A 'vicarage' is a specific type of residence for Christian clergy, particularly in the Anglican Church. The new vicar moved into the vicarage last week.
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British Term
'Vicarage' is mainly used in British English, and less common in American English where 'parsonage' or 'rectory' might be used. We'll have tea at the vicarage after the service.
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Proximity to Church
Often, the vicarage is located close to the church to allow easy access for the vicar. The vicarage is just behind the churchyard.