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Which of the following statements is false?

A An octagon has $20$ diagonals

B A hexagon has $9$ diagonals

C A hexagon has $4$ more diagonals than a pentagon

D A pentagon has the same number of diagonals as it has sides

E A quadrilateral has twice as many diagonals as it has sides

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

*This problem is taken from the UKMT Mathematical Challenges.*