How can template identify what page it's being used on?

How can template identify what page it's being used on?

I have a template that is used for many different sections of my site
(About, News, etc.). The nav bar needs to show a little graphic arrow
below the section you're in. I was passing this via the template:
$page = 'about';
require('header.php');
...but then I realized that won't work since the template file I was
editing is used for a bunch of different sections, not just 'about'.
So I need something on the 'page' interface screen where I can somehow
indicate the page that the user is on. Some ideas for solutions:
Idea #1: does WordPress would set some kind of class on the 'body' tag to
indicate the page you're on (like if it generated a slug based on the page
name and added that as a class to the body tag)? Then just do everything I
need to do in CSS without complicating things.
Idea #2: some kind of built-in field on WordPress's 'page' admin that
would let my header know what page it is on.
Idea #3: I could make copies of my template for each major section. Very
stupid way to go about it.
I would REALLY prefer to not go the whole extra-fields route, manually
adding a metadata field to my pages, pulling it from the db on page load,
etc. That seems like super overkill for just having a way to let my
template code identify what page it is being used on.
Would love any tips.