So, we want to know the total of the indicated angles on that pentagram?

Well, let's look at A. A is part of the top triangle, so it will be 180° minus the other two angles.
The other two angles can combine with one of the angles in the inner pentagon to make a straight line.

So they are both 180° minus an angle on the inner pentagon. This means that angle A is 180° - ((180° - p) + (180° - q)), which is equal to p + q - 180° (where p and q are the appropriate angles in the pentagon).

Similar reasoning can be applied to B, C, D and E, and so if we label the inner angle of the pentagon as p, q, r, s and t, then the total of A, B, C, D and E is found by 2(p+q+r+s+t) - 900°.

We know that the angles in any pentagon will always total 540°, which means that the wanted angles add up to 2*540° - 900° = 180°.

I've added some more angles to the diagram to show what I'm talking about.

So, there's a triangle above the central pentagon made up of angles A, x and y. Because these make a triangle, we know that A + x + y = 180°.

However, angle x shares a straight line with angle p, and so x + p must also equal 180°. Rearranging gets us that x = 180° - p. Similar reasoning tells us that y = 180° - q.

So putting these values into the triangle's equation tells us that A + (180° - p) + (180° - q) = 180°. Rearranging for A gives that A = p + q - 180°.

Now we repeat this process for angles B, C, D and E.
The same reasoning will tell us that B = q + r - 180°, C = r + s - 180°, and so on.

Adding all of these tells us that A+B+C+D+E = 2(p+q+r+s+t) - 900°.

p+q+r+s+t is the total of all the angles in the pentagon, and we know that the total of angles in any pentagon is always 540°. Substituting this value in tells us that A+B+C+D+E= 2*540° - 900°, which is equal to 180°.

If you still don't follow, tell me which parts of my explanation you don't follow and I'll try to explain them better.

Thankyou very much!

I worked the pentagon fact out from the more general fact that the interior angles in a polygon with n sides will always total (n-2)*180°.

So, angles in a triangle add to (3-1)*180° = 180°, angles in a quadrilateral add to 360°, angles in a pentagon add to 540°, etc.

That's odd. I can see it fine, and presumably John could. Maybe imageshack was down for a while. Can you see it now (or when you next view this topic)?

And when you say that it's really clear, does that mean that you can follow it without a picture anyway?