Gemma says that a regular polygon cannot have an exterior angle of 25 degrees. Is she correct? Explain your answewr.
Yes. 360 is not divisible by 25.
http://regentsprep.org/Regents/math/geometry/GG3/LPoly3.htm
To determine whether Gemma is correct in saying that a regular polygon cannot have an exterior angle of 25 degrees, we should first understand the concept of a regular polygon and its properties.
A regular polygon is a shape with equal side lengths and equal interior angles. Its interior angles can be found using the formula:
Interior angle = (n - 2) * 180 / n
where "n" represents the number of sides of the polygon. For example, a triangle (n=3) has interior angles of 60 degrees, a square (n=4) has interior angles of 90 degrees, and so on.
Now, an exterior angle of a polygon is formed by extending one of the sides of the polygon outward, creating an angle that lies outside the polygon. The exterior angles of any polygon sum up to 360 degrees.
The relationship between the interior and exterior angles of a polygon is given by the equation:
Exterior angle + Interior angle = 180 degrees
From this equation, we can deduce that the sum of the interior and exterior angles of any polygon is always 180 degrees, as they form a linear pair.
Now, let's consider Gemma's statement that an exterior angle of 25 degrees cannot exist in a regular polygon.
Since the sum of the interior and exterior angles of any polygon is always 180 degrees, we can say:
Interior angle + 25 degrees = 180 degrees
By rearranging this equation, we get:
Interior angle = 180 degrees - 25 degrees
Interior angle = 155 degrees
This means that the corresponding interior angle of the regular polygon in question would have to be 155 degrees.
To verify if this is possible, we'll apply the formula mentioned earlier:
Interior angle = (n - 2) * 180 / n
By substituting 155 degrees into the formula, we have:
155 = (n - 2) * 180 / n
Cross-multiplying, we get:
155n = (n - 2) * 180
Expanding this equation further, we have:
155n = 180n - 360
Combining like terms, we find:
180n - 155n = 360
25n = 360
Finally, by dividing both sides of the equation by 25, we can solve for n:
n = 360 / 25
n ≈ 14.4
This result indicates that n is approximately equal to 14.4, which is not a whole number. However, since a polygon must have a whole number of sides, we can conclude that there is no regular polygon with an exterior angle of 25 degrees.
Therefore, Gemma is correct in stating that a regular polygon cannot have an exterior angle of 25 degrees based on the properties of regular polygons and their interior-exterior angle relationships.