Geometry Test For Triangle Similarity
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This is Only a Test
Test For Triangle Similarity


 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.
 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


 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

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