# Solution to the Angles in a Circle Puzzle +-- {.image} [[AnglesinaCircle.png:pic]] > What's the sum of the three marked angles? =-- ## Solution by [[Angles in the Same Segment]], Angles in a [[Cyclic Quadrilateral]], [[Angles at a Point on a Line]], and [[Angles in a Triangle]] +-- {.image} [[AnglesinaCircleLabelled.png:pic]] =-- 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]].