The House of Plantagenet

The last monarch of the House of Plantagenet was actually Richard III.

The name Plantagenet originated as a nickname for Geoffrey, Count of Anjou (1113–1151). He married the 'Empress' Matilda and was the father of Henry II, who thus became the first monarch in what became known retrospectively as the House of Plantagenet. Henry was the father of kings Richard I and John; these three were known at the time as the Angevin kings, and the dynasty as the House of Anjou.

The first person to use Plantagenet as his family name was Richard of York, the third Duke of York (1411–1460). He was descended from King Edward III through both his father and his mother. It is not known why he adopted this name, but (as Wikipedia points out) during the Wars of the Roses it would have emphasised his status as a patrilineal descendant of Geoffrey of Anjou.

It was during the Tudor era that the retrospective use of the name for all of Geoffrey's male–line descendants became popular – perhaps encouraged by the further legitimacy it gave to Richard's great–grandson, Henry VIII. It was only in the late 17th century that it passed into common usage among historians.

What the question setter has failed to understand is that the houses of Lancaster and York were 'cadet branches' of the Plantagenet dynasty – that is, male–line descendants of younger sons (of Edward III). As Wikipedia explains: "the reign of the Plantagenets and the English Middle Ages both met their end with the death of King Richard III. Henry VII, of Lancastrian descent, became king of England; five months later, he married Elizabeth of York, thus ending the Wars of the Roses, and giving rise to the Tudor dynasty."