Constructions

Parallel Lines

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.

Figure 3.1: Parallel Lines

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.

Figure 3.2: The parallel postulate

In the above figure, we have line AB and a point C not on the line. The Parallel Postulate states that there exists one line through C which is parallel to line AB. As you know, an infinite number of lines can be drawn through point C, but only one of them will be parallel to line AB.

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.

Figure 3.3: Parallel lines cut by a transversal

Lines AB and CD are parallel. Line EF, the transversal, is parallel to neither, so it intersects with each. This intersection creates eight angles, numbered one through eight. The special pairs of angles are as follows:

Corresponding Angles

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.

These relationships form the bases of many a geometric proof, so it's important to understand them.