Transfer WordPress Login Session to an Extended Webpage on the Same Domain

First of all include wp-load.php from WP root installation to those PHP files which you want to display and once you have included to your PHP files then you can use WP functions. You can use is_user_logged_in() function to check if user is logged in or not.

// Your PHP file that can be viewed only to logged in user
require_once '/path/to/wordpress/wp-load.php';
if ( is_user_logged_in() ) {
    // your php file code...

}

And same you can do for HTML by converting them to PHP files and include those above line of code.