Answer:
1.
a. 720 degrees.
b. 3780 degrees.
2. decagon
3. 360 degrees
Explanation:
1.
a. The sum of the measures of the interior angles of a hexagon can be found using the formula:
Sum of interior angles = (n-2) x 180 degrees
where n is the number of sides of the polygon. For a hexagon, n = 6, so we have:
Sum of interior angles = (6-2) x 180 degrees = 4 x 180 degrees = 720 degrees
Therefore, the sum of the measures of the interior angles of a hexagon is 720 degrees.
b. The sum of the measures of the interior angles of a 23-gon can be found using the same formula:
Sum of interior angles = (n-2) x 180 degrees
where n is the number of sides of the polygon. For a 23-gon, n = 23, so we have:
Sum of interior angles = (23-2) x 180 degrees = 21 x 180 degrees = 3780 degrees
Therefore, the sum of the measures of the interior angles of a 23-gon is 3780 degrees.
2.
The sum of the measures of the interior angles of a convex polygon with n sides can be found using the formula:
Sum of interior angles = (n-2) x 180 degrees
We are given that the sum of the measures of the interior angles is 1440 degrees. Substituting this value into the formula, we have:
1440 = (n-2) x 180 degrees
Simplifying and solving for n, we get:
n-2 = 1440/180 = 8
n = 8+2 = 10
Therefore, the polygon has 10 sides, and it is a decagon.
3.
The sum of the measures of the exterior angles of any polygon is always 360 degrees.
For a polygon with 32 sides, the sum of the exterior angles can be found using the same formula:
Sum of exterior angles = 360 degrees
Divided by the number of sides in the polygon, which is 32:
Sum of exterior angles = 360 degrees / 32 = 11.25 degrees
Therefore, the sum of the measures of the exterior angles of a 32-gon is 360 degrees, and each exterior angle of a regular 32-gon measures 11.25 degrees.