Hi Noobmaster69,
Let O and H be the circumcenter and orthocenter of acute triangle ABC, respectively.
If the sum for the degrees of angle AHC (orange) and angle AOC (pink) is 240 degrees
what is the measure of angle ABC in degrees? (black)
Here is the pic
The pink plus orange angles add up to 240 degrees and I have to find the size of the black angle.
My pic in not to scale, I mean my pink and orange ones do not add up to 240.
Other than that I think my pic is a proper indication of the problem at hand.
Let the black angle be called alpha \(\angle CBA=\alpha\)
Consider minor arc CA
< COA is the angle at the centre standing on arc CA .... that is the pink angle
< CBA is the angle at the circumferecne standing on arc CA .... that is the black angle
so since the angle at the centre = 2 times the angle at the circumference when both are staning on the same arc,
BLACK \(\angle CBA=\alpha\)
Pink \(\angle COA=2\alpha\)
NOW look at the new image below.
Consider the green triangle \(\triangle IBA\) and the yellow triangle \(\triangle KHA\)
angle AIB = angle AKH = 90 degrees
angle at A is a common angle
therefore these two triangles are similar which means that \(\angle IBA=\angle KHA = \alpha\)
Now I have
\(\angle CHA+\angle KHA = 180\\ \angle CHA+\alpha = 180\\ orange + \alpha =180 \;degrees\\ orange \;angle=(180-\alpha) \;degrees\\\)
I also know that
\(orange+pink=240\\ (180-\alpha)+(2\alpha)=240\\ 180+\alpha=240\\ \alpha=60\;degrees\)
Therefore, angle ABC=60 degrees
I know my logic is correct but i hope i have not made a mistake during the presentation.
Plus
There may well have been a much simpler way to do it.
+118613 
