Definitions and Theorems (Kay):
You need to know the following definitions and theorems from Kay, from the second half of the
material. Although the focus is on the second half, the material builds on the previous – you still
need to know the definitions and main results from the first half of the course, particularly the
major theorems from absolute geometry that appeared in chapter 3. The new material picks up
at section 3.7 (quadrilaterals). Definitions and theorems below are in the order covered in class
(and not necessarily the order in Kay).
Definitions
quadrilateral terminology (p 183)
Axioms/Theorems
properties of convex quadrilaterals (p 183)
convex (p 183; defined precisely in lecture)
congruence theorems for convex
quadrilaterals – SASAS, ASASA, etc (pp 184185)
rectangle (p 186)
“theorem 2” and the other properties of
Saccheri quadrilaterals on p 187
Saccheri quadrilateral (p186)
Lambert quadrilateral (p 187)
circle terminology (p 194)
the circle properties on p 195
arc terminology (p 196)
additivity of arc measure (p 197)
tangent and secant (circles) (p 198)
the tangent theorem (p 198) and its corollary
(p 199)
the possible cases for parallel lines (pp 425 426) Know which case defines spherical, which
hyperbolic, and which Euclidean geometry
the equivalent forms of Euclid’s Fifth, given
on page 427
defect in hyperbolic geometry (p 438)
hyperbolic parallel postulate (p 436)
area in hyperbolic geometry (p 438)
unnamed hyperbolic theorem 1 and its
corollary on p 437
Poincaré disk model components (p 446)
parallel (lines) (p 211)
parallelism in absolute geometry (p 212) and
equivalent forms for corresponding, same side
interior, etc.
Euclid’s Fifth Postulate (p 213) and equivalent
forms for corresponding, same side interior,
etc.
Euclidean Parallel Postulate as stated as
axiom P-1 in Kay (p 214)
(unnamed) theorem 2 on p214, which is the
same as Euclid’s Fifth
transversal theorems (shorthand: C, F, and Z
properties) (p 215)
Euclidean exterior angle theorem (p 217)
midpoint connector theorem (p 218)
parallelogram, rhombus, square (p 224)
(unnamed) theorem 1, and the properties of
parallelograms on p 225
trapezoid (p 226)
midpoint connector theorem for trapezoids (p
218)
side-splitting theorem (p 229)
(more circle/arc material)
inscribed angle theorem (p 270) and
corollaries on pp 272-273
two chord theorem (p 273)
secant-tangent theorem (p 273)
two secant theorem (p 273)
regular polygon terminology (p 254)
angle sum of convex n-gon (theorem 2) (p
255)
Euclidean area axioms (1-6) (pp 286-287), and
particularly Cavalieri’s Principle
similar (polygons, esp. triangles) (p 236)
AA, SAS, and SSS similarity criteria for
triangles (pp 238-239)
Note: material on polyhedral and volume mostly not from Kay. See also the sheets on
“Formulas and computations” and “General stuff you should know.