The interior angle of a regular polygon is five times the size of its exterior angle. Identify the polygon.

To identify the polygon, we need to understand the relationship between the interior and exterior angles of a regular polygon.

In a regular polygon, all interior angles are equal and all exterior angles are also equal. Let's assume that the measure of the exterior angle is x degrees.

According to the given information, the interior angle is five times the size of the exterior angle. So, the measure of the interior angle would be 5x degrees.

In any polygon, the sum of the interior angles is given by the formula (n-2) * 180 degrees, where n is the number of sides of the polygon.

Using this formula, we can write:

(n-2) * 180 = Sum of the Interior Angles

Since all interior angles of a regular polygon are equal, the sum of the interior angles would be the product of the number of sides and the measure of each interior angle.

n * (5x) = Sum of the Interior Angles

Setting these two equations equal to each other, we have:

(n-2) * 180 = n * (5x)

Now, we need to find the value of n, the number of sides of the polygon.

Simplifying the equation:

180n - 360 = 5nx

Dividing both sides by 5x:

180n / (5x) - 360 / (5x) = n

Simplifying further:

36n / x - 72 / x = n

Combining like terms:

(36n - 72) / x = n

Cross-multiplying:

(36n - 72) = nx

Dividing both sides by n:

36 - 72/n = x

From the above equation, we can deduce that the value of the exterior angle x depends on the number of sides n. For a regular polygon, x should be a whole number.

By examining the answer choices, we can check if any of them result in a whole number for x. Calculating the value of x for each option:

: Dodecagon

n = 12 (a dodecagon has 12 sides)

x = 36 - 72/12 = 36 - 6 = 30 degrees

: Enneadecagon

n = 19 (an enneadecagon has 19 sides)

x = 36 - 72/19 = 36 - 3.79 ≈ 32.21 degrees

: Icosagon

n = 20 (an icosagon has 20 sides)

x = 36 - 72/20 = 36 - 3.6 = 32 degrees

: Hendecagon

n = 11 (a hendecagon has 11 sides)

x = 36 - 72/11 ≈ 29.45 degrees

From the calculations above, we can see that only (Dodecagon) results in a whole number for the measure of the exterior angle (x = 30 degrees). Therefore, the polygon in question is a dodecagon.