Does a reach weapon allow you to threaten squares 10 feet away or not?

In D&D 5e, does a reach weapon allow you to threaten squares up to 10 feet away for the purpose of Opportunity Attacks or does it just allow you to attack squares up to 10 feet away?

There is some confusion because PHB p.147 says that a reach weapon adds 5 feet to your reach when you attack with it, which can be interpreted to mean that, if an enemy is 10 feet away on his turn, he isn’t actually in your reach (since you aren’t attacking with the weapon yet) so you don’t get an Opportunity Attack. Is this interpretation correct and you only threaten 5′ away, or does a reach weapon always increase your reach for the purpose of threatening distance?