In this explainer, we will learn how to name and identify angle pairs formed by parallel lines and transversals and recognize their relationships to find a missing angle.
Before looking at the situation of two parallel lines cut by a transversal line, let us recall what vertically opposite angles are and some of the properties of angles between two intersecting lines.
Recap: Angles between Intersecting Lines
If two lines intersect at a point, then we call the pair of angles opposite to each other “vertically opposite angles.”
Any pair of vertically opposite angles are congruent (they have the same measure).
Since the adjacent angles combine to make a straight angle, we can note that the sum of their measures must be .
We call a pair of angles whose measures sum to give supplementary angles. We have shown that any adjacent angles in a pair of intersecting lines are supplementary angles.
We can extend these ideas and properties to a transversal of parallel lines. Recall that a transversal of parallel lines is a line that intersects both lines.
We see that and is a transversal of these lines. We can identify 4 pairs of vertically opposite angles by considering the angles around the points of intersection between and , and and .
We have that and , and , and , and and are pairs of vertically opposite angles. This is not the only relationship between these angles. Since , we can note that the angles that lie in the same position at the intersection between the transversal and each line are congruent. These are called corresponding angles.
Definition: Corresponding Angles
The angles that lie in the same position at the intersection between the transversal and two lines are called corresponding angles.
If the pair of lines connected by the transversal are parallel, then the corresponding angles have equal measures.
Let us now see some examples of using the congruence of corresponding and vertically opposite angles to find some missing angle measures in diagrams.
Example 1: Identifying the Measure of an Angle
In the figure below, find .
Answer
We first note that we are given two parallel rays:
We can extend these rays to get a pair of parallel lines and extend to get the following.
We have a pair of parallel lines cut by a transversal. We see that angles and occupy the same position at the intersections between the transversal and the parallel lines. Thus, these angles are corresponding angles.
We recall that when a pair of parallel lines are cut by a transversal, any corresponding angles have equal measures. Hence,
Since vertically opposite angles and corresponding angles have equal measures when the pair of lines are parallel, we can combine these two results to see more pairs of corresponding angles.
Thus, these two angles have the same measure.
We call these “alternate exterior angles” since they lie on either side of and on the outside of and . We can form “alternate interior angles” in the same way.
Thus, the following angles have the same measure.
These are called “alternate interior angles” since they lie on either side of and on the inside of and . We can define these formally as follows.
Definition: Alternate Interior and Exterior Angles
Alternate interior angles are pairs of angles from either set of the four angles formed by a transversal line cutting two (here parallel) lines and that are on either side of and on the inside of and .
Alternate exterior angles are pairs of angles from either set of the four angles formed by a transversal line cutting two (here parallel) lines and that are on either side of and outside and .
If the transversal cuts a pair of parallel lines, then the alternate interior angles are equal in measure. Similarly, if the transversal cuts a pair of parallel lines, then the alternate exterior angles are equal in measure.
These angle congruences are usually all referred to as the parallel line theorems.
Let us now see some examples of using alternate interior and exterior angles to determine the measures of angles given in a diagram.
Example 2: Finding the Measure of an Angle
Work out the value of in the figure.
Answer
We start by noting that we are given a transversal and a pair of parallel lines. In particular, the two angles given are alternate interior angles since they are inside the two parallel lines (therefore, they are interior angles) and on either side of the transversal line.
We can recall that the alternate interior angles of a pair of parallel lines cut by a transversal have an equal measure. Hence,
Example 3: Determining an Angle Measure from a Diagram
Find the value of in the figure.
Answer
We first note that we are given a pair of parallel lines and a transversal of these parallel lines. We can also see that the two given angles lie on either side of the transversal and outside the parallel lines.
We call angles like this alternate exterior angles. In particular, we can recall that any alternate exterior angles in a transversal of parallel lines are congruent.
Hence, we have
In our next example, we will determine the relationship between two given angles that are not vertically opposite, corresponding, alternate interior, or alternate exterior angles.
Example 4: Finding an Unknown Angle in Problems Involving Parallel Lines
From the information in the figure below, find .
Answer
We first notice that we have three parallel lines, with each pair cut by a transversal line. We can mark on the diagram to help us visualize the angle whose measure we are trying to determine.
We can see that is composed of two angles and . Therefore,
We can determine the measures of these angles using the parallel lines. We see that and are interior angles on opposite sides of the transversal.
We can recall that the alternate interior angles of a pair of parallel lines cut by a transversal have equal measure. Thus,
We can follow a similar process for . Let us start by marking an angle corresponding to on the diagram.
Since alternate interior angles have the same measure and the measures of angles on a straight line add to , we have
We can now add the measures of these angles together:
In our next example, we will find an unknown angle using the parallel lines theorems.
Example 5: Finding an Unknown Angle in Problems Involving Parallel Lines
In the figure, intersects and at and respectively. Find .
Answer
We first note that we have two parallel lines cut by a transversal line. However, and are not corresponding, alternate interior, or alternate exterior angles. So, we need an extra step to allow us to find the measure of . We can mark this angle on the diagram as shown.
There are several methods we can use to find the measure of this angle. One is to notice that and are on either side of the transversal and on the outside of the parallel lines. So, they are alternate exterior angles.
