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. Click any
of the links below to start understanding parallel lines better!
How to tell if lines are parallel
Using parallel lines to find information about triangles
Quiz on Parallel Lines
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 intersect
and that are on different planes. Intersecting lines
are lines that have a point in common.
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.
1. Given: r is parallel to s angle 1 = 60 degrees Find the measures of the other seven angles in the accompanying figure (below).
There are four different way you can see if lines are parallel. Each one is outlined below.
Problem: If angles 2 and 3 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. r is parallel to s by the above rule.
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:
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:
1. Problem: Show that lines a and b in the 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
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 us do this.
1. Problem: Find the measure of each numbered angle in the figure below. 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 because 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. (Very useful to review or to see if you've really got this topic down.) Do it!