Geometry Heavy Duty Theorems and Postulates for Line Systems
Math-Prof HOME Geometry Table of Contents Ask A Question PREV NEXT

Double Cross Theorems & Postulates
Heavy Duty Theorems and Postulates for Line Systems

 

 1) If two parallel lines are crossed by another line,

 alternate interior angles are the same size.
 

 
 We could turn this puppy upside down and say:
 "If alternate interior angles are the same size,
 the two lines are parallel."
 This is called a corollary.
 
 2) If two parallel lines are crossed by another line,
 alternate exterior angles are the same size (congruent).
 

 
 This one has a corollary too!
 "If alternate exterior angles are the same size,
 the two lines are parallel."
 
 Example:
 Are the two lines that look parallel really parallel?
 

 
 NO!
 The corollary to either #1 or #2 says that alternate interior 
 or alternate exterior angles must be the same size 
 for the lines to be parallel.
 
 3) If two parallel lines are crossed by a third line,
 corresponding angles are the same size (congruent).
 

 
 So what's the corollary to this one?
 "If two corresponding angles are the same size,
 the lines are parallel."
 
 Example:
 Are the two lines that look parallel, really parallel?
 

 
 Hmmm, the two angles we were given aren't alternate interior angles,
 they aren't alternate interior angles,
 and they aren't corresponding angles.
 What do we do?
 
 WE FIGURE OUT SOME MORE ANGLES!
 
 We know  the angles on the other side of each line cross
 are the same as the angles we have.
 So we can write this ...
 

 
 We still don't have alternate interior, alternate exterior,
 or corresponding angles.
 That means we need to find more angles!
 A while ago we had a theorem that went like this:
 "If the sides of adjacent angles that are away from each other 
 make a straight line, the angles add up to 180 degrees."
 We can use this one to find the other angles.
 

 
 So the whole thing looks like this:
 

 
 Now we can use any one of the three corollaries.
 The lines that look parallel,
 ARE NOT PARALLEL
 
 At this point, most people would think
 that we've beaten up this double cross thing enough.
 
 Not math people.
 
 4) If two parallel lines are crossed by another line,
 interior angles on the same side add up to 180 degrees (supplementary).
 

 
 Everybody else had a corollary, so #4 wanted one too!
 Here it is ...
 
 "If interior angles on the same side add up to 180 degrees,
 the two lines are parallel."
 (If we had this one a minute ago, 
 we could have solved the example with one less step.)
 
 5) If two parallel lines are crossed by a third line,
 exterior angles on the same side add up to 180 degrees
 (supplementary).
 

 
 Here's a big shock.
 This one has a corollary too!
 
 "If exterior angles an the same side add up to 180 degrees,
 the two lines are parallel."
 
 So far, when two parallel lines were crossed by another line
 we have looked for two angles arranged in a certain way
 that were either the same or added up to 180 degrees right?
 
 So here's the last Theorem:
 
 6) If two parallel lines are crossed by a third line,
 ANY two angles are either the same size or add up to 180 degrees.
 
 This one does NOT have a corollary.
 
 Example:

 

 
 Any two angles are the same or add up to 180 degrees BUT:
 Alternate Interior angles are not the same.
 Alternate Exterior angles are not the same.
 Same Side Interior angles don't add up to 180 (Aren't supplementary)
 So since none of our known corollaries work,
 this one doesn't have a corollary.
 

   copyright 2005 Bruce Kirkpatrick

Math-Prof HOME Geometry Table of Contents Ask A Question PREV NEXT