how to fix 27in. screen hinge? won't stay up

Unfortunately, you'll need to contact Apple - here is another thread with several people having the same problem:

Design fault? 27" 2013 iMac stand just gave up!

If your AppleCare has expired, or if Apple can't help you, you can effect a permanent fix yourself if you are confident in your abilities. See my reply on this thread: https://discussions.apple.com/thread/7072205

Good luck!

Same thing happened today to our 27" iMac ! 😠