Notes
a semi-circle and a quarter circle solution

Solution to the A Semi-Circle and a Quarter Circle Puzzle

Solution by Angle in a Semi-Circle, Angle at the Centre is Twice the Angle at the Circumference, and Angles in a Triangle

With the points labelled as above, angle $B \hat{C} D$ is half of the reflex angle $B \hat{O} D$ since the angle at the circumference is half the angle at the centre. As it is the outside of a quarter circle, angle $B \hat{O} D$ is $270^\circ$ so angle $B \hat{C} D$ is $135^\circ$.

Therefore, angles $B \hat{C} A$ and $A \hat{C} D$ add up to $135^\circ$. Since angles $A \hat{C} D$ and $A \hat{B} C$ are the same, this means that angles $A \hat{B} C$ and $B \hat{C} A$ add up to $135^\circ$. Then since the angles in a triangle add up to $180^\circ$, angle $B \hat{A} C$ is $45^\circ$.

Finally, angle $O \hat{A} B$ is $90^\circ$ as it is the angle in a semi-circle. Hence angle $O \hat{A} C$ is $45^\circ$.