Using deductive proof, prove that the sum of the angles at the tips of a n-sided star is 180.?

Question

Only apply it to a regular star, and the number of points on the star must be odd

This is TRUE. It only works for regular polygons though! PLEASE HELP MEEEE

Ok. I've aske dmy teacher over and over..and it is still right

this works when the angles at the tip of the star are even and there is a regular polygon shown on the bottom

Answer 1

Let n be the number of points on the star.
Then the center of the star is an n-sided polygon.

By drawing lines from one corner of the polygon to each other corner, this center polygon can be divided into (n-2) triangles. Each triangle has 180 degrees of angle.
Hence, the center polygon has 180*(n-2) degrees of angle.

Since the center polygon has n corners, each corner of the polygon has an angle of (180*(n-2)/n) degrees.

Now let's look at one of the triangles formed by a point on the star and the two closest corners of the polygon. This is an isosceles triangle. The two corners of this triangle that *aren't* a point of the star will have angles equal to 180 - (180*(n-2)/n), since these angles are supplementary to interior angles of the polygon.

Grinding through some algebra:
180 - (180*(n-2)/n)
180*(1-(n-2)/n)
180*((n-(n-2))/n)
180*(2/n)
360/n

Hence, one point of the star will have an angle of:
180 - 2*(360/n)
180 - 720/n

And since the star has n points, the sum of the points' angles is:
n*(180 - 720/n)
(180*n - 720) degrees

I've double checked, the formula (180*n-720) degrees is correct. The problem statement is wrong. You only get an answer of 180 degrees if you're dealing with a 5-point star.

It was a fun problem, though!

Oh yeah, the formula works for even-pointed stars as well as odd-pointed ones.

Answer 2

regular or irregular polygon..... the problem is wrong....

just check... if n=3, sum of angles = 180
if n=4, sum of angles = 360

before stating a problem, please do your homework....at least to state the problem right...