You shouldn’t do that and it wouldn’t be considered rude if you had lowered your gaze as they will know that you didn’t see them.

At my school/college i have some female teachers for my subjects so i usually say hello to them when i walk by. Also i say hello to my teachers who used to teach me before when i was younger because i don’t want to be rude. I do not think there is any fitnah as my teachers are usually much older. Is this permissable? Please clarify this topic of lowering the gaze for me.