Geometry Systems of Three Lines
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The Double Cross
Systems of Three Lines

 

 Now let's put this line stuff together.

 Take two parallel lines ...
 

 

 
 add a line that crosses the two ...
 

 

 
 and some special things happen.
 
 The cross on the top
 

 

 
 makes the same angles as the cross on the bottom.
 

 

 
 There's a lot of math shenanigans that show why this is so.
 You really don't need any of that stuff right now.
 You can probably see that when you have two lines that cross ...
 

 

 
 moving the horizontal line up and down
 doesn't change the size of any of the angles ...
 

 

 
 So one line that is moved up,
 or two parallel lines are
 REALLY THE SAME THING!
 

 

 
 That means angles 1 and 3 AND 
 5 and 7 are all the same size!
 
 For that same reason we can say that angles
 2 and 4 AND 6 and 8 are all the same size.
 
 Here's how problems you might get with this work.
 
 Example:
 Say you have two parallel lines and a line that crosses them.
 Say the angles are labeled with letters from A to H.
 They tell you that the measure of angle A is 119 degrees
 and ask you to find the measure of the other angles.
 

 

 
 F = H = D = 119 degrees
 

 

 

 

 
 More Examples:
 Another trick that you might have noticed is that
 any two angles that are next to each other
 add up to 180 degrees
 

 

 
 Or an even trickier thing is that 
 two angles on the same side of the crossing line 
 closest to each other add up to 180 degrees
 

 
 and that two angles on the same side of the crossing line 
 farthest from each other add up to 180 degrees too!
 

 
 In fact, we can pick any two angles.
 They will either be equal to each other 
 or add up to 180 degrees.
 
 As you can see, all of this is no big deal.
 But we know that math people LOVE to think up names for stuff.
 So they took this one little trick 
 and thought up about a zillion names for it.
 

   copyright 2005 Bruce Kirkpatrick

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