



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 
