How to determine (and explain) the sum of angles without measuring?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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

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