Angles Between Lines and Transversals
When you have a set of two lines with a third line (called a transversal) crossing them, it forms a group of 8 angles - four angles formed by the first line and the transversal, and four made by the second line and the transversal.
When we look at pairs of those angles, some of them have special relationships, so mathematicians have given names to some of the different pairings you can have. There are five specific types of angle pairs that we talk about when we're working with lines and transversals, so let's talk about each of them.
The first type of angle pairs is alternate interior angles. "Alternate" means that the two angles are on opposite sides of the transversal, and "interior" means that they're inside the two main lines. There will be two pairs of these: in our example scenario, angle C and angle F are alternate interior angles, and angles D and E are also alternate interior angles.
WARNING! Keep in mind that each of these terms is describing the relationship between two angles—the term only applies to the angles relative to each other. Think of it this way: If Bob and Fred are brothers, and José and Raúl are also brothers, that doesn't mean that Bob and Raúl are brothers too. Similarly, even though C and F are alternate interior angles, and D and E are alternate interior angles, C and F are not alternate interior angles with each other.
The next type of angle pairs is alternate exterior angles. Again, "alternate" means that they're on opposite sides of the transversal; "exterior" means that they're outside the two main lines. There will be two pairs of this type as well. In the example image, angles A and H form one pair of alternate exterior angles, and angles B and G form the second pair.
Next up is corresponding angles. Unlike the other relationships, this one's name isn't quite as self-explanatory. Angles are "corresponding" to each other if they're in the same position relative to the transversal and the main lines. For example, "upper left" on one line with "upper left" for the other line, or "down to the right" with "down to the right." Two lines with a transversal will have four of this type of pairs: in our example, the pairs are angle A with angle E, angles B and F, angles C and G, and angle D with angle H.
Those first three pair types are the ones you'll need to use most often, but there are two more that also tend to come up pretty regularly.
The next type is same-side interior angles. Once again, you can pretty much identify their position from the name: both angles are on the same side of the transversal, and "interior" means they're inside the two main lines. There will be two pairs of these: in our example, Angle C and angle E form one pair of same-side interior angles, and angle D and angle F are the second pair.
Last, but not least, are same-side exterior angles. If you've been following along with the previous ones, you can probably guess what this one means: both angles are on the same side of the transversal, and "exterior" means they're outside the two main lines. There will be two pairs for this type as well. In our example, Angle A and angle G form a pair of same-side exterior angles, and angle B and angle H form the other.
So, why do we even care? Why do you need to know all these different types of angle pairs anyway?
The reason that these angle pairs matter is because of what happens with them when the lines are parallel. The relationships between them stop being just names - when the lines are parallel, these angle pairs have relationships in terms of their size as well. Because of this, these relationships can be useful for finding the sizes of unknown angles, or they can be used as evidence when you're trying to prove things about the lines or the angles.
If the two main lines are parallel, then alternate interior angles will be congruent to each other. Alternate exterior angles will also be congruent to each other, as will corresponding angles. For example, in the image below, lines m and n are parallel. As a result, we can say that R=U and S=T because they're alternate interior angles; P=W and Q=V because they're alternate exterior angles; and P=T, Q=U, E=V, and S=W because they're corresponding angles.
What about same-side interior and same-side exterior angles? You can probably tell from looking at them that these pairs aren't going to be congruent like the others. Instead, when the lines are parallel, these pairs are supplementary - they add up to 180 degrees. In our example, C+E=180 and D+F=180 because they are same-side interior, angles and A+G=180 and B+H=180 because they are same-side exterior angles.
One last note: occasionally, you may be asked about the relationship between a pair of angles that doesn't fit any of the five types we just talked about. In some cases, you may be able to use other Geometry rules, such as vertical angles or linear pairs, to describe their relationship (such as with angles A and D or angles E and G in our first example diagram). However, some of the pairs of angles just don't have any special relationship to each other and can only be connected by their relationship to other angles. For example, in our diagram above with the parallel lines, angles P and U don't form any type of special angle pair, but we can still determine that they are supplementary — we know that angle P is congruent to angle S (vertical angles), and angle S and angle U add up to 180° (same-side interior), so angle P and angle U must add up to 180° as well.
Here are some practice problems to try!
- What type of angle pair is formed by angles D and J in the diagram below?
- If lines a and b are parallel in the image on question 1, which of the angles pictured are congruent to angle K? How do you know?
- According to the diagram below, are lines u and v parallel? Why or why not?
- In the diagram below, lines BD and EH are parallel. Prove that triangle CFG is a right triangle. (HINT: there are multiple approaches to this proof, but all of them will involve using one or more of the rules about angles formed by parallel lines and transversals!)
