Review outline 2

Neutral geometry (4.3 and on)

• Remind yourself about triangle congruence conditions
• Definitions
  • Transversal
  • Alternate interior angles
  • Angle sum
  • Defect
  • Quadrilateral (we will always assume convex)
  • Parallelogram
  • Rhombus
• Theorems
  • Scalene inequality
  • Triangle inequality
  • Hinge theorem
  • Alternate interior angles theorem
  • Saccheri-Legendre theorem
  • Additivity of defect
  • The universal hyperbolic theorem
• Double perpendicular construction

Euclidean geometry

• Equivalents of the EPP
  • Converse to the alternate interior angles theorem
  • Angle sum postulate
  • There exists a triangle with defect 0
  • Clairaut’s axiom
• Theorems
  • Properties of Euclidean parallelograms
  • The parallel projection theorem
  • The fundamental theorem on similar triangles
  • The Pythagorean theorem
• The foundations of trigonometry
  • Definitions of sine and cosine
• Euclidean triangle geometry
  • Medians and centroid
  • Altitudes and orthocenter
  • Circumcenter
  • Euler line theorem

Hyperbolic geometry

• Polygonal model construction
• The idea of positive and negative curvature
• Definitions
  • Saccheri quadrilateral
  • Lambert quadrilateral
  • Parallels admitting a common perpendicular
  • Limiting parallels
  • Critical angle (angle of parallelism)
• Theorems
  • Properties of Saccheri quadrilaterals
  • Porperties of Lambert quadrilaterals
  • AAA congruence criterion
  • Existence of the critical angle
• Poincaré disk model