What Are Parallel Lines?
We say that two lines are parallel if they are on the same plane and if they are always the same distance apart. In other words, two parallel lines will never touch each other no matter how far they extend. We can also say that any two lines that have the same slope are parallel.
Parallel lines also define some shapes, like squares, rectangles, and parallelograms. In fact, all shapes with four angles and opposite parallel sides are called parallelograms.
Testing To See If Two Lines Are Parallel
To find out whether two lines are actually parallel, there are several tests we can perform. They all start with a single straight line drawn through both parallel lines; this line is called a transversal.
Test #1: Corresponding Angles
Picture two parallel horizontal lines that are an inch or two apart. Now picture another line that crosses them both, at any angle. Where the transversal meets each line, four angles are created. The corresponding angles are the ones located in the same position on each line, for example, the upper right angle of each quadrant. If any pair of corresponding angles are equal, then the lines are parallel.
Test #2: Alternate Interior Angles
Interior angles are the angles that are between the two parallel lines. There are two pairs of alternate interior angles: the lower right of the top quadrant and the upper left of the bottom quadrant, and the lower left of the upper quadrant and the upper right of the lower quadrant. If either pair of alternate interior angles are equal, then the two lines are parallel.
Test #3: Alternate Exterior Angles
The exterior angles are on the ones on the outside of the two parallel lines. There are two pairs of alternate exterior angles: the upper right of the top quadrant and lower left of the bottom quadrant, and the upper left of the top quadrant and lower right of the bottom quadrant. If either pair of alternate exterior angles are equal, then the two lines are parallel.
Test #4: Consecutive Interior Angles
The consecutive interior angles are the ones between the two parallel lines and on the same side of the transversal. There are two pairs of consecutive interior angles: the lower right of the top quadrant and the upper right of the bottom quadrant, and the lower left of the top quadrant and the upper left of the bottom quadrant. If either of these pairs add up to 180 degrees, then the two lines are parallel.
Practice
Given the following information about each pair of lines, determine whether they are parallel.
- Two lines are crossed by a transversal. Two corresponding angles are 70 degrees and 110 degrees. Are these lines parallel?
Answer: No, they are not parallel because the corresponding angles are always equal.
- Two lines are crossed by a transversal. Two consecutive interiors angles are both 30 degrees. Are these lines parallel?
Answer: No, they are not parallel because two consecutive interior angles of two parallel lines always add up to 180.
- Two lines are crossed by a transversal. Two alternate interior angles are both 60 degrees. Are these lines parallel?
Answer: Yes, these lines are parallel.