Proposition 29 says the following:
A straight line falling on parallel straight lines makes the alternate angles equal to one another
Euclid proves this using the Parallel Postulate. I don’t understand why that’s necessary. I can prove it without the postulate:
Let the straight line intersect the parallel straight lines in A and B. I claim that angle CAB= angle ABD. Bisect AB in E and draw the circle with center E and A and B lying upon its circumference. The Circle intersects the straight parallel lines in F,A,B,G. Draw FG. Since angle AEF and angle BEG are vertical angles of two intersecting lines they are equal. Since FE=AE=BE=DE and angle FEA=angle BEG we conclude that ΔFAE=ΔBGE and ΔFAE,ΔBGE are isosceles. Equal isosceles triangles have equal angles at their bases, so we have: angle FAE=angle EBG and thus: angle CAB=angle ABD
Can anybody help me understand it please? I’ve heard that Proposition 29 is the first one in Book I that is depending on the 5. Postulate and thus doesn’t hold in hyperbolic geometry.
I think that this theorem is not true because you have assumed that
and that F,E,G will be collinear but if you give a reason then this theorum might be true
credits to @Intelligenti pauca and @ N.S. in the comments