How to determine (and explain) the sum of angles without measuring? Below is a photo of the angles/triangles in which I am working on determining the sum of the angles without measuring. The angles are a,b,c,d,e,f.

I understand that angles are formed my intersecting lines and I see many intersecting lines in this image.  The angles in a triangle add to 180 degrees and I see 3 labeled triangles.  Also, I know that if parallel lines are cut by transversal lines, then the corresponding angles are equal.  
I'm confused on how to determine the sum of the angles a+b+c+d+e+f without measuring.  I also don't know how I would explain my reasoning. I know the theorems but combining all of the knowledge into a reason is difficult. Where do I even begin?

Call the angles of the three outer triangles $a, b, g$; $c, d , h$; and $e, f, i$, respectively. Then
$$a + b + c + d + e + f = (a + b + g) + (c + d + h) + (e + f + i) - (g + h + i) = 3 \times 180^{\circ} - 180^{\circ} = 360^{\circ},$$ 
where we've used the fact that $g, h, i$ are also the angles of the inner triangle.
