You get a bonus - 1 coin for daily activity. Now you have 1 coin

Sign of parallel planes

Lecture



Two planes are called parallel if they do not intersect.

Theorem

If two intersecting straight lines of one plane are respectively parallel to two straight lines of another plane, then these planes are parallel.

Sign of parallel planes

Evidence

Let α and β be the given planes, a1 and a2 are straight lines in the α plane, intersecting at point A, b1 and b2 are, respectively, straight lines parallel to them in the β plane.
Suppose that the planes α and β are not parallel, and therefore intersect along some straight line c. According to the theorem on the parallelism of a straight line and a plane, straight lines a1 and a2, as parallel straight lines b1 and b2, are parallel to the plane β, and therefore they do not intersect the straight line c lying in this plane. Thus, in the plane α through lines A pass the lines a1 and a2, parallel to the line c. This is impossible according to the parallel axiom. That contradicts the assumption. The theorem is proved.

Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Stereometry

Terms: Stereometry