There may be two broad motivations for leaving an angry emoji; either they are angry at the article or the article made them angry. However categorizing these two groups as "liberal" or "their base" isn't justified. It may be the case that somebody, perhaps a conservative, is angry at Fox News for a particular spin or editorialization but generally likes other Fox News articles. It may also be the case that somebody who considers themselves something other than "liberal" or "Fox News' base" finds a Fox News article in their facebook stream and is either angry at Fox News or because of Fox News. It's also possible, although perhaps less likely, that a liberal might become angry not at Fox News but because of what Fox News is reporting. They might not be amused to see Fox News in their stream, but it's certainly possible that the substance of the story has them angered even more.
To give an example, consider this current Fox News headline:
> New Zealand shooting's a social media wake-up call -- YouTube, others, must stop amplifying violent crimes
[A 'fox news conservative' sees that headline and leaves an angry emoji.] Is it because he's angry about the shooting? Or angry at Youtube/etc? Or angry because he thinks Fox news is cynically exploiting the tragedy to take pot shots at a tech company?
[A liberal sees that headline and leaves and angry emoji.] Is it because he's angry that Fox News is making it into his news feed? Or because he's angry at the shooting? Or angry at youtube/etc? Or angry because he thinks Fox news is cynically exploiting the tragedy to take pot shots at a tech company?
You really can't say. Any particular individual may have their own reason for posting an angry emoji at a headline like that. For a headline like that, it's quite possible for a liberal and a fox news conservative to have the exact same reason for leaving an angry emoji.