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He happened to glance over the passage before class, the only portion of last night’s assigned reading that he’d actually looked at.

Should this sentence be "He had happened to glance over" or "He happened to have glanced over"? Are both of those correct? Does it even need to be changed at all?

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He happened to glance is simple past: we use this to talk about something completed in the past. The word before places this activity before the class. The sentence is grammatically correct, but probably not the best.

He had happened to glance is past perfect: we use this to talk about something completed in the past that took place before some other event in the past (for example before class). Because the word before specifies when the action took place, it is not necessary to use past perfect.

He happened to have glanced is present perfect: we use this to talk about something completed in the past which has some lasting effect, or some thing that started in the past and is still happening. Having glanced at this part of the text, he later, in class had some idea what it was about- that's a lasting effect. This is probably the best option of the three.

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