Geometry: Parallel Lines

On this page we hope to clear up problems that you might have with parallel lines and their uses in geometry. Parallel lines seem rather innocent, but are used in some complex geometry situations to help you solve problems. Follow any of the links below to start understanding parallel lines better!

Transversals
How to tell if lines are parallel
Using parallel lines to find information about triangles
Quiz on parallel lines

Definitions

Before you start working with parallel lines, you have to know what parallel lines are. Parallel lines are coplanar lines that do not intersect. There are two other kinds of lines you need to know about. One is skew lines, which are lines that do not interset and are on different planes. Intersecting lines are lines that have a point in common.

Transversals

A transversal, or a line that intersects two or more coplanar lines, each at a different point, is a very useful line in geometry. Transversals tell us a great deal about angles.

There is a special rule used in geometry (the Transversal Postulate) that involves angles and transversals. It says that if two parallel lines are intersected by a transversal, then the corresponding angles are congruent.

Example:

1. Given: r is parallel to s
          angle 1 = 60 degrees
          Find the measures of the other seven angles in the given figure.
   Solution: Angle 2 = 120 degrees since it is supplementary to
             angle 1.  Supplementary angles are any two angles
             whose sum is 180 degrees.
             Angle 3 = 60 degrees since angle 1 and angle 3 
             are vertical angles.  Vertical angles are two nonadjacent
             angles formed by two intersecting lines.
             Angle 4 = 120 degrees since it is supplementary to angle
             1.
             Angle 5 = angle 1 by the Transversal Postulate.
             Angle 6 = angle 2, angle 7 = angle 3, and angle 8 = angle 4
             by the Transversal Postulate.

How to Tell if Lines are Parallel

There are four different ways you can see if lines are parallel. Each one is outlined below.

1. If two lines are cut by a transversal, and the corresponding angles are congruent (congruent angles have the same measure), the lines are parallel. Example:

Problem:  If angles 2 and 3 in the given figure are congruent, are lines
          r and s parallel?
Solution: Angle 2 = angle 3 - Given.
          Angle 1 = angle 2 - Vertical angles are congruent.
          Angle 1 = angle 3 - Transitive Property: If a = b and b = c, 
                              then a = c.  In this case, a would be
                              angle 1, b would be angle 2, and c
                              would be angle 3.

          r is parallel to s by the above proof.

2. If two lines are cut by a transversal, so that alternate interior angles are congruent, the lines are parallel. In the example below, line e is parallel to line f because angle 4 is congruent to angle 3. Example: Example Figure

3. If two lines are cut by a transversal so that interior angles on the same side of the transversal are supplementary, the lines are parallel. Example:

Problem:  Show that lines a and b in the given figure are parallel.
Solution: Since angle 1 and angle 2 are both 90 degrees, they are
          supplementary.  By the statement above, they (lines a and b
          are parallel.

4. If two lines are perpendicular to another line, and they (the two lines) are in the same plane, then they are parallel.
The example problem above is a perfect example of this.
Be careful with this rule because it is possible to have two lines that are perpendicular to a third line that are not in the same plane. This is illustrated in this accompanying figure, in which lines AB and CG are skew.

Triangles

Since you know that triangles consist of three angles that have measures that equal 180 degrees when added together, you will sometimes be asked to find the measures of angles in triangles. Parallel lines help you do this.

Another special rule concerning triangles is outlined below.

1. The measure of an exterior angle is equal to the sum of the measure of its remote interior angles. Example: Example Figure

Example:

1. Problem:  Find the measure of each numbered angle in the given figure.
   Given:    Line GH is parallel to ray DK.
             Angle 6 = 75 degrees.
             Angle 2 = 30 degrees.
   Solution: Angle 5 = 105 degrees since it is supplementary to
             angle 6.
             Angle 4 = 45 degrees becuase of the rule outlined above.
                  (Angle 4 + angle 2 = angle 6, so angle 4 = angle 6 -
                  angle 2.)
             Angle 1 = 45 degrees since angles 1 and 4 are
             alternate interior angles.
             Angle 3 = 105 degrees since angles 3 and 5 are
             alternate interior angles.

Take the quiz on parallel lines. The quiz is very useful for either review or to see if you've really got the topic down.

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