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The type of valve that is normally closed and only allows flow in one direction is the check valve. Check valves are specifically designed to prevent backflow in a piping system. They automatically open to allow flow in the designated direction when sufficient pressure is applied, and they close when the flow reverses, preventing potential damage or contamination to the system.

In contrast, a ball valve can be either open or closed depending on the position of the ball inside; it is used to control flow but is not defined as normally closed without additional specification. A gate valve, primarily used for on/off service, does not inherently prevent backflow. A back check valve is less common and may refer to a type of check valve with additional features but is not the standard term used to describe valves that allow flow in one direction while remaining normally closed.

Thus, the check valve's design directly aligns with the requirement for a valve that is normally closed and restricts flow to one direction.