Proposition 22 in Book XI of Euclid's Elements states that "If there are three plane angles such that the sum of any two is greater than the remaining one, and they are contained by equal straight lines, then it is possible to construct a triangle out of the straight lines joining the ends of the equal straight lines.".
In essence, this means that given 3 isosceles triangles whose equal sides equivalent in all 3 and with the property that the sum of two of the angles is greater than the remaining one; then the unequal side of the remaining angle is also smaller than the sum of the unequal sides of the other two.
Euclid's proof goes through the following construction, which transposes one of the angles to the side of another: 
From it, we clearly see that the segment $HJ$ is greater than $BC$ because $\angle HGJ > \angle BAC$, and , therefore, $HI + IJ > HJ > BC$.
The problem, however, is that this construction does not work if $\angle HGI + \angle IGJ > \pi$: 
If that is the case, then we cannot say that $\angle HGJ = \angle HGI + \angle IGJ$ and, therefore, the construction is invalid.
I have tried to find answers about this, but there seems to be barely any content regarding even prop. 22 itself - and nothing about this inconsistency. Can somebody help?