Iago's betrayal is a central theme in Shakespeare's Othello, and, therefore, the play is full of lines portraying his betrayal. While we are limited to space, below are a few ideas of types of quotes to look for that help prove and demonstrate his betrayal.
One good type of quote to look for is anything relaying the reasons for his betrayal. Iago betrays Othello out of jealousy because Othello chose not to promote him to lieutenant, despite his merit, but instead chose another officer. Therefore, any quote explaining what Iago perceives as Othello's offense will help prove that Iago betrayed Othello by showing us the reasons why. For example, in his first speech, Iago argues that he knows he would be valuable as a lieutenant and should have been chosen:
I know my price, I am worth no worse a place.
But he, as loving his own pride and purposes
Evades them with a bombast circumstance
And, in conclusion, ... says he,
"I have already chosen my officer." (I.i.11-17)
The second type of quote would be any quote discussing Iago's plans for betrayal. In Act II, Iago observes Desdemona and Cassio flirting and concocts his plan to make Othello jealous. Not only that, he believes that Othello has slept with his own wife and therefore wants to avenge himself, not just for failing to promoting him to lieutenant but also for sleeping with Iago's wife, as we see him state:
And nothing can or shall content my soul
Till I am even'd with him, wife for wife;
Or failing so,, yet that I put the Moor
At least into a jealousy so strong
That judgement cannot cure. (II.i.306-310)
All of these types of lines will help convey and prove Iago's betrayal of Othello.