When lines and planes are perpendicular and parallel, they have some interesting properties. You can use some of these properties in 3-D proofs that involve 2-D concepts, such as proving that you have a particular quadrilateral or proving that two triangles are similar.
**Three parallel planes:** If two planes are parallel to the same plane, then they're parallel to each other.

**Two parallel lines and a plane:**
- If two lines are perpendicular to the same plane, then they're parallel to each other.

- If a plane is perpendicular to one of two parallel lines, then it's perpendicular to the other.

**Two parallel planes and a line:**
- If two planes are perpendicular to the same line, then they're parallel to each other.

- If a line is perpendicular to one of two parallel planes, then it's perpendicular to the other.

And here's a theorem you need for the example problem that follows.
**A plane that intersects two parallel planes:** If a plane intersects two parallel planes, then the lines of intersection are parallel. Note: Before you use this theorem in a proof, you usually have to show that the plane that cuts the parallel planes is, in fact, a plane.

Here's the proof diagram.

Here's the final proof: