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. The following diagram shows examples and nonexamples of vertically opposite angles.

We see that opposite angles are two angles *between two secant lines* (“secant
lines” simply means two lines that cross each other) that share a vertex (that is why
they are called “vertical” angles). We see also that they are not adjacent
(which means next to each other) but opposite each other. That is why they are sometimes
called “opposite” angles. Associating both gives vertically opposite angles.

Once we are able to identify vertically opposite angles, the important fact to remember is that vertically opposite angles are equal (they have the same measure). We say that they are congruent.

Looking more closely at the four angles between two secant lines, we see that we have two pairs of equal angles—the vertically opposite angles—while the adjacent angles are supplementary angles since they form a straight angle: the sum of their measures is .

Let us now look at the case of two parallel lines crossed by a third line. We have eight angles and we recognize the four pairs of vertically opposite angles: and and and , and and .

Because the lines and are parallel, the two sets
of four angles between and and between
and are congruent. In this case, we call each
pair of congruent angles *corresponding angles*.

### Corresponding Angles

Corresponding angles are in corresponding position with respect to the transversal line and one of the (here parallel) lines or .

We see that we have then other pairs of congruent angles. We use different names to identify angles formed by a transversal line that passes through the two lines: interior alternate angles and exterior alternate angles.

### 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 between 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 .

Now that we have learned what these pairs of angles are called, we can state all the pairs of angles that are congruent—this is the parallel lines theorems.

### Parallel Lines Theorems

When two parallel lines are cut by a transversal line, then the pairs of corresponding angles are congruent, the pairs of alternate interior angles are congruent, and the pairs of alternate exterior angles are congruent.

The converse is true: if corresponding angles, or alternate interior angles, or alternate exterior angles are congruent, then the lines cut by a transversal are parallel.

Let us now look at some questions to see how to apply what we have learned about corresponding, alternate interior, and alternate exterior angles.

### Example 1: Identifying Corresponding, Alternate Interior, and Alternate Exterior Angles

Work out the value of in the figure.

### Answer

The two angles shown in the figure are alternate interior angles: they are inside the two parallel lines (therefore, they are interior angles) and on either side of the transversal line. Pairs of alternate interior angles are congruent, which means that they have the same measure. Hence, we have

### Example 2: Identifying Corresponding, Alternate Interior, and Alternate Exterior Angles

In the figure below, find .

### Answer

Even if we have here two parallel rays ( and ), we can extend them across to recognize better the situation of two parallel lines cut by a transversal line.

The angles and are corresponding angles. So, they are congruent, which means that their measures are equal. Hence, we have

### Example 3: Identifying Corresponding, Alternate Interior, and Alternate Exterior Angles

Work out the value of in the figure.

### Answer

The two angles shown in the figure are alternate exterior angles: they are outside the two parallel lines (therefore, they are exterior angles) and on either side of the transversal line. Pairs of alternate exterior angles are congruent, which means that they have the same measure.

Hence, we have

Now that we have looked at two examples of questions where we needed to identify corresponding and alternate angles and use the fact that they are congruent, let us look at some more difficult questions.

### Example 4: Finding an Unknown Angle in Problems Involving Parallel Lines

Work out the value of in the figure.

### Answer

We have here two parallel lines cut by a transversal line. However, the angles of measures and are neither corresponding angles nor alternate interior angles nor alternate exterior angles. So, we need an extra step to allow us to find the value of .

This step consists in realizing that the corresponding angle to the angle of measure is adjacent to the angle of measure and that both are supplementary angles since they form a straight angle together. Hence, we can write

Subtracting 131 from each side gives us

### Example 5: Finding an Unknown Angle in Problems Involving Parallel Lines

In the figure, intersects and at and respectively. Find .

### Answer

We have here two parallel lines cut by a transversal line. However, the angles and are neither corresponding angles, nor alternate interior angles, nor alternate exterior angles.

So, we need an extra step to allow us to find the measure of .

There are several possibilities to find the answer. One is to consider the fact that and are alternate exterior angles and, as such, are congruent. Therefore, . Then, since and are supplementary angles (they form a straight angle together), we can write

that is,

Subtracting 84 from each side gives us

We could have considered another angle related to both and , being congruent to one of them, and we would have found that it is supplementary to the other.

Let us now look at a last question involving a pair of parallel lines and two transversal lines.

### Example 6: Finding an Unknown Angle in Problems Involving Parallel Lines

Answer the questions for the given figure.

- Find the value of .
- Find the value of .

### Answer

We have here two parallel lines cut by two transversal lines.

Let us start with finding . We need to focus only on the transversal line that forms the angle of measure with one of the parallel lines. We can ignore the other transversal line. The angles of measures and are between the two parallel lines, so they are interior angles but not alternate interior angles because they are on the same side of the transversal line. However, the angle that is in the pair of alternate interior angles with the angle of measure is adjacent to the angle of measure ; these two angles are supplementary because they form a straight angle together. Since pairs of alternate interior angles are equal, we can write

We find that

To find , we now look at the angles formed by the other transversal line with the two parallel lines. The angles of measures and are corresponding angles, so their measures are equal. Hence, we have

### Example 7: Finding Unknown Angles Formed by Three Parallel Lines and Two Transversals

Find .

### Answer

Here, we have three parallel lines and two transversal lines cutting them. We want to find .

Let us try to see how this angle is linked to the angles whose measures are known. Let us consider the two parallel lines and cut by the line . According to the parallel lines theorems, and are consecutive interior angles; therefore, they are supplementary (the sum of their measures is ). We have that .

Considering now the two parallel lines and cut by the line , we see that and are consecutive interior angles as well; therefore, they are supplementary. Hence, we have that .

We can now use this equation to find first with and . Plugging these in, we get

that is,

Plugging this into our first equation, , we find that

that is,

### Key Points

- Corresponding angles are in corresponding position with respect to the transversal line and one of the parallel lines or .
*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 between 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 .- The parallel lines theorems state that when parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent, the pairs of alternate interior angles are congruent, and the pairs of alternate exterior angles are congruent.
- The converse of the theorems is true: if corresponding angles, or 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.