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Geometry: Constructions

Parallel Lines

Problems

Parallel Lines, page 2

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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 %: 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 %: 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 %: 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.

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