(i) To find angle ABC, use the sine function in the right triangle ABC:

\(\sin \angle ABC = \frac{AC}{BC} = \frac{8}{10}\)

\(\angle ABC = \arcsin\left(\frac{8}{10}\right) \approx 0.927 \text{ radians}\)

(ii) First, find AB using the Pythagorean theorem:

\(AB = \sqrt{BC^2 - AC^2} = \sqrt{10^2 - 8^2} = 6 \text{ cm}\)

Calculate the area of triangle BCD:

\(\text{Area of } \triangle BCD = \frac{1}{2} \times 8 \times 6 = 48 \text{ cm}^2\)

Calculate the area of sector BCD:

\(\text{Area of sector } BCD = \frac{1}{2} \times 10^2 \times 2 \times 0.927 = 92.73 \text{ cm}^2\)

Calculate the area of segment BCD:

\(\text{Area of segment } BCD = 92.73 - 48 = 44.73 \text{ cm}^2\)

Calculate the area of the semicircle with radius 8 cm:

\(\text{Area of semicircle} = \frac{1}{2} \times \pi \times 8^2 = 100.53 \text{ cm}^2\)

Finally, find the shaded area:

\(\text{Shaded area} = 100.53 - 44.73 = 55.8 \text{ cm}^2\)