Dr. Peterson, I thought about that but didn't know how to prove it was. However, I now realise that the sides of the pentagon are the same length as the sides of the hexagon. I should have noticed that
I think the original diagram (as well as mine) is deceptive because the line BG should be a rotation of the line BC. I assume the original is drawn that way to make it less obvious. Now it is straight forward to work out:
The angle GBC is 12 (interior angle of 6gon - interior angle 5gon) and the base angles of the isosceles are 84. So angle HGC will be 360 -( interior angle 5gon + 84) = 168.
Very deceptive original diagram!