Oops. I made an error in my previous analysis.

His actual statement in the 2nd paragraph works just fine.

But his final paragraph has a MAJOR leap of logic that just isn’t true at all.

Here it is:

(Bolding mine)

In absolute geometry (the geometry you can do with just the first 4 postulates), this statement is not supported.

In absolute geometry, the interior angle sum is **at most** two right angles. The existence of a single example of a triangle with interior angle sum of two right angles does not contradict the possibility of the existence of other triangles with interior angle sum less than 180 degrees.

It’s only if you assume some kind of Parallel postulate (0, exactly 1, > 1 parallels) that the interior angle sum can be used to define a geometry (Elliptic, Euclidean, Hyperbolic, respectively). But, of course, that simply leads to a tautology.

While this error is bad, it doesn’t necessarily prevent the possibility that Hyperbolic geometry is invalid while leaving Euclidean, Elliptic, and absolute geometry alone.

Fortunately, he made an earlier error with a similar generalization that isn’t backed up by the work he already did and is not necessarily true, unless one assumes a Euclidean framework.

The statement he makes is:

His set up is designed to prove this. But what he actually showed was the existence of a specific construction for two such new equal/parallel lines which joined the original parallel lines. The construction relied on a transversal.

The generalization is that such equal/parallel lines can be generated even without the conditions used for the construction (though he clearly didn’t understand he was doing this).

Even though it will work in a Euclidean world, that generalization fails if you don’t have the Parallel Postulate.