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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.

Respuesta :

mathmate
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

In order for a 25-gon to be regular, true statement is given by all interior angles and all sides are congruent.

What is regular polygon?

" Regular polygon is defined as the 2 dimensional closed figure joining by three or more lines , with all the sides and angles equal."

According to the question,

For 25 - gon regular polygon

a.  The interior angles are congruent to their exterior angles. : If interior angles are congruent to exterior each angle is of measure 90°, which is true in square and rectangle not for all regular polygon.

Given statement is false for 25-gon regular polygon.

b. All the diagonals are congruent: It is true for all regular polygon, but vice versa is not true if diagonal are congruent it is a regular polygon.

example : Rectangle

It is not a true statement.

c. Opposite pairs of sides are parallel.

This statement is true only for regular polygon with even number of sides. In odd number of sides there is no opposite pair of sides.

25 is an odd number.

It is not a true statement.

d. All interior angles and all sides are congruent.

In regular polygon all sides are congruent and interior angles are also congruent.

Hence, Option(D) is the correct answer.

Learn more about regular polygon here

https://brainly.com/question/1217245

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