



Sometimes
we have two triangles that are the same shape,



but one
is bigger than the other. 


When we
have this, 


the
angles will be the same in each triangle. 


The
length of the sides, however, will not be the same. 











The
triangle on the right has the same angles 


as the
triangle on the left, but is twice as big. 


These
triangles are similar. 











In the
triangle on the right, the sides are 1.5 times as long 


as the
sides in the triangle on the left. 


These
triangles are similar. 





So in
similar triangles, the angles are the same. 


Each of
the sides in the bigger triangle 


are equal
to the length of the sides in the smaller triangle 


times
some number. 


That
number is the ratio of the length of the sides. 


An easy
way to find the ratio of the side length 


between
two similar triangles is to take the length 


of one of
the sides in the bigger triangle 


and
divide it by the length of the CORRESPONDING side 


in the
smaller triangle. 











If two
triangles with the same angles are the same size, 


we can
call them similar if we want. 


But we
say more if we call them congruent, 


so that's
what we usually do. 





(Maybe
you can guess what's coming next ...) 





Usually
we don't know the length of all of the sides. 


We
usually don't know the measure of all of the angles 


So just
what do we need to know? 





Well. any
of the theorems that showed that two triangles are congruent 


also show
that they are similar, but there are others ... 








ANGLE
 ANGLE  ANGLE 


If all
three angles of one triangle 


are the
same as all three angles of the other triangle, 


the
triangles are similar. 





ANGLE
 ANGLE 


If we
know that two angles of one triangle 


are the
same as two of the angles of the other triangle, 


the
triangles are similar. 


(This one
is kind of cheezy. If we know 2 angles we actually know all 3) 





SIDE 
ANGLE  SIDE 











When we
know two sides 


and the
angle between them in two triangles and ... 


1) The
angles are the same 


2) The
ratio of the corresponding sides are the same 


Then the
triangles are congruent. 





copyright 2005 Bruce Kirkpatrick 
