There is a well-known proverb found in the book of Proverbs which seems to contradict itself, about how to answer a fool.  It is Proverbs 26:4-5.

Pro 26:4  Do not answer a fool according to his folly, Lest you also be like him.
Pro 26:5  Answer a fool according to his folly, Lest he be wise in his own eyes.

Rather than contradict each other these two verses complement each other.  Perhaps the best way to explain this is through an example.

If a child says something rude to a parent such as, “smelly face”, how should the parent respond?  In this case the child by its very nature is speaking foolishly.  If the parent responds back with something similar, such as, “stinky face”, then the parent is answering the child according to the child’s folly and is behaving foolishly like the child.  If the parent says nothing to the child, then child “wins”, and thinks it’s smarter than the parent. 

What is the solution?  The parent has to answer the child, but appropriately.  The parent needs to address the issue and say something like, “That’s not nice.  We don’t say things like that.”

With this example in mind, the verses can be paraphrased to help elucidate their meaning.

Don’t answer a on the same level as a fool, lest you become foolish too.
Answer the fool by addressing the issue, lest he thinks himself wise.

There is a good example of this in how God answered Job.  Job felt that he was righteous and that God had treated him unjustly. 

Job 19:6  Know then that God has wronged me, And has surrounded me with His net.

Job 34:5  “For Job has said, ‘I am righteous, But God has taken away my justice;

This is a very understandable human reaction; however, it is nevertheless foolish when you consider that God created humanity and that God’s thoughts are so much higher than man’s thoughts.  God responded, but not on the same level.  God addressed the issue.

Job 40:8  “Would you indeed annul My judgment? Would you condemn Me that you may be justified?

God then proceeded to ask Job a lot of questions which made Job realise he had no idea how the physical creation worked and that God’s wisdom far exceeded his wisdom.  In other words, God’s response made Job realise that God knows what He’s doing.

These two verses do not contradict each other but complement each other.  They instruct us to not to respond foolishly to a foolishness but to address the underlying issue.