Aregular polygon has possible angles or rotational summetry of 20, 40, and 80 degrees. how many side does the polygon have?

The number of sides regular polygon have = 18

Step-by-step explanation:

The possible angle or rotational symmetry is 20° , 40° , 80° of any regular polygon .

now,

Symmetrical under 20° Rotation is also symmetrical on rotation of any multiple of 20°

So, number of sides = n =

I.e                                  n =  

So,                                 n = 18

Hence the number of sides the polygon have 18    Answer  

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4