Who is asking: Student
Level of the question: Secondary
Question: I was wondering how I would prove the following theorem. I am completely lost at this point and all my other ideas have been extremely complicated and somewhat blurry.
"Through a point outside a given plane, there is at least one line parallel to the given plane."
(Note: I say "at least one" instead of "exactly one" because uniqueness is not important for the purpose this theorem will serve in another proof. Thanks kindly in advance.)
To help you, we would really have to know what axioms and theorems you are allowed to use. My guess is that you will want to drop a perpendicular line p from your point (say, P) to the plane, which intersects the given plane at Q. Next, draw a line m through Q in your given plane. Finally, in the plane determined by the lines p and m draw the perpendicular to p through P. That line will be parallel to m, so it will be parallel to the given plane.
Does this help?