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# 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

• 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