Relate inclined plane's angle of incline to FBD's weight COM

[Tyrannosaurus_]

Please show me the relationship that proves how an inclined plane's angle of incline, θ, is equal to the angle used to find the components of gravity force.

I know that the angles are the same, but I'd like a way to show it.

Thanks!

 

[billy_joule]

If you extend a few lines two similar triangles will appear:

https://www.mathsisfun.com/geometry/triangles-similar.html

 

[Tyrannosaurus_]

Thanks for the reply! I do appreciate your help, and support.
I understand similar triangles.

I am looking for a relationship that is simple to communicate with students. Currently, I use a physical model with a weighted string, & some cardboard arrows to show the x & y axis.

All help & discussion is appreciated.
I would appreciate it a little more if you could find, or show me the relationship explicitly; instead of, implying the relationship is there, & I'm too stupid to figure it out. I'd like the proof. I expect parallel line theorem is used. Is this true? Can someone show this to me, please.

Thank you!

 

[A.T.]

I'd like the proof.

http://math.stackexchange.com/quest...ese-angles-equal-for-object-on-inclined-plane

 

[Tyrannosaurus_]

Elegant & Exact.
A sincere Thank You!