Take part or become involved

• Deal with or interact with
• Draw someone into conversation

Attract and hold attention

• Interest deeply or engross
• Keep someone busy or occupied

Hire or contract for services

• Arrange the use of something
• Book a place or resource

Enter into combat

• Engage an enemy or opposing force

Promise or commit to do something

• Bind by agreement or obligation
• Be betrothed to marry (passive)

Of parts, to mesh or interlock

• Put a mechanism into operation, as a clutch or brake