What is this weed???

For the life of me, I cannot figure out what this weed is.

It is extremely invasive in my garden, with very very long roots that creep line a vine throughout the garden. I feel like the only way to get rid of it, is to sell my house. lol I also notice that it irritates me - my body is so sore after touching it, and my lungs feel weaker. I have attached the front and back view of it. PS - this is in Ontario. 

Please help!