I thought uploading pictures of the problem would be the easiest way to show the whole problem.
I'm assuming the flaw is going to have to do with one of the figures being drawn inaccurately, but when I read through the proof as a whole I'm having trouble identifying where the flaw could be other than just with the figures. Any tips would help!
The book is Foundations of Geometry, 2nd edition, by Gerard Venema.
In figure 1.8 the one on the right, is what's wrong with it has to do with the fact it doesnt show at DMB is a right triangle?
Hint: In one of the figures, on one of the lines on which three points are identified, the point that is drawn between the other two is not actually between the other two.
Second hint: Two of the three figures for the "$\ell$ is not perpendicular to $\overline{BC}$" case are based on false assumptions and serve only as a distraction. That leaves just one figure in which to look for the error.