Two different lines will either intersect at a single point or they will never intersect. If lines never intersect, we say they are parallel.
If a pair of lines are not marked with chevrons, how can we tell if they are parallel or not? Is the point of intersection off the edge of the diagram or do the lines not intersect at all? To tell these cases apart we introduce a third line, called a transversal, that intersects both of the original lines.
The transversals (in blue) are drawn to test whether the lines they intersect are parallel. The transversal forms angles at the intersection points, and there are three useful pairs of angles that are created.
Select the diagram showing a pair of parallel lines:
Parallel lines are lines that never intersect and are marked with chevrons like this: >.
A transversal is a line that intersects a set of parallel lines.
The first kind of angles are called cointerior, the two angles formed between the original lines, and on the same side of the transversal:
The second kind of pair are called alternate angles, lying between the original lines on opposite sides of the transversal:
The alternate angles are congruent and are marked the same way on a pair of parallel lines.
We can investigate these angle relationships more using the applet below.
Click and drag any blue points to adjust the lines.
Check the boxes to highlight different angle pairs.
Do the relationships change as the points on the lines are moved?
Alternate angles are equal while the cointerior angles are supplementary or have a sum of 180\degree. These relationships do not change as the points on the lines are moved.
Cointerior angles formed on parallel lines are supplementary.
Alternate angles formed on parallel lines are equal.
The final type of angles, called corresponding angles, lie in the same relative position on each line:
The corresponding angles are marked the same way because they are equal.
We can investigate this angle relationship using the applet below.
Click and drag any blue points to adjust the lines.
Check the box to show all corresponding angles.
Do the relationships change as the points on the lines are moved?
All pairs of corresponding angles on parallel lines are equal and this relationship does not change as points on the line are moved.
We can tell whether lines are parallel using one of these criteria:
Cointerior angles formed on parallel lines are supplementary. If they are not supplementary, the lines are not parallel.
Alternate angles formed on parallel lines are equal. If they are not equal, the lines are not parallel.
Corresponding angles formed on parallel lines are equal. If they are not equal, the lines are not parallel.
Note: This lesson refers to lines, but all of these ideas also apply to rays and segments. Rays and segments are parallel if the lines through their defining points are parallel.
If we know two lines are parallel, we can use the criteria mentioned to find the values of other angles.
Consider the diagram below:
Which of the following are true statements?
Is there a pair of parallel lines in the diagram?
Solve for the value of x.
Corresponding angles, lie in the same relative position on each parallel line.
We can tell whether lines are parallel using one of these criteria:
Cointerior angles formed on parallel lines are supplementary. If they are not supplementary, the lines are not parallel.
Alternate angles formed on parallel lines are equal. If they are not equal, the lines are not parallel.
Corresponding angles formed on parallel lines are equal. If they are not equal, the lines are not parallel.