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?
 A: 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.
A: On the picture below:
$$\color{red}{\text{red angle}} + \color{orange}{\text{orange angle}} + \color{blue}{\text{blue angle}}=180^\circ$$
And
$$\begin{align}
\color{red}{\text{red angle}}+a+b=180^\circ&\implies
\color{red}{\text{red angle}}=180^\circ-a-b\\
\color{blue}{\text{blue angle}}+c+d=180^\circ&\implies
\color{blue}{\text{blue angle}}=180^\circ-c-d\\
\color{orange}{\text{orange angle}}+e+f=180^\circ&\implies
\color{orange}{\text{orange angle}}=180^\circ-e-f
\end{align}$$
Hence
$$180^\circ-a-b+180^\circ-c-d+180^\circ-e-f=180^\circ$$
$$a+b+c+d+e+f=360^\circ$$

A: 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.
