Important Stuff From Geometry
   
 Most people know kind of what an angle is.
Related Chapters
   
 You can see them at the point of a slice of pizza ...  
 
 
 The "lift" in a skateboard jump ramp ...
 
 
 and stuff like that all have angles we know about.
 
 To get really technical,
 if you take two different rays that start at the same point
 you get an angle ...
 
 
 There are lots of ways to measure angles.
 The way that gets used the most is called degrees.
 
 Why do they call it degrees?
 Who knows! 
 And who really cares?
 
 They could have called it anything,
 degrees is just a name.
 
 The important thing is how it works.
 
 It works kind of like a crazy watch that runs backwards.
 
 Example:
 Start with a ray at 3 O'clock ...
 

 

 
 Now put another ray at 12 O'clock.
 

 

 
 The angle measure from the 3 O'clock ray 
 to the 12 O'clock ray ...
 

 

 
 Now take away the ray at 12 O'clock 
 and put in one at 9 O'clock ...
 

 

 
 There are 3 hours from 12 O'clock to 3 O'clock
 and 90 degrees.
 There are 6 hours from 9 O'clock to 3 O'clock.
 Guess how many degrees it is?
 

 

 
 Now take away the ray at 9 O'clock 
 and put one in at 6 O'clock.
 
 It's 9 hours from 6 O'clock to 3 O'clock
 How many degrees do you think it is?
 

 

 
The degrees are the measurement of how far you must spin
 (math types say rotate)
 to get from pointing in the direction of one line to pointing
 in the direction of the other.
 The next thing you need to know is a bit about is triangles.
 What makes a triangle a triangle? 
 Three straight lines all connected end to end to end. 
 All three lines might be the same length, 
 two might be the same length or none of the lines might be the same length. 
 
 In geometry class, you probably had to learn all kinds of names
 like equilateral, and obtuse, and iscoceles, and stuff like that. 
 Well, in trig we couldn't care less about those names and postulates 
 and all those things. 
 
In fact, there's only 3 things we really need from geometry.
 
  1) If you add up the 3 angles on the inside of a triangle you get 180 degrees. 
No matter what the triangle looks like.
 
  2) If one of the angles on the inside of the triangle is 90°,
  the triangle is called a right triangle. 
 The triangle is drawn with a little square on the 90° angle 
 to show that it is 90°
 The side of the triangle across from (called opposite) the 90° angle 
 (labeled c here) is the longest side of the triangle. 
 It gets a special name. It's called the hypotenuse.
 If you multiply the hypotenuse times itself,
 the answer you get is always the same number that you get
 when you multiply the other two sides times themselves
 and then add that together.
 That little trick is called the Pythagorean Theorem.
 
3) (The most important one.)
Say we have a small right (one 90° angle) triangle.

and we make a really big copy of it.

 now even though the copy is much bigger, it's still exactly the same shape. 
   That means the angles are still the same. 
 The pointy little angle on the right 
 is the same number of degrees in the little triangle as it is in the big copy.
 The other angle (in the upper left) is still the same size, 
 and the 90° angle is still 90°. 
 OK, Big deal, so what?
  Here's the deal. 
 Let's label the sides of the little triangle a, b, and c, 
 and let's label the sides of the big copy A, B, and C.
 Suppose the ratio of the length of b to the length of c
 (which is the fraction b/c) is:
THEN ON THE BIG TRIANGLE:

 
 That means, 
 that if we have a right triangle of any size with these same angles, 
 the length of the side across from the little pointy angle on the right (B) 
 divided by he length of the hypotenuse equals 1/3.
 
 In geometry, this idea is called the law of similar triangles. 
 When it was first discovered, average people said 
 "Oh, that's sort of neat, I guess" but math types said: 
 "Wow! I see big time job security here. 
 We can take that little puppy and build an entire area of math out of it. 
 We'll call it trigonometry, and with all the shenanigans we can do with it, 
 we'll all have jobs for life!" And they did. 
 All of trigonometry comes from the law of similar triangles.
 

copyright 2008 Bruce Kirkpatrick