According to what I see in your question, it appears that you are right, and the lecture slide is wrong. It is correct that the area between the mean and z(0.9) is 0.3159, and the area beyond z(0.9) = 0.1841. That is the answer that you got, and, as long as there is not more to the question, that is where you stop. What they did on the slide next doesn't make sense as an answer to the question P(X>45). There is no reason to add 0.5 to the 0.1841 unless the question is the probability of randomly selecting a value greater than 45 or less than 36! As I said earlier, it appears that you are correct and the lecture slide wrong, unless I'm missing something.
