Buying car off craigslist. Is this a good deal? Are there any other safety...

...measures we should take? My Husband and I have already made an offer on a 05' Toyota Corolla today from an ad that was posted on craigslist. We saw the car yesterday, and met the family who's selling it. Apparently, the parents are selling the car because their college-aged daughter did not keep up her end of the bargain with the payments. The car is in great shape--50K miles, it's been serviced last week by a dealership mechanic, and checked to be in great working order, grey in color, and clean. It does need new tires. Upon checking the carfax report, which the family encouraged us to do, it said that there was only one minor rear impact accident in 05', but the car was still driveable after. The car was used as a rental for the first year of its life, and then it was sent to a dealership, where this family bought the car for their daughter. They were asking $9,000 for the car, but they agreed for us to only pay them $8,000.

IS THIS A GOOD DEAL?? EVERYTHING SOUND LEGIT? Are there other safety measures we should take to make sure this transaction won't bite us in the butt later on? I've never bought a car from a private party. Honestly, this is the best deal we've found on this car. The dealerships were selling even older cars that were way more $.

Thanks for your responses!

CarFax reports are not accurate.