It's all about trust at the end of the day. The easiest is if you have a position of trust already. I had an older brother, and while he was a dick a lot, he also looked out for me, and I could tell when he was being serious about something. He also loved science, and he wanted me to love science too. So I studied hard and learned how it works, and now I think he was right.

He probably saved me, since I myself have always found more rebellious positions very appealing. But once you learn what the scientific method is, how it works and why it was needed in the first place, that sticks with you. It's a method, not a belief, and just like your method for tying your shoes or riding a bike, it just sticks, because it works.

If you don't have a position of trust, you have to lean on things that don't require it. Clean rationality/logic can be used, and make sure to very thoroughly go step by step. If A, then B. If B, then C. If C, then D. Don't assume any like you're talking with a peer, instead go extremely by-the-book.

If they're religious, you can use bible quotes. It does just so happen that Jesus himself was a rather opinionated fellow and left us a lot of material to work with. The man was even downright reasonable sometimes.

You can also use history, if you're strong at it. It gives numerous routes for dealing with these arguments.

None will work reliably, but I think it's a numbers game, where we just try to save what we can. First and foremost though, if you have a younger sibling, please be a good older brother/sister. They're probably not ready for this messy of a world.