Question

Which theorems belong in Neutral Geometry, Euclidean Geometry, or both?

Answer 1

Exterior Angle Theorem - Both
Alternative interior angle theorem - both
angle side angle theorem - euclidean geometry
big angle big side theorem for triangles - euclidean geometry
sum of any 2 angles of a trianlg is < 180 degrees - both
sum of any 3 angles of a triangle is = 180 degrees - both
Triangle ABC is similar to triangle DEF if angle A = angle D and AB/DE = AC/DF - Euclidean Geometry
Circle A is orthogonal to circle B if and only if inversion in circle A sends circle B to itself - Euclidean Geometry
inverse in a circle sends lines and circles to lines and circles - Euclidean Geometry

Answer 2

only I swear inversions belong to Neutral Geometry only and the sum of any 3 angles of a triangle is strictly Euclidean Geometry

Answer 3

@perl

Answer 4

he left shortly after I did this question OMG!

Answer 5

sum of any 3 angles of a triangle is = 180 degrees that should be just Euclidean Geometry... though that's Hilbert's Postulate.

Answer 6

Euclid is a stealer he didn't do that much really *rolls eyes*

Answer 7

I'm confident on the first four though. and the one with the triangle abc triangle def

Answer 8

sum of any 2 angles of a trianlg is < 180 degrees -
sum of any 3 angles of a triangle is = 180 degrees -
rethink these

Answer 9

sum of any 3 angles is hilbert's postulate... <A+<B+<C =180
why is this even in neutral geometry?

Answer 10

Isn't it true that every theorem in neutral geometry is a theorem in Euclidean geometry? Why the option that a theorem that is only true in neutral geometry even exists?

Answer 11

it was either both, neutral geometry, or euclidean geometry... though I have to admit I did see most of the theorems in neutral and euclidean but angle side angle is only in Euclidean?

Answer 12

. <A+<B+<C =180 is Hilbert's postulate... so isn't that just strictly Euclidean geometry?

Answer 13

Yes that is strictly Euclidean even though I suck at geometry.

Answer 14

http://en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle

Answer 15

ok .. yes it is woo. so what about inversions though? is it just in Euclidean geometry as well?

Answer 16

I think many triangle theorems are in Euclidean Geometry only though :/

Answer 17

I have no idea where other theorems belong to.

Answer 18

@rational

Answer 19

@ikram002p

Answer 20

@perl

Answer 21

so you want to know which axioms belong to which geometry

Answer 22

YES!

Answer 23

but I am stuck @perl

Answer 24

which one are you stuck on

Answer 25

sum of any 2 angles of a trianlg is < 180 degrees - both - stuck
sum of any 3 angles of a triangle is = 180 degrees - both (though this one is Euclidean Geometery)
Triangle ABC is similar to triangle DEF if angle A = angle D and AB/DE = AC/DF - Euclidean Geometry (answert is correct)
Circle A is orthogonal to circle B if and only if inversion in circle A sends circle B to itself - Euclidean Geometry - stuck
inverse in a circle sends lines and circles to lines and circles - Euclidean Geometry -s stuck

Answer 26

inversion is what I'm stuck on the most follow by the sum of 2 angles of the triangle < 180

Answer 27

Circle A is orthogonal to circle B if and only if inversion in circle A sends circle B to itself
inverse in a circle sends lines and circles to lines and circles
sum of any 2 angles of a trianlg is < 180 degrees

Stuck on those three

Answer 28

In neutral geometry, the angle sum of a triangle is less than or equal to 180 degrees.
Theorem 1 (Exterior Angle Inequality) The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

Answer 29

so netural geometry is sum of 2 angles < 180.. it belongs there right? not in euclidean geometry?

Answer 30

in a euclidean triangle the sum of the triangles is exactly 180 degrees.

Answer 31

what about the inversion theorems?

Answer 32

so in neutral geometry sum of two angles < 180 but in euclidean geometry angle A + angle B + angle C = 180

Answer 33

it is both true in neutral and euclidean geometry that the sum of two angles is less than 180

Answer 34

but for euclidean geometry only it's angle A + angle B + angle C = 180

Answer 35

note that euclidean geometry is a type of neutral geometry , with an extra postulate (parallel postulate)

Answer 36

thats for 3 angles

Answer 37

all the neutral geometry axioms, theorems, and lemmas hold in euclidean geometry.
but not the other way. there are some theorems in euclidean geometry which do not hold in neutral geometry

Answer 38

but angle a + angle b + angle c = 180 has to be in euclidean geometry only right?

Answer 39

??? wahhh lost!

Answer 40

yes but your question asked about 2 angles, not 3 angles

Answer 41

are you talking about the next question, oh

Answer 42

sum of any 3 angles of a triangle is = 180 degrees is one of the questions

Answer 43

yeah that's Euclidean only right? since that Hilbert's Postulate

Answer 44

sum of any 2 angles of a trianlg is < 180 degrees - both neutral and euclidean
sum of any 3 angles of a triangle is = 180 degrees - euclidean

Answer 45

right

Answer 46

whew... ok... what about the 2 inversion theorems

inversion in a circle sends lines and circles to lines and circles. isn't that just Euclidean too?

Answer 47

Circle A is orthogonal to circle B if and only if inversion in circle A sends circle B to itself

inversion in a circle sends lines and circles to lines and circles

Answer 48

hmm, thats a good question. it looks like that you need a normal plane to get inversions,

Answer 49

what does euclidean geometry have that neutral does not, which would allow us to use inversions

Answer 50

ok i am going with euclidean geometry

Answer 51

that's what I have too.

Answer 52

i have a pdf which says its a result of euclidean

Answer 53

can I have a link to it?

Answer 54

or maybe I saw it already somewhere not sure...

Answer 55

Exterior Angle Theorem - Both
Alternative interior angle theorem - both
angle side angle theorem - euclidean geometry
big angle big side theorem for triangles - euclidean geometry
Triangle ABC is similar to triangle DEF if angle A = angle D and AB/DE = AC/DF - Euclidean Geometry

but these are correct right?

Answer 56

yes thats correct

Answer 57

yay

Answer 58

Exterior Angle Theorem _both
Alternative interior angle theorem - both
angle side angle theorem -neutral geometry
big angle big side theorem for triangles - neutral geometry
sum of any 2 angles of a trianlg is < 180 degrees - both
sum of any 3 angles of a triangle is = 180 degrees - eclids
Triangle ABC is similar to triangle DEF if angle A = angle D and AB/DE = AC/DF - neutral Geometry
Circle A is orthogonal to circle B if and only if inversion in circle A sends circle B to itself - Euclidean Geometry(Hilbert)
inverse in a circle sends lines and circles to lines and circles - Euclidean Geometry (Hilbert to be exactly)

Answer 59

Good luck @!