The alpha and beta planes are parallel, and the alpha plane intersects some straight

| September 20, 2021 | education

The alpha and beta planes are parallel, and the alpha plane intersects some straight line a. Prove that the beta plane also intersects the straight line a.

Let us prove by contradiction. Let the straight line a and the plane b do not intersect, then they are parallel. Let’s call the point of intersection of the plane a and the straight line a – M. Through the point of the plane b – C, as well as the straight line a, we draw a plane that intersects the plane a along the straight line c, and the plane b along the straight line p. Straight lines p and c are parallel (by the properties of parallel planes and cutting plane). Lines a and c are also parallel (lie in the same plane and do not intersect according to our assumption). We get that two straight lines parallel to c – a and p pass through the point M, and this contradicts the axiom of parallelism. Our assumption is incorrect, straight line a and plane b intersect.



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