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I think you should generally have the idea that single bonds are longer than double bonds which are longer than triple bonds and the ionic/covalent bond trend for the periodic table. For exact numbers, I highly doubt he would have us memorize exact numbers unless he specifically tells us to.
The main things you need to know are that single bonds are longer than double bonds and double bonds are longer than triple bonds. For resonance structures, bond length will be an average of the possible bonds within the resonance structures of the molecule.
I think all you need to do is have a general understanding of bond lengths and what affects them. For example, in lecture Dr. Lavelle talked about how larger atoms have longer bond lengths (due to electron shielding etc.) and single bonds are generally longer than double bonds which are longer than triple bonds etc. So using that general knowledge, you should be able to explain why certain bonds are the lengths that they are.
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