Parallel Lines        Congruent Tri.        Congruent R. Tri.        Isosc. and Equil.        Quadrilaterals        Parallelograms        Ratios        Similar Polygons        Special Triangles       Circles        Area        Coordinate Geo.        Triangle Ineq.        Solids        Computer Fun 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! Transversals 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. 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. ```1.   Given: r is parallel to s angle 1 = 60 degrees Find the measures of the other seven angles in the accompanying figure (below). 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 Transveral Postulate.``` There are four different way 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 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 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 the following figure in which lines AB and CG are skew. 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. Another special rule concerning triangles is outlined below. The measure of an exterior angle is equal to the sum of the measure of its remote interior angles. ```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!

Math for Morons Like Us - Geometry: Parallel Lines
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