Is the page (SPA) coming from the same server the API's are hosted? (same domain?) If that's the case, then everything should "just work".
WAAD is not involved in knowing whether the user is logged on or not. That should generally be up to your site. Token expiration does play a role though, if the supplied token is rejected by the API you would have to re-authenticate the user as usual.
If the APIs are on a different site, but same hi-level domain (app.yoursite.com and api.yoursite.com), you can still set cookies and use WIF underneath. Make sure you set the same machine key on both sites.
If the APIs are on a completely different domain, then you have to use a different approach. My recommendation in that case is usually to use Json Web Tokens (JWT) and CORS.
I think WAAD can give you a JWT if you configure an ACS namespace in your WAAD instance. Take a look at this document that describes the process for securing WebAPI with JWTs (it is not specific to WAAD, but the mechanics would be the same).