if you want to increase your community engagement you can connect you Twitter and Facebook profile by providing special application to show latest news from your Twitter profile.
I know there is a way to publish tweets directly in your timeline, but this is not available for company pages and it’s boring to read 50 tweets a day via Facebook timeline, so why don’t you help your fans to get what they need.
Add the application to your page’s menu:
show the info like this
Oh yes, it’s good idea to check if the current user likes your page or not and to show a different header like this: