In order for a 25-gon to be regular, which of the following statements must be true? a. the interior angles are congruent to their exterior angles. b. all the diagonals are congruent. c. opposite pairs of sides are parallel. d. all interior angles and all sides are congruent.

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

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