The sides of ∠BAC are AB and AG
The sides of ∠GEC are EB and EG

AB⊥EB
AG⊥EG

I'll just tell you how to prove it.  You will have to write
your own two-column proof.

It's easy to prove ᐃABC∼ᐃEGC
because they are right triangles and 
∠ACB = ∠FCG, so the third angles 
∠BAC = ∠GEC.

So that's the case when they are both
acute angles.   

The reason the "or supplementary" has to be in the theorem is because:
 
1. what is true about the sides being perpendicular for acute angle
∠BAC is also true for its supplementary obtuse angle ∠DAC.

and similarly

2. what is true about the sides being perpendicular for acute angle 
∠GEC is also true for its supplementary obtuse angle ∠FEG.

If you have any questions, ask me in the thank-you note and I'll get
back with you.

Edwin