Each of these figures can be solved using the formula
Interior angle = 180*(1 - 2/n). [Google equiangular polygons to get the formula). If the polygon is regular then all its interior angles are equal (and so are the sides). So all of the questions above can be done using the following formula. I'll do one of them carefully and you can do the rest substituting the given numbers in for the left had side.
interior angle = 128 4/7
n = ???
128 4/7 = 180(1 - 2/n) Divide by 180
128.5714 = 180(1 - 2/n)
128.5714/180 = (1 - 2/n)
0.71428 = 1- 2/n Bring the 2/n to the left side.
0.71428 + 2/n = 1 Subtract 0.71428 from both sides.
2/n = 1 - 0.71428
2/n = 0.285714
Multiply both sides by n
2 = 0.285714 * n divide by 0.285714
n = 2/0.285714
n = 7 sides.. Note you should leave each result in your calculator as you do it. If you do, will get exactly 7 sides.
The answers to the other three are
140 gives 9 sides
144 gives 10 sides
160 gives 18 sides
