Notes
a square and a hexagon solution

Solution to the Square and a Hexagon Puzzle

Solution by Angle in a Semi-Circle, Angles in the Same Segment, and angles in a Regular Hexagon

In the diagram above, the orange circle is the circumcircle of the regular hexagon. The chord $A B$ joins opposite vertices so is a diameter of the circle. Angle $A \hat{G} D$ is a right-angle and so since the angle in a semi-circle is a right-angle, $G$ lies on the circle. Then since angles in the same segment are equal, angle $B \hat{G} D$ is the same as angle $B \hat{A} D$, which is half the interior angle of a regular hexagon and so is $60^\circ$.