# Solution to the [[Three Regular Hexagons IV]] Puzzle +-- {.image} [[ThreeRegularHexagonsIV.jpeg:pic]] > Three regular hexagons. What’s the angle? =-- ## Solution by [[Angles in Parallel Lines]], Angles in a [[Regular Hexagon]], [[Isosceles Triangles]], and [[Congruent Triangles]] +-- {.image} [[ThreeRegularHexagonsIVLabelled.jpeg:pic]] =-- 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 $150^\circ$.