So I started going in the other direction and saying well if \(\displaystyle \alpha+\gamma+\beta\prime \ge 180\) then also \(\displaystyle \beta+\gamma+\alpha\prime \ge 180\) and started working out that

\(\displaystyle \alpha+\beta\prime \ge 180-\gamma\) and \(\displaystyle \beta+\alpha\prime \ge180-\gamma\)

so \(\displaystyle \alpha+\beta\prime +\beta +\alpha\prime \ge 2(180-\gamma)\)

\(\displaystyle \alpha +\beta +\alpha\prime +\beta\prime + 2\gamma \ge 360\)

\(\displaystyle \alpha +\beta +\gamma \ge 360-(\alpha\prime +\beta\prime + \gamma)\)

But this is as far as I've gotten and I'm not sure if I'm going in the right direction.

I know I was given that it is less than 180 so I need to show greater than 180 so then I can conclude it must equal 180.