A table of the given information is as follows

Number of spokes | 4 | 6 | 8 | 10 | 12 |

Angle between a pair of consecutive spokes | 90° | 60° | x_{1} | x_{2} | x_{3} |

From the given table, we obtain

4 × 90° = 360° = 6 × 60°

Thus, the number of spokes and the angle between a pair of consecutive spokes are inversely proportional to each other. Therefore, 4 × 90° = x_{1} × 8

similarly, x_{1} = (4 x 90°)/8 = 45°,

x_{2} = (4 x 90°)/10 = 36° and

x_{3} = (4 x 90°)/12 = 30°

Thus, the following table is obtained.

Number of spokes | 4 | 6 | 8 | 10 | 12 |

Angle between a pair of consecutive spokes | 90° | 60° | 45° | 36° | 30° |

(i) Yes, the number of spokes and the angles formed between the pairs of consecutive spokes are in inverse proportion.

(ii)Let the angle between a pair of consecutive spokes on a wheel with 15 spokes be

x. Therefore,

4 × 90° = 15 × x

x = (4 x90°)/15 = 24°

Hence, the angle between a pair of consecutive spokes of a wheel, which has 15 spokes in it, is 24°.

(iii) Let the number of spokes in a wheel, which has 40º angles between a pair of consecutive spokes, be y.

Therefore,

4 × 90° = y × 40°

y = (4x90)/40 = 9

Hence, the number of spokes in such a wheel is 9.