Question 14476
Wow! That's a ton of information for one small problem.

A regular pentagram (AKA Pentacle, but not Penotgram) is a five-pointed star formed by five straight lines connecting the vertices of a regular pentagon and enclosing another regular pentagon in the completed figure.
 The triangles of the pentagram thus formed are, as you point out, isosceles triangles, whose base angles, as you also point out, are congruent.

Now, if you know the measure of these congruent base-angles of the isosceles triangeles, then you can find the measure of the third angle.

Start with the regular pentagon on the inside of the figure.  The measure of each interior angle is:

{{{A = (5-2)(180)/5}}} = 108 degrees.

You will note that this angle is the supplement of a base angle of one of the triangles of the pentagram.
Therefore, the base angle is {{{180 - 108 = 72}}} degrees.
The other base angle is also 72 degrees (congruent angles).
The third angle is then: {{{180 - 2(72)) = 180 - 144}}} = 36 degrees.

You are asked to find the measure of each interior angle of the regular pentagram. This is the same as the third angle of the triangle which is 36 degrees.