Properties of parallel planes

Theorem

If two parallel planes intersect the third, then the direct intersections are parallel.



Evidence

According to the definition, parallel straight lines are straight lines that lie in the same plane and do not intersect. The straight lines lie in the same plane - the cutting plane. They do not intersect, since parallel planes containing them do not intersect. Therefore, straight lines are parallel. The theorem is proved.

Theorem

The segments of parallel lines enclosed between two parallel planes are equal ..



Evidence

Suppose α1 and α2 are parallel planes, a and b are parallel lines intersecting them, A1, A2 and B1, B2 are intersection points of lines with planes. Draw a line through a and b. It intersects the planes α1 and α2 along parallel straight lines A1B1 and A2B2. The quadrilateral A1B1B2A2 is a parallelogram, since its opposite sides are parallel. And the parallelogram has opposite sides. Therefore, A1A2 = B1B2. The theorem is proved.