Question

Corollary 6.1.10. In a Saccheri quadrilateral, the length of the summit is greater than the length of the base.

please proof

Answer 1

If the fourth angle were obtuse, our quadrilateral would have an angle sum greater
than 360◦
, which cannot happen. If the angle were a right angle, then a rectangle would
exist and all triangles would have to have defect 0. Since there is a triangle with angle sum
less than 180◦
, we have a triangle with positive defect. Thus, the fourth angle cannot be a
right angle either.

  if M is the midpoint of AB and N is the midpoint of CD, then
✷AMND is a Lambert quadrilateral. Thus, AB > MN and, since BC ∼= AB, both sides
are greater than the altitude.
Also, applying upper part  DN > AM. Since CD ∼= 2DN and AB ∼= 2AM it follows
that CD > AB, so that the summit is greater than the base.