physics (geometry bit)
posted by physics (geometry bit) on .
h t t p : / / i m g 4 3 . i m a g e s h a c k . u s / i m g 4 3 / 4 7 4 1 / 9 4 7 4 7 1 1 6 . j p g
ok the two read angles are equal to each other because they are complementary
if i remeber correctly the only thing we can conclude is that when we add the two angles togther we get 90 degrees... how does that necessarily mean they are equal? How do we even know they are complementary? I remeber in geomentry we did statement and reason proofs like so
statements  reasons
________________________
were the statements would be just statements and the reasons are why the statements are true...
also I still don't see the triangle =[, even looking at it from the red thetas that i've been told are complemetary,
i labeled the forces with the names of the parts of the triangles they must be in order for us to resolve into it's components...
however i still don't see the triangle...

http://img43.imageshack.us/img43/4741/94747116.jpg
The two red angles are not equal. They indeed are complementary. They add to 90 degrees.
Set something flat like a piece of cardboard in front of you horizontal on your desk or table.
Set something on it like a paper clip.
Presently the slope, theta, is zero and the angle the gravitational force makes with the cardboard is 90 degrees.
Now tilt the cardboard up a small angle theta.
You can see that the gravitation force vector now gets a little off perpendicular to the cardboard, in fact by theta.
It now has a component perpendicular to the cardboard of mg cos theta
and a component in the plane of the cardboard of mg sin theta down the slope.
To draw a triangle with this, draw a line parallel to the cardboard through the tip of the downward pointing m g vector.
Then you have a hypotenuse down of mg, a component almost down for small theta of m g cos theta, and an opposite side parallel to the slope of mg sin theta. 
By the way your drawing is not very accurate. The mg sin theta arrow should be parallel to the slope and the mg cos theta arrow should be perpendicular to the slope. If you draw it more carefully the little triangle will make more sense.