1. John's father caught a big fish at the lake.

2. John's father caught a big fish in the lake.

3. John's father caught a big fish on the lake.

4. John's father caught a big fish near the lake.

5. John's father caught a big fish by the lake.
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Which preposition should we use?

I'd use "at" and possibly "on" if he were fishing from a boat.

The word "in" would work if he were standing in the lake, probably with waders on.
http://www.mychamplain.net/sites/default/files/fly_fishing%202.jpg

To determine the correct preposition to use in a sentence, it is helpful to understand the meaning and usage of each preposition in context. Let's analyze the given options:

1. "at the lake" suggests that John's father caught the fish while physically outside the lake, potentially near the shore or on the banks, but not necessarily in the water.

2. "in the lake" implies that John's father caught the fish while submerged in the water of the lake itself, likely using some form of fishing equipment, such as a fishing rod.

3. "on the lake" typically denotes that John's father caught the fish while on a boat or some other floating object on the surface of the lake.

4. "near the lake" indicates that John's father caught the fish in close proximity to the lake, possibly in an area adjacent to the lake but not specifically in the water.

5. "by the lake" implies that John's father caught the fish close to the lake, potentially near the shoreline, using a variety of fishing methods.

Ultimately, the correct preposition to use depends on the intended meaning and the specific context of the situation.