The order of annexation shall be entered in the minutes of the board and certified copies shall be filed with the secretary of state and with the county clerk and county auditor of each county in which the district or any part of it is situated.

From and after the date of the filing and recording of the certified copies of the order, the territory described in the order is a part of the district, with all the rights, privileges, and powers set forth in this act and those necessarily incident thereto.

After the annexation of territory to a district, the district board shall consist of the number of members and shall be appointed in the manner prescribed by this chapter for a district formed originally with boundaries embracing the annexed territory. However, the members of the district board in office at the time of the annexation shall continue to serve as members during the remainder of the terms for which they were appointed.