I was given this exercise but unlike the other questions, this didn't have the title of the song in it. I thought it was just something I had to analyze and try to put chords that fit with the melody. Apparently I was correct, except for the 75th measure. I'm not sure what chords to put there. I tried V-I (E - A) but it preempts the ending. Unless there are slash chords here? How would you know what chords would work?
The "how would you know" part comes with more study and especially with practical experience. In this case, use V and VI — a deceptive cadence.
You're correct to observe that the melody suggests a cadential quality and also that V - I undermines the actual ending. That is something of the "purpose" of a deceptive cadence — a "fake out" ending before the "real" one.
For starters, there's rarely one way in which to harmonise a melody. That's been proved thousands of times. Right now, I'm playing Summertime with four different bands, all with very different chords underneath !
Without getting into PACs etc., it makes sense that the chord tones and the melody lines match up. Obviously not totally, that would maybe mean a chord change every note, but by and large. Let's face it - if the melody doesn't match the chords, or the chords don't match the melody, the piece just won't sound right.
Since in 4/4 time, usually the 1st and 3rd beats are the predominant ones, that's a good place to start considerations. As an example, the 1st and 3rd beats of bar 70 are C♯ and A - both important components of the A major triad. So A would fit happily there. However, the B and G♯ in that bar (beat 2) fit better to an E chord - quite o.k., as that's the V of A. So either A or E will fit on the 2nd beat.
You ask what might fit in bar 75. I'm not certain whether that is an allowable question here! But, bearing in mind what I've already said, perhaps you can come up with a solution. There's one obvious, one not so, at the level of this question on the sheet, and some others that work nicely in a jazz situation, perhaps inappropriate here.
For the how-do-you-know part -- I was never good at this stuff so I used the brute force method. Here's what you do. You look at the notes that are sounding on that beat and check if they are part of some triad. There might be two or three choices. At that point you can narrow things down by playing it and checking if one or the other sounds better in the context. If it's a very pedagogical exercise, as yours seems to be, then you can pick something that seems similar to the examples and exercises you've been doing for your class.
But you might have to watch out for appoggiaturas, passing tones, etc.
There's often no 'right answer to this sort of question'. V - I would be fine. There's nothing wrong with 'pre-empting the ending'. (Call it 'emphasizing the ending' if it makes you feel better!) V - VI would be another option - and I suspect that's the one your teacher is looking for. It would complete the exercise with one of each of the four main cadence types.