What is the relationship between Golden Section with a polygon?

Good question, but I think you meant pentagon.

If you had a regular pentagon with a side of 1, then the triangle formed any two adjacent sides of the pentagon, the third side equals Phi.

To check we can use the law of Sines.

Sin A / a = Sin B / b = Sin C / c

Each interior angle of a regular polygon is found by this formula:

180(n - 2) / n, when n = the number of sides the polygon has.

So, for a regular pentagon

180(5 - 2) / 5
180(3) / 5
540 / 5
108 degrees

Since the triangle we formed is an isoscles one, the other 2 angles can be found out by subtracting 108 from 180 and dividing the result by 2.
(180 - 108)/2 = 72/2 = 36 degrees

Sin 108 / Phi = Sin 36 / 1

The formula for Phi is:
5 ^ .5 * .5 + .5

ArcSin(Sin 108 / (5 ^ .5 * .5 + .5)) =
= ArcSin(0.9510565163 / 1.618033989) =
= ArcSin(0.5877852523) = 36