Question

select the statement that is the answer of the following statement: if a polygon is equiangular, then all its exterior angles (measured in degrees) must measure... answer options include: if a polygon’s exterior angles don’t have the same measure, then it isn’t equiangular; a polygon isn’t equiangular, then all its exterior angles don’t have the same measure; if all a polygon’s exterior angles have the same measure, then it is equiangular; if all a polygon’s exterior angles don’t have the same measure, then it isn’t equiangular (paraphrased from deltamath problem on polygon exterior angles and equiangularity)

Explanation:

To solve this, we analyze the concept of a regular polygon and its exterior angles:

Key Concept:

In a regular polygon, all exterior angles (when one is drawn at each vertex) are equal, and their sum is always \( 360^\circ \). For a regular polygon with \( n \) sides, each exterior angle measures \( \frac{360^\circ}{n} \), so they are congruent (equal in measure).

Analyzing the Options:

• Option 1: "If a polygon’s exterior angles don’t have the same measure, then it isn’t regular."

This is the contrapositive of the original statement ("If a polygon is regular, then all its exterior angles have the same measure"). The contrapositive of a true statement is always true. If a regular polygon requires all exterior angles to be equal, then a polygon with unequal exterior angles cannot be regular.

• Option 2: "If a polygon isn’t regular, then all its exterior angles don’t have the same measure."

This is false. A non - regular polygon could still have exterior angles with the same measure (e.g., a rectangle is regular, but a parallelogram with equal - length sides but non - right angles is non - regular, yet its exterior angles still sum to \( 360^\circ \) and can be equal in some cases).

• Option 3: "If all a polygon’s exterior angles have the same measure, then it is regular."

This is false. For example, a rectangle (a regular quadrilateral) has exterior angles of \( 90^\circ \), but a non - regular quadrilateral (like a parallelogram with equal - length sides and \( 60^\circ \) and \( 120^\circ \) interior angles) has exterior angles of \( 120^\circ \) and \( 60^\circ \), but a different non - regular polygon (e.g., a rhombus) has equal exterior angles but is regular? No, a rhombus is regular in terms of side lengths but not necessarily angles? Wait, no, a rhombus has equal - length sides and its exterior angles are equal (since interior angles are equal). Wait, actually, the correct counterexample: consider a convex polygon with 4 sides where all exterior angles are \( 90^\circ \), but it’s a rectangle (regular) or a square (regular). Wait, maybe a better example: a regular polygon has both equal sides and equal angles. A polygon with equal exterior angles (hence equal interior angles) but unequal side lengths would be non - regular but have equal exterior angles. Wait, no—if interior angles are equal, by the formula for interior angles \( I=(n - 2)\times180^\circ/n \), if \( I \) is equal, \( n \) is fixed, and for a polygon with \( n \) sides, if interior angles are equal, the polygon is equiangular. A regular polygon is both equilateral and equiangular. So a polygon that is equiangular (equal exterior angles) but not equilateral is non - regular but has equal exterior angles. For example, a rectangle with length 4 and width 5: interior angles are \( 90^\circ \) (so exterior angles \( 90^\circ \)), but sides are not equal, so it’s non - regular. Thus, equal exterior angles do not guarantee a regular polygon.

• Option 4: "If all a polygon’s exterior angles don’t have the same measure, then it isn’t regular."

This is the same as Option 1? Wait, no—Option 1 is "If not (all exterior angles same), then not (regular)". Option 4 is also that? Wait, maybe a typo in the options, but from the analysis, the contrapositive (Option 1) is the correct logical equivalent.

Answer:

The statement that is the answer (logical equivalent) is: "If a polygon’s exterior angles don’t have the same measure, then it isn’t regular" (the first option among the given choices for the answer).