Why do the NOR and AND gates have 3 lines going in them?

I think if you look closer, you'll see that in each type of gate,

two lines are the inputs and the third one is the output.

The signal that appears on the output line is decided by

the signals that you apply to the two input lines. The 'rule'

that's used to make the decision is different for the two

types of gates ... they apply different 'logic'.