Geometry Two Column Proofs
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Two Column Proofs


 In geometry, we need to prove stuff is true.

 Most people think this proof stuff is really weird and wacky.
 Here's what's really going on.
 It works like a detective story.
 Before they get to be real detectives,
 recruits in Math Town go to detective school.
 In school they learn lots DEFINITIONS.
 Stuff like:


If something is missing and nobody knows 
where it went.

 ALIBI -  

If somebody was away when the thing was stolen, 
they have an alibi.
 There are lots of other definitions they have to learn,
 but that' s enough for us.
 There are special detective words that are used to tell
 detectives stuff that is always true.
 Stuff like:
 When something gets stolen, 
 the crime of robbery has been committed.
 If somebody was away when the crime happened,
 they have an alibi so they didn't do it.
 These things are called POSTULATES.
 They are given special names so the detectives can talk about them to each other
 without it taking all day.
 The names of the two we just talked about are:
 The crime of robbery postulate.
 The alibi/didn't do it postulate.
 Then the detective goes to work.
 When the detective gets to a crime scene
 they take down some information about what happened.
 Stuff like:
 1) Many people saw the missing stuff 
 in the place where it is kept last night at midnight.
 2) Lots of people saw that the stuff was gone
 this morning at 8 AM.
 3) Nobody knows where the stuff is.
 This information that the detectives are given at the scene
 is called THE GIVEN INFORMATION (big surprise eh?)
 Now the fun begins.
 The detective says "I learned in class that since the stuff is gone
 and nobody knows where it is, it was STOLEN!
 That means that the crime of robbery has been committed."
 Then the detective gets really tricky and figures out:
 "The stuff was here last night at midnight
 and gone at 8 AM today.
 That means the stuff was stolen
 between midnight and 8 AM!"
 We call that last bit of detective work a theorem.
 In general, we could say:
 "The time something is stolen is between
 the last time somebody saw it where it was supposed to be
 and when somebody first noticed it was gone."
 We can also say that:
 Theorems are stuff we can figure out 
 from definitions, postulates, and other theorems
 that we already know.
 People in Math Town love to name stuff.
 The detective squad has written down:
 When We Were Robbed Theorem:
 If we were robbed, 
 the robbery happened between the last time we saw the stuff
 and when we find out it's gone.
 We could even write down what we know so far 
 as a formal Math Town proof:
 1) The stuff was here at midnight last night.
 2) The stuff was gone at 8 AM today.
 3) Nobody knows where the stuff is.
 Our stuff was stolen between midnight last night
 and 8 AM today.

 Definitions, Theorems and Postulates 
 are the tools the detective uses to figure out what happened.
 They work any place the detective goes.
 As time goes by, and the detectives figure out more stuff,
 they add more Theorems to their crime solving tool kit.
 Once a detective figures out a new theorem,
 all the detectives can use it just like the postulates,
 definitions and other theorems they already have
 to figure out even more theorems 
 and solve more crimes.
 To be good at their job,
 a detective needs a lot of tools.
 That means having a big list
 of theorems, postulates and definitions to use.
 The very best detectives don't even need to read their lists.
 They use them so often 
 that they know them all from memory.
 The given information are the facts of a case.
 Things like what was stolen, when we noticed it was gone,
 and stuff like that.
 The given information goes with one particular case.
 Theorems, postulates and definitions
 work with every case.

   copyright 2005 Bruce Kirkpatrick

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