Axiomatic Geometry

Front 1

Euclid's First Postulate

Back 1

If A and B are points there is a unique line between them.

Front 2

Euclid's Second Postulate

Back 2

If AB and CD are line segments, there is a unique point E so that B is between A and E and BE=CD

Front 3

Euclid's Third Postulate

Back 3

For Distinct points O and A there is a circle with center O and radius OA.

Front 4

Euclid's Fourth Postulate

Back 4

All right angles are congruent

Front 5

Euclid's Fifth Postulate

Back 5

Parallel Postulate: Given a line l and a point P not on l, there is a unique line m that is parallel to l and goes through P

Front 6

I1

Back 6

For every point P and every point Q (not equal to P) there is a unique line l incident with P and Q

Front 7

I2

Back 7

For every line l there are at least two distinct points incident with l

Front 8

I3

Back 8

There exists three distinct points with the property that no line is incident with all three points

Front 9

B1

Back 9

if A*B*C the A*B*C are all distinct points on the same line and C*B*A

Front 10

B2

Back 10

if B doesn't equal D then there are A,C,D so A*B*D, B*C*D, and B*D*E (tells us there are infinitely many points. Many models fail this)

Front 11

B3

Back 11

If A,B,C all lie on a line, exactly one A*B*C, B*C*A, C*A*B is true

Front 12

B4

Back 12

For every line l and points A,B,C not on l: a) if A,B are on the same side of l and B,C are on the same side then A,C are on the same side. b) if A,B are on opposite sides and B,C are on opposite sides then A,C are on the same side of l

Front 13

C1

Back 13

If A not equal to B are distinct points and A' is a point and r is a ray from A', then there is a unique point B' on that ray so that A'B' congruent AB

Front 14

C2

Back 14

if AB congruent CD and AB congruent EF, then CD congruent CF. Also Ab congruent AB

Front 15

C3

Back 15

A*B*C & A'*B'*C' and AB congruent A'B' and BC congruent B'C', then AC congruent A'C'

Front 16

C4

Back 16

Given any angle ABC and another ray A'C' there is a unique ray B'C' on either side of A'B' such that angle ABC is congruent to angle A'B'C' (we can move angles)

Front 17

C5

Back 17

If angle A congruent angle B and angle A congruent angle C then angle B congruent angle C. Also angle A congruent angle A

Front 18

C6

Back 18

SAS

Front 19

AIAT

Back 19

If two lines are cut by a transversal and have a congruent pair of alternate interior angles, then the lines are parallel

Front 20

EAT

Back 20

An exterior angle to a triangle is larger than either of its remote interior angles