We can then recall that alternate exterior angles in a transversal of parallel lines are congruent. So, .
We can now note that and combine to make a straight angle, so they are supplementary angles. Thus,
We know that , so
Subtracting from each side gives us
We could have also considered any other angle related to both and , being congruent to one of them, and then we would have found that it is supplementary to the other.
In our previous two examples, we were able to show that some angles in a transversal of parallel lines are supplementary. In general, it is useful to note that the interior angles on the same side of the transversal are supplementary.
We can prove this result by considering a transversal of parallel lines and with angle measures as shown. We can also mark the congruent angles.
We know that the adjacent angles are supplementary, so
We also know that the angles of measures and are corresponding angles. Thus
We can substitute this into the supplementary equation to get
We can use a similar proof to show that
We can also show that the exterior angles on the same side of the transversal are also supplementary. We have proven the following.
Theorem: Cointerior and Coexterior Angles of a Transversal of Parallel Lines
The cointerior angles (i.e., on the same side of the transversal) of parallel lines are supplementary:
The coexterior angles on the same side of the transversal of parallel lines are supplementary:
Let us now see an example of using this property to determine the measure of an angle in a given figure.
Example 6: Finding an Unknown Angle in Problems Involving Parallel Lines
Find .
Answer
We are only given the measure of a single angle, so we should start by trying to use this angle together with the parallel lines and transversals to determine the measures of other angles in the diagram. We can note that is an alternate interior angle to .
We can recall that alternate interior angles on the transversal of a pair of parallel lines have an equal measure, so .
We can then determine by noting that it is a cointerior angle with since they both lie inside a pair of parallel lines on the same side of a transversal.
We then recall that the sum of the measures of cointerior angles is , so
In the diagram, we are told that bisects this angle, so each of these angles has a measure of .
We can then determine by recalling that the measures of angles on a straight line sum to . Thus,
We now have the measures of two of the interior angles of quadrilateral . We can determine the measure of by finding the measure of and using the sum of the interior angle measures in a quadrilateral, .
We can find by noting that and is a corresponding angle to .
Therefore, they both have a measure of .
We can now find by using the sum of the interior angles in . We have
We have shown a number of relationships involving the angles in a transversal of parallel lines. However, we can also consider the reverse. For example, if a transversal has congruent corresponding angles with a pair of lines, will these lines be parallel? Take, for example, a transversal of two lines and with congruent corresponding angles.
It appears as though and are parallel; however, this is not enough to prove the statement. We can prove this by instead assuming the lines are not parallel. Since we assume they are not parallel, they must intersect at a point that we will call . If we label the points of intersection between the lines and transversals and , then is a triangle.
We can determine the measures of two of these angles by using supplementary angles and vertically opposite angles.
We see that the angles on the inside of the parallel lines have measures that sum to . Since the measures of the internal angles in a triangle must sum to , we cannot have two supplementary angles as two of the internal angles in a triangle. This means there cannot be a point on both and , so the lines are parallel. We have shown that any two lines with congruent corresponding angles are parallel.
The same is true if they have congruent alternate interior or exterior angles or if the interior angles are supplementary. We can follow the same proof or show that in either case the lines will have congruent corresponding angles. We have shown the following result.
Property: The Congruence of Corresponding Angles of Two Lines with Another Line Implies They Are Parallel
If two lines have congruent corresponding, alternate interior, or alternate exterior angles with the same line, then they must be parallel.
If two lines have supplementary cointerior or coexterior angles, then they must be parallel.
In our final example, we will use this property to identify that two lines are parallel.
Example 7: Identifying That Two Lines Are Parallel Because the Alternate Interior Angles Are Congruent
Fill in the blank: In the diagram, we can conclude that is .
Answer
We see that and are transversals of the two lines and . We can find several angle measures using the given angles. For example, we can determine the measure of the angle alternate interior to the angle of measure as follows.
We know that angles on a straight line sum to , so the following angle has a measure of .
The measures of the interior angles in a triangle sum to , so the final angle in the triangle has a measure of .
We now see that we have congruent alternate interior angles of the transversal of the lines and .
We can then recall that if two lines have congruent alternate interior angles with a transversal, then they must be parallel.
Hence,
Let us finish by recapping some of the important points from this explainer.
Key Points
- The angles that lie in the same position at the intersection between a transversal and two lines are called corresponding angles.
- Corresponding angles in a transversal of parallel lines are congruent.
- Alternate interior angles are pairs of angles from either set of the four angles formed by a transversal line cutting two lines and that are on either side of the transversal and on the inside of and .
- Alternate exterior angles are pairs of angles from either set of the four angles formed by a transversal line cutting two lines and that are on either side of the transversal and on the outside of and .
- If the transversal cuts a pair of parallel lines, then the alternate interior angles are equal in measure. Similarly, if the transversal cuts a pair of parallel lines, then the alternate exterior angles are equal in measure.
- If the transversal cuts a pair of parallel lines, then the interior angles on the same side of the transversal are supplementary.
- The converse of the theorems is true: if corresponding angles, alternate interior angles, or alternate exterior angles formed by a transversal cutting two lines are congruent, then the lines cut by the transversal line are parallel.
