POLITE – Try to appreciate that the "angry client" is likely at that state, not because of you personally, but rather, they are more likely to be angry because of repeated frustrations that they have encountered. A combination of things have not gone the way they anticipated. It will be helpful to remember that causing further agitate for the client is not going to improve the situation. It will be important to listen to the client, recognize what they have experienced, reassure them that you want to help them find a solution, or next step (as may be appropriate).
POSITIVE – While it may be tempting to argue with an "angry client", a more successful outcome is likely to be achieved through positioning yourself as someone who wants to help the client find a satisfactory solution. In a very real sense, we should be thankful that they have brought the problem to our attention. Take the approach that you and your client are on the same team - that by working together, a better solution will be found for your clients, today, and well into the future!