Parallel Lines and Transversals Solving for Angle Measures

Hey, everyone! It's Justin again. Are you, like Mia, feeling little suspicious or maybe feeling like you're experiencing déjà vu right now? Don't worry — know this looks awfully similar to your last lesson, but promise that it is different. Similar? Definitely. In our prior lesson, we learned about transversals crossing over any old lines. Today, we're going to learn about transversals crossing parallel lines, which creates some special relationships. By the end of this lesson, you should be able to use theorems to find angle measurements when parallel lines are cut by transversal. First, we'll learn transversal theorems, then we'll complete two practice problems. So, at this point, you are familiar with this situation. We know there are these eight angles created by the transversal crossing each line. We can name all of their relationships to each other — we learned all of that in the previous lesson. Do you see how with these lines, the intersections where the transversal crosses each line create very different angles? Based on just the vertical angles, what angle congruences can you come up with in this diagram? Each of these pairs of angles are vertical angles and are therefore congruent, but that's as much as we can come up with in this type of diagram. When the two lines that transversal intersects are parallel, something interesting happens. Now these two lines are parallel — notice the arrows on each line. The vertical angles are still congruent, just like always. But now, remember how parallel lines move at the same rate? This means that they meet the transversal at the same angle. Take look at these two intersections now. What do you notice? Let's, bring them little closer together to make it even more clear. look at that! They're identical! Can you find some new congruences to add to our list? All of our obtuse angles in this diagram — 2, 4, 6, and 8 — are all congruent to each other. And all of our acute angles in this diagram — 1, 3, 5, and 7 — are congruent to each other, too. can't wait to see what we can do with this! Let's do ourselves favor here and add in the angle congruence marks. yeah, that's better! Think about these angle pairs you learned about in the previous lesson. Do you notice any specific relationship between the pairs now that we're using parallel lines? Did you notice that each of these angle pairs is congruent? Let's double-check and look for the congruent symbols for each pair: Alternate exterior angles, Alternate interior angles, And corresponding angles. Vertical angles are also congruent, but as you learned in Unit 1, this is true for any intersection of two lines. Each of these angle pairs has congruence theorem that goes along with it. theorem is math fact that can be proven. You can find the full text of the theorems in your Notes Template PDF. You do not need to memorize the full theorems, but you should be able to confidently identify and use the congruence of special angle pairs. bet you're thinking, "Wait second, there's one more angle pair we learned about." You're right! left consecutive interior angles off of that discussion. Let's take look at those now. That would be this pair and this pair. Wait second. These pairs look different than the ones we've been looking at so far. These pairs are not congruent, but they do have special relationship. Let's label them so we can talk about them little more easily. Take look here. