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?

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
$$180^\circ-a-b+180^\circ-c-d+180^\circ-e-f=180^\circ$$
$$a+b+c+d+e+f=360^\circ$$

