Geometry question

Question

1. The following polygons are given. All of the polygons are
regular polygons.

Polygon a. Convex 15-gon
Polygon b. Convex 16-gon
Polygon c. Convex 17-gon
Polygon d. Convex 18-gon
Polygon e. Convex 19-gon
Polygon f. Convex 43-gon
Polygon g. Convex 44-gon
Polygon h. Convex 45-gon
Polygon i. Convex 46-gon
Polygon j. Convex 47-gon

1. Which polygon(s) has (have) interior angles that are whole
numbers (a number that is not a fraction or a decimal)? Explain
why it is that way.
2. What happens to the value of the interior angles as the
number of sides of the polygon increases? Explain your answer.
3. What happens to the value of the exterior angles as the
number of sides of the polygon increases? Explain your answer.
4. Explain what happens to the total sum of interior angles
as the number of sides in the polygon changes?
5. Explain what happens to the total sum of exterior angles
as the number of sides in the polygon changes?

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Answer 1

The polygons REGULAR, therefore, they are equilateral (all sides are equal) and equiangular (all the interior angles are congruent) for each and every one of these. 

CONVEX: this tells us each polygon has all the points of its diagonals inside the polygon.

To answer each of your questions first let us calculate each of the interior angles and each of the exterior angles:

 

Interior Angle formula: (n - 2)(180)/n, where n is the number of sides

Exterior Angle formula: 360/n

         

a. Convex 15-sided. Interior Angle =156  Exterior Anlgle = 24

b. Convex 16-sided. Interior Angle =157.5 Exterior Anlgle= 39.375
c. Convex 17-sided.  Interior Angle =158 R 14 Exterior Anlgle =  21r3
d. Convex 18-sided.      Interior Angle =160.  Exterior Anlgle = 20                    
e. Convex 19-sided.    Interior Angle =161R1 Exterior Anlgle = 18r18 
f. Convex 43-sided.      Interior Angle =171R27. Exterior Anlgle = 8 r16      

g. Convex 44-sided.    Interior Angle =171R36. Exterior Anlgle = 4R8  
h. Convex 45-sided.      Interior Angle =172 Exterior Anlgle = 8       
i. Convex 46-sided.  Interior Angle = 172R8 Exterior Anlgle = 7r38
j. Convex 47-sided.   Interior Angle =172R16 Exterior Anlgle = 7r31

 

use these findings to state your answers

1. 15, 18, 45 Because these are factors of 180
2. increases Because the factor (n-2)/n  increases as n increases
3. decreases Because the divisor, n, increases
4. increases Because (n-2), with which 180 is multiplied, increases with n
5. as the number of sides in the polygon changes the  exterior angles add up to a total sum of 360

note that these questions are repetitions, and you will be able to solve several of these on your own if you abbreviate your question. 

As you can see, the formulas will be available to you in your textbooks or notes from your studies. As a suggestion,

A. Don't copy all the problems, choose one

B. Write a pointed question

C. use a new thread and ask a second or third question

D. If you receive multiple answers and they seem to contradict, my suggestion is take it to your MATH TEACHER; in other words, decide if any of the answers are not worth your time.