Search Results: liking a girl with a boyfriend

So, I was out eating with my boyfriend & he was trying to show my something on his phone when suddenly a snapchat message appears from (i'll change the name for anon purposes) Marla. He swipes it away quickly, and I ask, "who was that, whats her name?". He says just some chick, and then he tries changing the subject really fast. We get into the car, and I ask again, "so how do you know eachother, AND WHATS HER NAME?" He insists its just some girl he knows (and that he ONLY has her on snap), and after a while he says her name is Melissa. A lie. I let it go, but the next day I look up the name I saw (Marla) on facebook & instagram. I see he's consistently liking pics, including bikini pics. I ask him again saying, "you obviously know her, and you lied about her name, WHO IS SHE?" he says he has a class with her, so he talks with her through only snap (which is sketch, cus everything dissapears.
Okay, so skip ahead a bit, we're at his house and we took a nap, but I wake up first. I see he had a message from Marla through snap and text, and I know it was wrong, but I was anxious, so I look through his phone. I find out they had been going out a LOT (which he never mentioned), he would call her adorable, offer to buy her food a lot, and that they were planning on taking more classes together next semester. I got mad, because he is the most JEALOUS person I have ever been with, yet he's getting THIS close with another girl.
I confront him about it and get mad that I don't trust him. We almost broke up over it.
After all of this, he still insists on liking her photos and STILL ended up taking another class with her.
I just feel disrespected. How would you all feel?