Geometry Axioms, Postulates and Theorems
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The Real World
Axioms, Postulates and Theorems

 

 Say we made contact by radio or something

 with an alien race from another dimension.
 
 Because of some technical problems,
 they can't come here and we can't go there.
 
 We find out that space in their universe 
 doesn't work the same way as it does in ours.
 
 We want to tell them how space works in our universe,
 so we invent geometry.
 
 Lots of different math types with big egos
 work on the project.
 
 They come up with some basic ideas.
 Those are the definitions on the last page.
 
 Then they start using those terms to describe our space.
 Each one works alone.
 They each make two lists of descriptions.
 
 The first list are things that are just the way it is.
 These are things that we just have to accept as true.
 Some call this list Postulates, others call it Axioms.
 
 The second list are things about our space.
 The things on this list are made by putting things together
 from the other list into bigger ideas.
 The things on this list are called Theorems.
 
 Well everything was going along just great,
 until the math types started looking at each others lists.
 
 All of the lists were similar, 
 but none of the lists were exactly the same!
 Some of them had split one idea into a couple of pieces.
 Some had combined a few ideas into one big idea.
 Some put an idea on the Postulate list
 while others put the same idea on the Theorem list.
 
 They all got mad at each other 
 and wrote their own geometry books.
 All of them were similar,
 but no two were exactly the same.
 
 And that's where we are today.
 There are lots of geometry books out there.
 All have similar lists of Theorems and Postulates
 (some call them Axioms), but none are exactly the same.
 
 Here, we aren't going to make two different lists.
 Hey, all this stuff works the same way anyway 
 so we just put it all in one list,
 We'll let the big ego PhD's 
 worry about what goes on what list.
 
 Also, we've tried to write this stuff in plain English,
 so the definitions won't be as stuffy sounding 
 as they are in your book.
 
 OK, Here goes ...
 
 1) There is only one straight line that you can draw
 between any two points.
 (Don't believe it? Draw two points and try to draw more than one line between them)
 
 2) If you have 3 points that are not all in a line,
 there is only one plane that all three are in together.
 (This is how a tripod or 3 legged stool works)
 

 
 If the three points were all lined up,
 there would be lots of planes that they were in ...
 

 
 3) When we have adjacent angles (remember the definition?)
 they add up to an angle made from the outside rays ...
 

 
 4) If the sides of adjacent angles that are away from each other
 make a straight line, the angles add up to 180 degrees.
 

 
 5) If the sides of adjacent angles that are away from each other
 are perpendicular, the angles add up to 90 degrees.
 

 
 6) If angle A + B = 180 and angle B + C = 180
 then angle A = angle C.
 

 
 We can even solve this puppy with algebra
 

 
 There's nothing really special about 180 degrees.
 We could say:
 

 If A + B = 90 and B + C = 90 then A = C

 or

 If A + B = 37 and B + C = 37 then A = C

 
 Also the angles don't have to be touching (adjacent).
 

 
 7) We could even stretch that last one out a bit.
 

 If A + B = 85  and  C + D = 85  and  B = C, then A = D.
 We can solve this one with algebra too!
 

 A + B = 85   C + D = 85   B = C

 
 Substitute C for B in the first equation and we get:
 

 
 Again, there is nothing special about 85. 
 We could have said:
 
 If A + B = 103.5  and  C + D = 103.5  and  B = C, then A = D.
 
 OR
 
 If A and B are supplementary (equal 180) 
 and C and D are supplementary and B = C, then A = D
 
 8) All right triangles have the same measure (Yeah, that's a "well duh" one ...)
 

 
 9) If two angles have the same measure (are congruent)
 and add up to 180 (supplementary) then each equals 90.
 

 X + X = 180   so X = 90

 
 10) When 2 straight lines cross, they cross at only one point.
 

 
 11) Vertical angles are equal (congruent).
 

 
 12) When perpendicular lines cross, 
 they make four 90 degree (right) angles.
 

 
 Hey! wasn't that the definition of perpendicular?
 
 13) When 2 lines cross, if the angles touching each other (adjacent)
 are equal (congruent), the lines are perpendicular.
 

 
 The only way that this can work is if all of the angles are 90 degrees.
 
 14) Through any one point on a line 
 there is only one perpendicular line.
 

 
 15) Through any one point not on a line 
 there is only one line perpendicular to the first line.
 

 
 16) Through one point not on a line,
 there is only one line parallel to the first line.
 

 

   copyright 2005 Bruce Kirkpatrick

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