# Review outline 1

## Foundations

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

## Incidence geometry

- Undefined terms
- 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