Notes
two circles solution

Solution to the Two Circles Puzzle

Solution by Angles in the Same Segment, Opposite Angles in a Cyclic Quadrilateral, and Angle at the Centre is Twice the Angle at the Circumference

Label the points as above, with $O$ the centre of the blue circle.

Angles $E \hat{O} A$ and $E \hat{D} A$ are equal since they are angles in the same segment. Then $A D E C$ is a cyclic quadrilateral so angles $A \hat{C} E$ and $E \hat{D} A$ add up to $180^\circ$. But also angles $A \hat{C} B$, $C \hat{B} A$, and $B \hat{A} C$ add up to $180^\circ$ since they are angles in a triangle. Therefore, angles $C \hat{B} A$ and $B \hat{A} C$ add up to angle $E \hat{O} A$.

Then angle $E \hat{O} A$ is twice angle $C \hat{B} A$ since the angle at the centre is twice the angle at the circumference. Hence angles $C \hat{B} A$ and $B \hat{A} C$ add up to twice angle $C \hat{B} A$, so they must be equal. Thus angle $C \hat{B}A$ is $50^\circ$.