On 7.27 proceed similarly to 7.5. Try to write $\angle APB$ as the difference of two inscribed angles that open precisely to the arcs $\widehat{AB}$ and $\widehat{CD}$.

On 7.28 try finding some equilateral triangles in your diagram. (Extra hint: Since both cords $AB$ and $CD$ have the same length, what can you say about $AE$, $BE$, $CE$ and $DE$?)