SOLUTION: Sarah has some large, polygon-shaped cardboard cutouts that she needs to sort, because she wants to hang them in her geometry classroom. The first division she needs to make is to

Question 1118226: Sarah has some large, polygon-shaped cardboard cutouts that she needs to sort, because she wants to hang them in her geometry classroom. The first division she needs to make is to divide the regular polygons from the irregular ones. In order to save time measuring all angles and sides of each polygon, and owing to the size and number of sides/angles in each shape, Sarah decides to measure just one angle in each polygon. Consider the following angle measures for polygons A through T, and answer the following questions.
A B C D E F G H I J
10 15 20 30 40 45 50 60 72 75
K L M N O P Q R S T
80 90 100 108 120 144 150 210 240 360
a. If each angle measure represented an measure of its polygon, which of the polygons could be regular? Explain your answers.
b. If each angle measure represented an measure of a regular polygon, name each polygon by using its number of sides. (For any figure with more than 12 sides, just use the number: e.g. 45-gon.)
c. If each angle measure represented an measure of its polygon, which of the polygons could be regular? Explain your answers.
d. If each angle measure represented an measure of its polygon, which of the polygons do we know are convex? Explain your answers.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The sum of the exterior angles of any polygon is 360 degrees. So to find the measure of an exterior angle of a regular n-gon, divide 360 by n. Then the interior angle is 180 minus the measure of the exterior angle.

To see if a given integer angle measure x can be the measure of an interior angle of a polygon, subtract x from 180 and see if the result is a factor of 360.

We can quickly rule out many of the given angle measures by finding the interior angle measure of a regular n-gon for small values of n:

triangle (n=3): exterior angle 360/3 = 120; interior angle 180-120 = 60
square (n=4): exterior angle 360/4 90; interior angle 180=90 = 90
pentagon (n=5): exterior angle 360/5 = 72; interior angle 180-72 = 108
hexagon (n=6): exterior angle 360/6 = 60; interior angle 180-60 = 120

So the four smallest interior angle measures for regular polygons are 60, 90, 108, and 120. That rules out all the other numbers shown that are less than 120.

It should be obvious that an interior angle of a regular polygon can't be 180 degrees or more. If a polygon has an interior angle measure of more than 180 degrees, then the polygon is not convex

That leaves only two more given numbers to check -- 144 and 150:

interior angle 144; exterior angle 180-144 = 36; 360/10 = 36. A regular polygon with 10 sides (decagon) has an interior angle of 144 degrees.
interior angle 150; exterior angle 180-150 = 30; 360/12 = 30. A regular polygon with 12 sides (dodecagon) has an interior angle of 150 degrees.

To summarize, then, and answer all your questions....

An interior angle of 360 degrees makes no sense.
An interior angle of 210 or 240 degrees means the polygon is not convex and therefore cannot be regular.
If a triangle has an interior angle of 10, 15, 20, 30, 40, 45, 50, 72, 75, 80, or 100, then it cannot be a regular polygon.
The other angle measurements shown can be the interior angle measurements of a regular polygon:
60 degrees: equilateral triangle
90 degrees: square
108 degrees: regular pentagon
120 degrees: regular hexagon
144 degrees: regular decagon
150 degrees: regular dodecagon
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