Question

A polygon has two pairs of complementary interior angles in three sets of supplementary interior angles. The sum of the remaining interior angles is 1440°. How many sides does a polygon have? Explain 

Answer 1

To determine the number of sides of the polygon, we calculate the total sum of its interior angles using the given information about complementary and supplementary angles, and the sum of the remaining angles. After finding the total sum as 2340 degrees, we use the formula for interior angles of a polygon to deduce that the polygon has 15 sides.

Step-by-step explanation:

To find out how many sides a polygon has based on the given information, we first need to understand some basic properties of polygons and their interior angles. The sum of the interior angles of any polygon with n sides is given by the formula (n-2) × 180 degrees. Knowing that we have two pairs of complementary angles, which sums up to 180 degrees for each pair, and three sets of supplementary angles, which also sum up to 180 degrees for each set, we can begin to solve the problem.

Since the remaining interior angles sum up to 1440 degrees, we first add back the excluded angles (2 pairs of complementary and 3 sets of supplementary):

1440 degrees + (2 × 180 degrees) + (3 × 180 degrees) = 1440 degrees + 360 degrees + 540 degrees = 2340 degrees.

Now, we use the formula for the sum of interior angles of a polygon to find the number of sides (n):
2340 degrees = (n - 2) × 180 degrees

n - 2 = 2340 degrees / 180 degrees

n - 2 = 13

n = 13 + 2

n = 15

Therefore, the polygon has 15 sides.

Answer 2

A polygon with n sides has a total of (n-2) sets of interior angles, each with measure equal to 180(n-2)/n degrees.

Since two pairs of complementary interior angles in a polygon add up to 90 degrees each, the sum of these two pairs is 2 * 90 = 180 degrees.

Since three sets of supplementary interior angles in a polygon add up to 180 degrees each, the sum of these three sets is 3 * 180 = 540 degrees.

The sum of all interior angles in a polygon is (n-2) * 180 degrees.

Putting this all together, we have:

(n-2) * 180 = 180 + 540 + 1440

Expanding and simplifying the right-hand side, we get:

(n-2) * 180 = 1860

Dividing both sides by 180, we get:

n-2 = 10

Adding 2 to both sides, we get:

n = 12

So the polygon has 12 sides