I think pets (Dogs especially) become more than just pets. They become a family member. They are there to greet you when you come home or sit by the window and watch you the whole time as you leave.
When that day comes when there is no alternative and you have to agree to put down your dog or whatever pet for that matter, there whole life with you flashes right before your eyes. You think about all the good times and all the bad times but usually laugh about the bad times like when they ate your new shoes and how mad you were then but how trivial it is now.....
The pain of losing them is so intense at first but trust me, time does heal all wounds. Good luck to you and your family.