Draw the circumscribed circle and radii to the vertices:


The two angles marked 72° are 72° each because they are one-fifth 
of 360°.  

The two angles marked 18° are 18° because they are base angles of
an isosceles triangle whose vertex angle is twice 72° or 144° and
180°-144° = 36° and since the base angles of an isosceles triangle
are congruent, each has measure of half of 36° or 18°.


Similarly, you can show that these angles are as marked below, too:



So therefore one of the points of the star makes an angle
of 36° since it is twice 18°, as indicated below:




By the same reasoning, each of the points of the star makes
an angle of 36°, so therefore the sum of the angles made at
all five points of the star is 5×36° = 180°.

This is just an outline of how to prove it.  You have to write it 
up as a two-column proof yourself.  You'll have to label some of
the points with letters so you can talk about triangles such as
triangle ABC, and angles PQR, etc., or whatever lettering system
you want to use.

Edwin