What is a Transversal Angle?
Given a pair of transversal lines, the four angles formed by them are called transversal angles. Their properties are going to be the object of study of this lesson when the situation involves parallel lines and transversals and, for this reason, it was important to answer the question "what is a transversal angle?"
Transversal Angles
When a transversal line cuts a pair of lines, eight angles are formed, out of which some are congruent and some are supplementary. We categorize the transversal angles as corresponding, alternate, vertical and alternate angles. Consider the following image with two parallel lines intercepted by a transversal and the 8 transversal angles that they form.
Parallel lines cut by a transversal and labeled angles

Types of Transversal Angles
Using the last image from the previous section we are going to categorize the types of transversal angles in a list. Two angles can be:
 Corresponding: 1 and 5, 2 and 6, 3 and 7, 4 and 8
 Interior alternate: 3 and 5, 4 and 6
 Exterior alternate: 1 and 7, 2 and 8
 Interior collateral: 4 and 5, 3 and 6
 Exterior collateral: 1 and 8, 2 and 7
 Vertical: 1 and 3, 2 and 4, 5 and 7, 6 and 8
If we move line {eq}r {/eq} until it coincides with {eq}s, {/eq} four angles would be on the top of the other four. This is the idea of the corresponding angles. Alternate angles are the ones on different sides of the transversal, collateral, on the same side, and vertical are the ones that share the vertex, but the rays of one are opposed by the vertex to the rays of the other.
Some of the angles formed by parallel lines and transversals are congruent, namely: corresponding, interior alternate, exterior alternate, and vertical angles. Let's see why this happens. Because of how we described corresponding angles, it is clear that they are congruent. If we move one of the parallel lines until it coincides with the other, four angles would overlap the other four, and the pairs of overlapping angles are congruent. Now, to see why vertical lines are congruent, consider the two lines depicted in the sequence.
Intersecting lines r and s forming the four labeled angles

Because {eq}r {/eq} and {eq}s {/eq} are lines, they form {eq}180^{\circ}. {/eq} So, {eq}\alpha + \beta = 180, \beta + \gamma = 180, \gamma + \delta = 180 {/eq} and {eq}\delta + \alpha = 180. {/eq} The first and second equations imply {eq}\alpha = 180  \beta {/eq} and {eq}\gamma = 180  \beta. {/eq} Therefore, {eq}\alpha = \gamma. {/eq} The fact that {eq}\beta {/eq} and {eq}\delta {/eq} are congruent can be proven similarly.
Finally, let's see why alternate angles are congruent. For this end, consider the image with parallel lines {eq}r {/eq} and {eq}s {/eq} intercepted by transversal {eq}t {/eq} from the section "Transversal Angles". To see that 1 and 7 are congruent, note that 1 and 3 are vertical angles, implying that they are congruent. Now, 3 and 7 are congruent because they are corresponding angles. By transitivity, it follows that 1 and 7 are congruent. The same can be shown for the other pairs of alternate angles.
Collateral angles, though, are not congruent, but they have a nice property: they are supplementary, that is, the sum of two collateral angles is {eq}180^{\circ}. {/eq} To show, for instance, that angles 3 and 6 add up to {eq}180^{\circ}, {/eq} note that 3 and 5 are alternate angles, so they are congruent. Because 5 and 6 are supplementary, it follows that 3 and 6 are also supplementary.
Example 1: When Two Parallel Lines Cut are by a Transversal
Consider the following diagram where parallel lines with transversal are shown. Determine {eq}x {/eq} and {eq}y. {/eq}
Parallel lines cut by a transversal

First, observe that {eq}150^{\circ} {/eq} and {eq}y {/eq} are corresponding angles, so they are congruent. Next, {eq}x + y = 180. {/eq} Therefore, {eq}x = 30^{\circ} {/eq} and {eq}y = 150^{\circ}. {/eq}
Example 2: Transversal Congruent Angles
In this example, no two lines are parallel, but use the information given on the image to find {eq}x {/eq} and {eq}y. {/eq}
Three intersecting lines

We have that {eq}x {/eq} and {eq}160^{\circ} {/eq} are supplementary, whereas {eq}150^{\circ} {/eq} and {eq}y {/eq} are transversal congruent angles because they are vertical. So, {eq}y = 150^{\circ} {/eq} and {eq}x = 20^{\circ}. {/eq}
Lesson Summary
As we saw, transversal lines are lines that intercept each other, whereas parallel lines have no point in common. Two parallel lines cut by a transversal determine eight angles that can be categorized into groups. In what follows, we are going to state the categories of angles and give one example of each using as reference the image from section "Transversal Lines". Angles can be corresponding (1 and 5), vertical (1 and 3), alternate interior (3 and 5), alternate exterior (1 and 7), collateral interior (3 and 7), collateral exterior (1 and 8). A pair of collateral angles are supplementary, that is, their sum is {eq}180^{\circ}, {/eq} but corresponding, alternate and vertical angles are congruent.