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?

Triangle w/ angle a and b has remaining angle we will call x.
Triangle w/ angle c and d has remaining angle we will call y.
Triangle w/ angle e and f has remaining angle we will call z.
The 'middle triangle' has angle x, y and z by vertical angles.
The middle triangle tells us that x + y + z = 180.
We also know that 
a + b + x = 180
c + d + y = 180
e + f + z = 180
So it is elementary from here: 3 x 180 - (x + y + z) = 2 x 180 = 360.
