Because lines extend infinitely in both directions, every pair of lines either intersect once, or don't intersect at all. The pairs of lines that never intersect are called parallel lines. Although parallel lines are usually thought of in pairs, an infinite number of lines can be parallel to one another.
The Parallel Postulate
The most important thing to understand about parallel lines is the parallel postulate. It states that through a point not on a line, exactly one line is parallel to that line.
The parallel postulate is very important in doing geometric proofs. It is basically a way to formally say that when given one line, you can always draw another line somewhere that will be parallel to the given line. In the problem section we'll see how to use the parallel postulate to find the measures of unknown angles.
Parallel Lines Cut by a Transversal
Whenever you encounter three lines, and only two of them are parallel, the third line, known as a transversal, will intersect with each of the parallel lines. The angles created by these two intersections have special relationships with one another. See the diagram below.
Angles, 1 and 5, 2 and 6, 3 and 7, and 4 and 8 are pairs of corresponding angles. Each is on the same side of the transversal as its corresponding angle.
Alternate Interior Angles
Angles 4 and 5, and 3 and 6 are pairs of alternate interior angles. They are on opposite sides of the transversal, and between the parallel lines.
Alternate Exterior Angles
Angles 1 and 8, and 2 and 7 are pairs of alternate exterior angles. They are on opposite sides of the transversal, and on the exterior of the parallel lines.
When AB and CD are Parallel
These eight angles would exist even if lines AB and CD were not parallel. However, when lines AB and CD are parallel, we can draw conclusions about the special angle pairs.
- The corresponding angles (1 and 5, 2 and 6, etc.) are congruent.
- The alternate interior angles (4 and 5, 3 and 6) are congruent.
- The alternate exterior angles (1 and 8, 2 and 7) are congruent.
- Interior angles on the same side of the transversal (3 and 5, 4 and 6) are supplementary.