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A bridle is a piece of equipment used to direct a horse. As defined in the Oxford English Dictionary, the "bridle" includes both the headstall that holds a bit that goes in the mouth of a horse, and the reins that are attached to the bit.
Headgear without a bit that uses a noseband to control a horse is called a hackamore, or, in some areas, a bitless bridle. There are many different designs with many different name variations, but all use a noseband that is designed to exert pressure on sensitive areas of the animal's face to provide direction and control.