This is a one term course.
Chapter 1 begins with a reflection on Euclidean Geometry as a revolutionary step in human civilization, then on criticisms of logical structure of Euclid's Elements, and finally on reformulation of the axiom system given by D. Hilbert in 1899.
Chapter 2 starts with the so called Absolute Geometry-geometry without Euclid's parallel axiom, and a few equivalent forms of parallel axioms. It discuss Non-Euclidean Geometry before the fomulation of Lobachevsky and Bolyai.
Chapter 3 focus on the most remarkable invention of Lobachevsky and Bolyai, namely the idea of horocycles and horospheres, its use in Non-Euclidean geometry, including explicit formula for angle of parallelism etc.
Chapter 4 discuss Spherical Geometry in both Euclidean and Non-Euclidean case, problems on consistency and Beltrami-Klein model and Poincare model.
Chapter 5 shall study in some detail B. Riemann's famous lecture on "On the hypotheses which lie at the foundations of geometry" in 1854, generally own as Riemann' s Habilitation lecture.