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Skew Lines

Objective

In this lesson, we will define skew lines and show how to illustrate them.

Previously Covered:

  • If the sum of the measures of two angles is 180°, then the angles are called supplementary, and each angle is called the supplement of the other.
  • If the sum of the measures of two angles is 90, then the angles are called complementary and each angle is called the complement of the angle.
  • The angle bisector of angle BAC is the ray such that angle BAD angle DAC.

Skew Lines

We already know that in a plane, two lines can either be parallel or can intersect at one point. There is one more possibility for two lines in a plane: they can be coincident. This means that they are the same line. So there are three possibilities for defining a pair of coplanar lines: parallel, intersecting, or coincident.

What about lines in three-dimensional space?

There are actually four ways that two lines can be situated in three-dimensional space . We have all of the possibilities for coplanar lines, plus one more.

Two lines that are not coplanar and do not intersect are called skew lines.

The fact that this is the only other possibility in three dimensions is guaranteed by the following two theorems.

Two parallel lines lie in a unique plane. Two intersecting lines lie in a unique plane. If two lines are actually the same line, then they are automatically coplanar.

Therefore, by the theorems above, skew lines are distinct lines that do not intersect and are not parallel.

Question

Which choice describes a pair of skew lines in the figure below?

Skew lines

  1. L and M
  2. M and N
  3. N and O
  4. L and O

Reveal Answer

The correct answer is D . The pairs (L, M) and (M, N) are parallel. This fact is verified by the faces of the rectangular prism shown in the figure . The pair (N, O) intersects . L and O are therefore the only possible skew lines.

The example shows how one can use a rectangular prism to illustrate skew lines.

Other polyhedra can also be used to illustrate skew lines.

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