Download Study guide for the final exam - University of Toronto Mississauga

Study guide for the final exam.
Sources:
• Chapters 1–6 of Greenberg’s book.
Chapter 1: skip “Viète’s axiom. Chapter 3: exclude fields. Chapter 4: You
needn’t keep track of which continuity principle is used in which theorem: in
theorems that rely on a continuity principle, you may assume Dedekind’s axiom.
(Strengthening the assumption weakens the theorem but is harmless for our
purposes.) Also, exclude fields. Chapter 5: Throughout this chapter you may
assume Dedekind’s axiom. You need to know statements that are equivalent to
Euc-PAR. Chapter 6: Focus on the definitions and statements rather than the
proofs.
• The review exercises in the book.
• The handout from Coxeter’s book.
• Your notes from the lectures.
• The handout about complex numbers.
• The handout about the Poincaré upper half plane model.
• All the homework.
The exam.
• About half of the exam will be on the same material as the midterm.
• I will give you Hilbert’s axioms of incidence, betweenness, and congruence. You
need to know Hilbert’s axioms of continuity (except Aristotle’s). You need to
know Euclid’s axioms.
Tentative large list of terms. This is a list of terms that I find important. For each
of these terms, you should be able to write two sentences, in which you define it, state
it, explain it, or discuss it.
- Euclid’s definition of “right angle”
- Euclid’s definition of “Parallel lines”
- “All right angles are equal”.
- colinear points.
- concurrent lines.
- Euclid’s first postulate
- Euclid’s second postulate
- Euclid’s third postulate
- Euclid’s fourth postulate
- Incidence geometry.
- Model for a theory.
- Isomorphism of models.
- Three point model.
- Four point model.
1
2
-
Affine plane.
Great circle.
Examples and counterexamples for “straight” on S 2 .
Euclidean parallel property (=Euclidean parallel postulate)
Hyperbolic parallel property
Elliptic parallel property
S 2 /antipodes.
RP2 .
Independence of the Euclidean parallel postulate from the incidence axioms.
Transitivity of parallelism.
Projective completion.
Point at infinity.
Homogeneous coordinates.
Central projection.
Vanishing line.
Parallel projection.
Euclid’s reliance on diagrams.
The “betweenness” relation A ? B ? C
Definition of “segment”.
Definition of “ray”.
Opposite rays
Definition of “angle”.
Trichotomy for “betweenness”
Line separation property
Plane separation axiom
Pasch’s theorem
The crossbar theorem
Euclid’s reliance on diagrams.
Convex sets
Congruence of segments is an equivalence relation.
Congruence of angles is an equivalence relation.
“Segment addition”
Congruence of triangles.
“Congruence”
SAS
ASA
SSS
Base angles of an isosceles triangle.
Vertical angle theorem.
Ordering of segments
Ordering of angles
Continuity principles (circle-circle, line-circle, segment-circle)
3
-
A Hilbert plane
Dedekind cut of a line
Dedekind cut of a ray
Dedekind’s axiom of continuity
Segment ordering
Segment addition/subtraction
Interior of an angle
A ray between two other rays.
Angle ordering
Angle addition/subtraction
Supplements of congruent angles
Vertical angles
An angle that is congruent to a right angle
Existence of a perpendicular to a given line through a given point
The model R2
Archimedes’s axiom
Measurement theorem
Alternate interior angle theorem
Uniqueness of the perpendicular ot a given line through a given point
Existence of parallel lines
Why elliptic geometry is not part of Neutral geometry
Exterior angle theorem
SAA
Existence and uniqueness of the midpoint of a segment
Existence and uniqueness of an angle bisector
Existence and uniqueness of a perpendicular bisector.
Hypotenus-leg theorem
“the greater side opposite the greater angle”
Triangle inequality
Saccheri-Legendre theorem
Hilbert’s Euclidean axiom of parallelism
Euclid’s fifth postulate
Transitivity of parallelism
Angle sum of a triangle
Saccheri quadrilateral
Lambert quadrilateral
Angle inscribed in a circle (in Euclidean geometry)
“rectangles exist”
“there exist similar triangles of arbitrary magnitude”.
Saccheri (universal) angle theorem
Universal non-Euclidean property
“AAA”
4
- Divergently parallel lines
- Asymptotically parallel lines
- Limiting parallel rays
- Angle of parallelism
Poincaré upper half plane model:
- “h-points”, “h-lines”.
- Möbius transformations are conformal
- Möbius transformations form a group
- Möbius transformations take lines/circles to lines/circles
- Möbius transformations act simply transitively on triples
- “h-transformations”
- Congruence of h-angles
- Congruence of h-segments
- Examples of h-rays, h-angles, h-triangles
- Examples of limiting parallel rays
- Failure of Euc-PAR