Notes
angles in two circles solution

Solution to the Angles in Two Circles Puzzle

Solution by the Alternate Segment Theorem and Angles in a Triangle

With the points labelled as above, the alternate segment theorem gives that both angles $C \hat{B} D$ and $B \hat{C} D$ are both $a$ while $C \hat{A} D$ and $D \hat{C} A$ are both $b$. As angles in a triangle add up to $180^\circ$, considering triangle $A B C$ shows that $a + a + b + b = 180^\circ$, and so $a + b = 90^\circ$.