Geometry Building a Geometry Detective Kit
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Bag of Tricks
Building a Geometry Detective Kit


 Well the last page was kind of interesting,

 but what did it have to do with math?
 We prove stuff in math the same way 
 as those detectives work.
 We have our bag of definitions, theorems and postulates.
 We show up at the crime scene (the problems).
 We use our tools and the information given at the scene
 to solve the problem.
 You have already worked with most of these ideas,
 we just didn't get real fancy and call them 
 theorems and postulates back then.
 This time they're all dressed up.
 It' s math detective school.
 Just a place in space. So tiny that we could never see it.
 We just know where it is.
 A straight line coming out of a point
 in some direction. It goes forever ...

 Like a ray, but it goes forever in both directions.

 A piece of the line.

 Like a very flat sheet of paper.
 But the plane has no edges.
 It goes forever ...

 Say there were two identical twins
 that were so much alike that nobody could tell them apart.
 Everything about them was the same,
 but they aren't the same person.
 They are CONGRUENT.
 So in math, two line segments that are the same length
 or two angles that are the same number of degrees
 or two objects that are the same size and shape
 If the piece of a line segment on one side of a point
 is the same length as the piece of the line segment
 on the other side of the point,
 that point is the midpoint of the segment 
 (and the two pieces are congruent).


 This is like the midpoint deal,
 but this time we use another line segment (or ray or line)
 to cut the segment into two equal pieces.

 A ray that cuts an angle into 2 angles 
 that are equal to each other.
 For example:

 Any two angles that add up to 180 degrees.

 Any two angles that add up to 90 degrees

 Two angles that share one side but don't overlap

 When 2 lines cross, the angles across from each other
 are called VERTICAL ANGLES.
 They have the same measure.

 When two lines cross, if all the angles at the cross are 90 degrees
 the lines are PERPENDICULAR.

 We could also turn this definition upside down and say this.
 If two lines that cross are perpendicular 
 all the angles are 90 degrees.
 When we can turn something like this upside down
 and say something else,
 that other thing we can say is called a COROLLARY.
 Hey, we know what a bisector is.
 We just had what perpendicular means.
 Just put them together.

 If a line (or segment or ray) cuts another segment 
 into two equal pieces and also makes 90 degree angles, 
 it is the PERPENDICULAR BISECTOR of the other segment.
 The distance between two points
 is the length of the line segment between them.

 Two lines that are in the same plane 
 that never cross are parallel.

 Two line segments or rays that are always 
 the same distance apart are PARALLEL.


   copyright 2005 Bruce Kirkpatrick

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