Explainer: Parallel Lines and Transversals: Angle Relationships

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 180.

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: 𝑎 and 𝑐,𝑏 and 𝑑,𝑒 and 𝑔, and 𝑓 and .

Because the lines 𝐿 and 𝐿 are parallel, the two sets of four angles between 𝐿 and 𝐿 and between 𝐿 and 𝐿 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 𝐿 and one of the (here parallel) lines 𝐿 or 𝐿.

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 (𝐿) cutting two (here parallel) lines 𝐿 and 𝐿 that are on either side of 𝐿 and between 𝐿 and 𝐿.

Alternate exterior angles are pairs of angles from either set of the four angles formed by a transversal line (𝐿) cutting two (here parallel) lines 𝐿 and 𝐿 that are on either side of 𝐿 and outside 𝐿 and 𝐿.

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 𝑥 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 𝑥=61.

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

In the figure below, find 𝑚𝐴𝐵𝐶.

Answer

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

The angles 𝐸𝐴𝐷 and 𝐴𝐵𝐶 are corresponding angles. So, they are congruent, which means that their measures are equal. Hence, we have 𝑚𝐴𝐵𝐶=65.

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

Work out the value of 𝑥 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 𝑥=124.

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 𝑥 in the figure.

Answer

We have here two parallel lines cut by a transversal line. However, the angles of measures 131 and 𝑥 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 𝑥.

This step consists in realizing that the corresponding angle to the angle of measure 131 is adjacent to the angle of measure 𝑥 and that both are supplementary angles since they form a straight angle together. Hence, we can write 131+𝑥=180.

Subtracting 131 from each side gives us 𝑥=49.

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

In the figure, 𝐸𝑁 intersects 𝐴𝐵 and 𝐶𝐷 at 𝑀 and 𝐹 respectively. Find 𝑚𝐸𝐹𝐶.

Answer

We have here two parallel lines cut by a transversal line. However, the angles 𝐸𝑀𝐵 and 𝐸𝐹𝐶 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 𝐸𝐹𝐶.

There are several possibilities to find the answer. One is to consider the fact that 𝐸𝑀𝐵 and 𝑁𝐹𝐶 are alternate exterior angles and, as such, are congruent. Therefore, 𝑚𝑁𝐹𝐶=84. Then, since 𝐸𝐹𝐶 and 𝑁𝐹𝐶 are supplementary angles (they form a straight angle together), we can write 𝑚𝐸𝐹𝐶+𝑚𝑁𝐹𝐶=180;

that is, 𝑚𝐸𝐹𝐶+84=180.

Subtracting 84 from each side gives us 𝐸𝐹𝐶+96.

We could have considered another angle related to both 𝐸𝑀𝐵 and 𝐸𝐹𝐶, 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.

  1. Find the value of 𝑥.
  2. Find the value of 𝑦.

Answer

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

Let us start with finding 𝑥. We need to focus only on the transversal line that forms the angle of measure 𝑥 with one of the parallel lines. We can ignore the other transversal line. The angles of measures 60 and 𝑥 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 60 is adjacent to the angle of measure 𝑥; these two angles are supplementary because they form a straight angle together. Since pairs of alternate interior angles are equal, we can write 𝑥+60=180.

We find that 𝑥=120.

To find 𝑦, we now look at the angles formed by the other transversal line with the two parallel lines. The angles of measures 110 and 𝑦 are corresponding angles, so their measures are equal. Hence, we have 𝑦=110.

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

Find 𝑚𝐸𝐶𝐷.

Answer

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

Let us try to see how this angle is linked to the angles whose measures are known. Let us consider the two parallel lines 𝐶𝐷 and 𝐸𝐹 cut by the line 𝐶𝐸. According to the parallel lines theorems, 𝐸𝐶𝐷 and 𝐹𝐸𝐶 are consecutive interior angles; therefore, they are supplementary (the sum of their measures is 180). We have that 𝑚𝐸𝐶𝐷+𝑚𝐹𝐸𝐶=180.

Considering now the two parallel lines 𝐴𝐵 and 𝐸𝐹 cut by the line 𝐴𝐸, we see that 𝐴𝐸𝐹 and 𝐵𝐴𝐸 are consecutive interior angles as well; therefore, they are supplementary. Hence, we have that 𝑚𝐴𝐸𝐹+𝑚𝐵𝐴𝐸=180.

We can now use this equation to find first 𝑚𝐹𝐸𝐶 with 𝑚𝐴𝐸𝐹=𝑚𝐹𝐸𝐶+42 and 𝑚𝐵𝐴𝐸=103. Plugging these in, we get 𝑚𝐹𝐸𝐶+42+103=180;

that is, 𝑚𝐹𝐸𝐶=35.

Plugging this into our first equation, 𝑚𝐸𝐶𝐷+𝑚𝐹𝐸𝐶=180, we find that 𝑚𝐸𝐶𝐷+35=180;

that is, 𝑚𝐸𝐶𝐷=145.

Key Points

  1. Corresponding angles are in corresponding position with respect to the transversal line 𝐿 and one of the parallel lines 𝐿 or 𝐿.
  2. Alternate interior angles are pairs of angles from either set of the four angles formed by a transversal line (𝐿) cutting two (here parallel) lines 𝐿 and 𝐿 that are on either side of 𝐿 and between 𝐿 and 𝐿.
  3. Alternate exterior angles are pairs of angles from either set of the four angles formed by a transversal line (𝐿) cutting two (here parallel) lines 𝐿 and 𝐿 that are on either side of 𝐿 and outside 𝐿 and 𝐿.
  4. 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.
  5. 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.

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