The existence of a plane parallel to the plane

Theorem

Through a point outside this plane, you can draw a plane parallel to this one, and, moreover, only one.



Evidence

Let us draw in the given plane α some two intersecting straight lines a and b. Through this point A we draw parallel lines a1 and b1 parallel to them. The plane β passing through straight lines a1 and b1, according to the theorem on the sign of parallelism of planes, is parallel to the plane α.



Suppose that through the point A passes another plane β1, also parallel to the plane α. Let us mark on the plane β1 some point C which is not lying in the plane β. Draw the plane γ through the points A, C and some point B of the plane α. This plane intersects the planes α, β, and β1 along straight lines b, a, and c. The lines a and c do not intersect the line b, since they do not intersect the plane α. Therefore, they are parallel to the straight line b. But in the plane γ, only one straight line can pass through point A, parallel to straight line b. which contradicts the assumption. The theorem is proved.