Question

———————- that Anne sees the broken vase , just tell her the dog into it

Answer 1

The correct answer is: in the event

In the event that Anne sees the broken vase, ...

This sentence means that if Anne sees the broken vase, tell her that the dog did it.

In order for the first option to be correct, in case should be followed by that or no preposition at all. For example:

In case that Anne sees the broken vase, ...

OR

In case Anne sees the broken vase, ...

Answer 3 should be provided, if we wanted it to be correct. For example:

Provided that Anne sees the broken vase, ...

Finally, answer 4 is wrong because whether cannot be followed by that. It should be:

Whether Anne sees the broken vase or not, ...