Final answer:
The thermocouple is the safety feature that shuts down the furnace by detecting a lack of heat from an extinguished pilot light, causing the gas valve to close.
Step-by-step explanation:
The safety feature in a furnace that shuts down the system when it detects an issue is the thermocouple. The thermocouple is responsible for monitoring the pilot light. If the pilot light goes out, indicating a potential hazard such as gas leaks, the thermocouple will no longer detect heat from the flame. This causes the thermocouple to trigger the gas valve to close, thereby shutting off the gas supply to the furnace to prevent any dangerous situation. The thermostat typically controls the on/off cycles of the furnace based on temperature settings, but it is not the safety feature that shuts down the furnace in the case of the pilot light going out.