How can you turn a angry customer into a loyal customer ?

This is an easy one.  The key thing to remember is that it doesn't matter whether the customer is in the right or in the wrong, it's a good customer you want to keep.

1) Apologise, properly

2) Fix the problem completely

3) Give them a present

So if the widget was delivered late and was broken, then give them a new widget, all their money back, a genuine apology, all the addons or upgrades the widget can have (or a better widget) and an appropriate amount of money as compensation for their trouble.   (Remember that if the problem drags on it costs money to deal with it.)  Customer service at this level is so rare that the customer will tell everybody they know about how good you were in the end.

you need to satisfy his needs , and yo had to work on the source of this angry

Time is a major factor in this change

angry customer needs some extra time. we should get to know what he is angry about, tell him what situation has led you to this inconvenience... and make him/her realize you are doing your best to facilitate them...

Capacity to listen customers, Assurance about product and services, providing feel to customer that you are always with customers to solve problem related to product or services, making them to feel that you care for customers and you have patience to handle any type problems. Basically make customers to feel that you are creating a bonding with customers and your product or service is just a means of this bonding. These factors work to make customer loyal. But when you face an angry customer means you need a high level of patience to listen and art of customer handling that proves customer that he is listened and his problems taken care professionally. Then you will turn him loyal one. Presentation, service and support, these3 things makes difference in professional world.

A customer gets angry, when the expectation is not met. Or what has been advertised, is not what has been delivered. This gap in expectation and delivery is what  creates the customer to be dissatisfied. The angry customer can easily be changed into a loyal one, by simply filling in this gap. This process includes, listening to the problem carefully, apologizing, finding out the exact expectation of the customer,  taking action, and following up if the customer is customer is happy with the change of action/ plan.

For example, in hotels a honey moon couple has booked a certain type of room, but due to some reason it has not been able to be provided by the hotel. But if the Front desk agent/ Manager apologies, and explains them the situations and assures that they would be given a room requested as soon as its availability and provide them room when available next day  and even add an upgrade, along with something to surprise them like a bottle of champagne/ flowers/ chocolate/ breakfast . This would change their anger to loyalty. 

In my view, no one will be simply get angry for no reason, there may be some fault in providing service or in product, which was not dealt properly with the customer, made him angry. So surely there will be a wrong from one side, so the other party gets angry. only solution is be prompt and correct the wrong and provide sincere and prompt service, obviously make the other person happy and will be loyal for sure. The same is the case with the customer as well. If you provide sincere and prompt service without any latches at your end, surely the customer will be loyal and turn up again to make business with you. My2 cents. Thanks.

Turn what's negative about your company into a positive experience, but remember it's all about the equilibrium.