rotation of angle $120^\circ$ about point $F$. Therefore, angle $B \hat{F} D$ is $120^\circ$ and triangle $B F D$ is isosceles. This means that angle $D \hat{B} F$ is $30^\circ$.

Since cointerior angles in parallel lines add up to $180^\circ$, angle $B \hat{G} E$ is therefore $150^\circ$.