Review outline 1

Foundations

• Undefined terms
• Theory (postulates / axioms)
• Model
• Defined terms
• Theorems / propositions

Incidence geometry

• Undefined terms
  • point
  • line
  • lie on
• Theory
  • I1 every pair of distinct points lie on a unique line
  • I2 every line has at least two points
  • I3 there exist three points that are non-colinear
• Example models
• Defined terms
  • intersect
  • parallel
  • colinear
• Sample theorems / propositions

Logic

• Implies
• Quantifiers
• Negate

Neutral axioms 1-3

• Undefined terms
  • point
  • line
  • lie on
  • distance
  • angle measure
• N1 (EP) there are at least two points
• N2 (IP) two points determine a line
• N3 (RP) every line admits a coordinate function f such that |f(P)-f(Q)|=PQ
• Defined terms
  • coordinate function
  • between
  • ray
  • segment
  • congruence of segments
  • convex
• Theorems
  • Betweenness can be expressed in terms of coordinates
  • Ruler placement
  • Point construction

Neutral axioms 4-6

• N4 (PS) for every line l, the points not on l can be partitioned into two convex sets H1 and H2 such that if P is in H1 and Q is in H2 then the segment PQ meets l.
• Defined terms
  • two sides of a line
  • on the same side
  • on opposite sides
  • angle
  • interior of an angle
  • betweenness for rays
• Theorems
  • betweenness for points versus betweenness for rays
• N5 (PP) every angle has a measurement in [0,180); an angle measure of 0 means the two rays of the angle are the same; given any ray and angle measurement, there is a unique angle on each side of the ray with the given measurement; the measures of adjacent angles add up to the measure of the larger angle
• Defined terms
  • congruence of angles
  • acute angle
  • right angle
  • obtuse angle
• Theorems
  • betweenness theorem for rays
  • crossbar theorem
  • linear pair theorem
  • vertical angles theorem
• N6 (SAS) if AB is congruent to DE, angle ABC is congruent to DEF, and CD is conrguent to EF, then triangle ABC is congruent to DEF.
• Defined terms
  • triangle
  • isosceles triangle
  • congruence of triangles
• Theorems
  • isosceles triangle theorem

Sample models

• Euclidean distance
• Taxicab distance

Triangle angles and congruence

• Exterior angle theorem
• Existence and uniqueness of perpendiculars
• SAS is true because we said so
• ASA was proved using SAS and EAT
• AAS was proved using SAS and EAT
• SSA is false, we have counterexamples
• SSA is true for right triangles