On the one hand, the British Empire provided a vast and expanding market for British goods, which in turn stimulated the growth of British industry. The empire also provided a source of raw materials for British factories, and a dumping ground for surplus goods. In addition, the empire provided a training ground for British engineers and managers, who would go on to play a key role in the development of British industry.
On the other hand, the Industrial Revolution provided the technology and the capital that made possible the expansion of the British Empire. The development of steam power and the steamship made it possible for Britain to transport its goods and troops around the world more quickly and efficiently than ever before. The development of new weapons, such as the Maxim gun, also gave Britain a significant military advantage over its rivals.
In short, the British Empire and the Industrial Revolution were two sides of the same coin. The empire provided the market, the raw materials, and the labor force for British industry, while industry provided the technology and the capital for the expansion of the empire.