give someone a line

Meaning 1
to give a fish a line in order to catch it; to give someone a line/rope in order to pull them in
Sentence 1
I hope to catch a nice German Brown with this line, hook, bait, and sinker. Throw out a line to that boat, so they can dock at the pier.
Meaning 2
to tell someone something favorable so they become interested in you; usually the person wants a date
Sentence 2
That line you gave Amy, that you had never been in love before you met her, is not true.