1.Use Gauss Theorem to prove that an angle of 20 degree is not contructible.
2.Use Gauss Theorem to decide whether or not an angle of 6 degree is constructible.
Thanks very much guys.....
Gauss theorem states that an $\displaystyle n-gon$ is constructible $\displaystyle \mbox{iff}$ $\displaystyle n=2^k.p_1,\dots, p_r$,Originally Posted by suedenation
where $\displaystyle k \in \mathbb{Z_+}$ and $\displaystyle p_1,\dots, p_r$ are distinct Fermat primes (note $\displaystyle r$ can be $\displaystyle 0$).
If an angle of $\displaystyle 20$ degrees were constructible so would a $\displaystyle 360/20=18$ sided
polygon.
The first three Fermat primes are $\displaystyle 3,\ 5,\ 17$, clearly $\displaystyle 5$ and $\displaystyle 17$
do not divide $\displaystyle 18$, so for the $\displaystyle 18-gon$ to be constructible
$\displaystyle 18$ would have to be a power of $\displaystyle 2$, or $\displaystyle 6$ (as it is $\displaystyle 18/3$) would
have to be a power of $\displaystyle 2$. They are not so the $\displaystyle 18-gon$ is not
constructible and so an angle of 20 degrees is not constructible.
RonL
The $\displaystyle 6$ degree angle is constructible $\displaystyle \mbox{iff}$Originally Posted by suedenation
the $\displaystyle 360/6=60$ sided polygon is constructible.
$\displaystyle 60=4.3.5=2^2.3.5$
so the $\displaystyle 60-gon$ is constructible and so the $\displaystyle 6$ degree
angle is constructible.
RonL