Lines and planes in space

The following is a list of basic facts about lines and planes in space. No attempt is made to deduce all of them from a few basic principles; instead we take a purely descriptive approach. We don't repeat facts from plane geometry. Words or phrases being defined are in italics.

Some terminology: Let A be a point, let L be a line, and let P be a plane. If A is an element of the set L, we say A is contained in L. One also may say A lies on L, or L passes through A. The same terminology is used if A is an element of the set P: A is contained in P, or A lies on P, or P passes through A. And if the line L is a subset of the plane P, we say that L is contained in P, or L lies on P, or P passes through L.

If A, B, and C are three noncollinear points, there is exactly one plane P such that A, B, and C are contained in P. (One also says there is a unique plane P such that A, B, and C are contained in P, or there is one and only one plane P such that A, B, and C are contained in P. Informally, one says "Three points determine a plane.")

If two distinct points A, B are lie on the plane P, then every point of the line L through A and B lies on P. Thus if a line L intersects a plane P, either L intersects P in exactly one point, or L is contained in P.

If L is a line and A is a point not in L, there is a unique plane containing A and L. If L and M are distinct lines intersecting at one point, there is a unique plane containing L and M.

Two lines are parallel if they do not intersect and they lie in the same plane. Two lines that do not intersect and do not lie in the same plane are called skew. The line L is parallel to the plane P if L does not intersect P. If two distinct planes intersect, then their intersection is a line. Two planes are parallel if they do not intersect.

If the point A is not in the plane P, there is a unique plane through A parallel to P.

If two planes P and Q are parallel, and R intersects P, then R intersects Q, and the line of intersection of R and P is parallel to the line of intersection of R and Q.

The angle between two rays in space with the same endpoint is measured just as in plane geometry, since the two lines containing the rays determine a unique plane containing the rays.

The line L is perpendicular to the plane P if L intersects P at a point A, and every line in P through A is perpendicular to L. If L intersects P at A, and L is perpendicular to two distinct lines in P through A, then L is perpendicular to P. Given a point A and a plane P, there is a unique line L through A perpendicular to P. Given a point A and a line L, there is a unique plane P through A perpendicular to L.

Two distinct planes P and Q are perpendicular if the intersection of P and Q is a line L, and for every point A of L, if M is the line in P perpendicular to L at A, and N is the line in Q perpendicular to L at A, the M and N are perpendicular. If this hold for just one point A of L, then it holds for all points A of L. If the line L is perpendicular to the plane P, then every plane containing L is perpendicular to P. If P and Q are perpendicular, with intersection L, then every line in Q perpendicular to L is perpendicular to P.

Let P and Q be distinct planes with intersection a line L. Let P' be one of the half-planes of P determined by L, and let Q' be one of the half-planes of Q determined by L. The angle (also called the dihedral angle) between P' and Q' is measured as follows. Let A be a point of L, and let R be the plane through A perpendicular to L. Then the intersection of R and P' is a ray p' with endpoint A, and the intersection of R and Q' is a ray q' with endpoint A. The angle between P' and Q' is the angle between the rays p' and q'. The planes P and Q are perpendicular if and only if the angle betweeen P' and Q' is 90 degrees. (If P and Q are the same plane, the angle between them is 0; the previous definition can be applied to every line L in P.)

Let P and Q be planes, and let R be a plane that is not parallel to P or Q. (R is a transversal of P and Q.) Then P and Q are parallel if and only if any one of the following conditions holds:

Corresponding dihedral angles of P and Q with the transversal R are equal.
Alternate interior dihedral angles of P and Q with the transversal R are equal.
Same side interior dihedral angles of P and Q with the transversal R are supplementary.

Let P be a plane and L a line such that the intersection of L and P is a point A, and L is not perpendicular to P. The angle between L and P is measured as follows. Let Q be the plane through A perpendicular to L, and let M be the intersection of P and Q. Let R be the plane containing L and M. The angle between L and P is the angle between R and P. (If L is perpendicular to P, the angle between L and P is 90 degrees; the previous definition can be applied to every plane Q containing L. If L is contained in P, the angle between L and P is 0; the previous definition can be applied to every point A on L.)

Many theorems of solid geometry are analogous to theorems of plane geometry. Here is an example:

Perpendicular Bisector Theorem. If A and B are distinct points, the set of points that are equidistant from A and B is the plane P that is the perpendicular bisector of the segment AB. In other words, P is perpendicular to the line L through A and B, and P intersects L at the midpoint of the segment AB of L.

There are theorems about tetrahedra (not necessarily regular) analogous to theorems about triangles, and there are theorems about spheres analogous to theorems about circles. It is an excellent mental exercise to look for these analogies!

January 21, 2008