He kissed her when he was drunk, but...

My partner recently got very drunk at a party and began kissing someone else. I caught them both and he apologised the next day. He says he doesn’t remember what happened and wants me to forgive him, but I don’t know what to do. He did this under the influence of alcohol, and I accept that. I have also never had any reason to doubt him, but I worry that this may happen again when I am not around. If I can’t trust him, how am I supposed to be in a relationship with him?
If you can’t trust him, end this relationship. If you think you can, and this was a slip, you can take some time to yourself and decide to forgive him. If you do, however, it has to be unconditional because without that, this incident will keep cropping up and preventing you both from moving on. Acknowledge that what he did hurt you and reflect on whether his apology is enough. If he is genuinely upset about what happened, and takes steps to ensure it won’t happen again, you can judge him by his actions instead of his words. If you believe a compromise may work, ask him to avoid drinking alcohol if he can’t handle it, or not drink if you aren’t around. This is a painful incident, but it needn’t end your relationship if you are both committed to fixing it.

I found out that my boyfriend deliberately lied about something a few months ago. When I confronted him, he said I was making a big deal out of nothing. I am angry because he thinks it’s okay to lie, which is why I want to break up with him. Am I being unreasonable?
It’s not about being reasonable as much as it is about respect. If he believes lying is not a problem, he doesn’t want a relationship that is truthful. Without that, your future with him is doomed even if you choose to forgive him. He needs to understand that.