



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 
