Notes
angles in a circle solution

Solution to the Angles in a Circle Puzzle

Solution by Angles in the Same Segment, Angles in a Cyclic Quadrilateral, Angles at a Point on a Line, and Angles in a Triangle

With the points labelled as above, angles $E \hat{A} G$ and $E \hat{B} G$ are equal as they are angles in the same segment. Then angles $A \hat{E} D$ and $D \hat{C} A$ add up to $180^\circ$ as they are opposite angles in a cyclic quadrilateral. Angles $F \hat{E} A$ and $A \hat{E} D$ add up to $180^\circ$ since the are angles at a point on a line. Putting these last together shows that angles $F \hat{E} A$ and $D \hat{C} A$ are equal. Therefore the three angles in the triangle $F A E$ are each equal to one of the marked angles, so the sum of the three marked angles is $180^\circ$ as that is the sum of the angles in a triangle.