(i) The area of sector AOB is given by \(\frac{1}{2} r^2 \times 2\theta = r^2 \theta\).

The area of the triangle AOT is \(\frac{1}{2} \times r \times r \tan \theta = \frac{1}{2} r^2 \tan \theta\).

Since the area of the sector is equal to the area of the shaded region, we have:

\(r^2 \theta = \frac{1}{2} r^2 \tan \theta\)

\(2\theta = \tan \theta\)

(ii) Given \(r = 8\) cm and the length of the minor arc AB is 19.2 cm, we find \(\theta\) using the arc length formula:

\(r \times 2\theta = 19.2\)

\(8 \times 2\theta = 19.2\)

\(\theta = 1.2\) radians

The area of sector AOB is \(\frac{1}{2} \times 8^2 \times 2 \times 1.2 = 76.8\) cm².

The area of the kite (or quadrilateral) is \(165\) cm².

The area of the shaded region is \(165 - 76.8 = 88.2\) cm², approximately \(87.8\) cm².