



On the
last page we defined similar triangles.



To
review, 


we said
that they were triangles 


that had
all the same angles, but might be different sizes. 


A good
question is, 


How much
info do we need about a couple of triangles 


to tell
if they are similar or not. 





If we
know all three angles of two triangles, 


we can
tell right off if they are similar. 





If we
know all three sides of two triangles, 


we can
figure out if they are similar 





But what
if we are told the lengths of one or two sides 


and the
lengths of one or two angles? 





Are there
ways to tell if they are similar triangles? 





CASE
1: 2 Angles 


If we are
told the measure of two of the angles 


in each
of two triangles our job is easy. 


The
angles inside a triangle add up to 180 degrees 


so we can
find out the other angle. 


If the
three angles are the same in each of the triangles 


then they
are similar triangles. 





Example: 


Find out
if any two or all three of these triangles 


are
similar triangles ... 











Here's
what we do. 


The angles on
the inside of a triangle 


add up to
180 degrees. 


That
means ... 











So the
triangles on the ends are similar. 





So if we
only have the measure of two angles, 


we can
find out if the triangles are similar. 


We don't
need any other info. 





What if
we only have the measure of 2 sides? 





CASE
2: 2 Sides 


If we are
only told the lengths of two sides we're stopped. 


For
example, which of these are similar triangles? 











B and D
look close, 


but there
is no way to know for sure. 


So when
we only know the lengths of two sides, 


we can't
tell if the triangles are similar. 





CASE
3: 2 Sides and 1 Angle 


So two
sides doesn't get it, 


what
about two sides and an angle? 





This
time, the answer is ... It depends. 





DEPENDS?
Depends on what? 





It
depends on where the sides and angles we know are 


in
relation to each other. 





If the
angle we know is BETWEEN the two sides we know, 


then we
can tell if the triangles are similar. 











Otherwise
we can't tell. 











How come? 





The whole
answer is kind of technical, 


but
here's the easy part. 


Look at
these two triangles 











We know
two sides and an angle, 


and they
are both the same. 


But it's
easy to tell that they are not similar 





It's like
there is a hinge at the top of the triangle 


that lets
us make two different triangles 


with
these same numbers ... 











When
there is something like this that is too technical to prove 


without a
whole lot of math shenanigans, 


we can
just give you a rule and say 


"Never
mind why for now, this is just how it works" 





That's
fine with most people, 


but math
people aren't most people. 





So they
invented a way to say these things in really fancy math talk called 





THEOREMS
and POSTULATES






And then
they string them together in things called proofs. 





Most
geometry books and courses 


hit you
with these things very early on in the course 


while you
are still trying to just figure out 


what all
this junk means. 





They are
included in these pages too, but way at the end. 


We want
tow wait until you understand this stuff as much as possible 


before
hitting you with that #&$@%. 





copyright 2005 Bruce Kirkpatrick 
