Question

In order for a 25-gon to be regular, which of the following statements must be true?

Answer 1

A regular polygon is, by definition, a polygon with congruent interior angles and sides.  Thus, if we rotate n-gon about its centre, we will repeat the polygon n times before it comes back to its original position.

a. The interior angles are congruent to their exterior angles.
    This means that all angles are 90 degrees, so
true for a square or rectangle,but FALSE for a 25-gon.

b.All the diagonals are congruent.
   It is true for all regular polygons, but not sufficient to make the polygon  regular.
   Counter example:  A rectangle has all diagonals congruent, but it's not a regular quad.  So FALSE.

c.Opposite pairs of sides are parallel.
   All even-number sided regular polygons will satisfy this property, but not sufficient to make a polygon regular.  Counter example:  A rectangle with different adjacent sides.  So FALSE.
In the case of a 25-gon, there are no "opposite" sides.

d. All interior angles and all sides are congruent.
This will undoubtedly create a regular polygon, since it satisfies the definition of a regular polygon.  If we rotate it about its centre, we can repeat the same 25-gon 25 times.  So TRUE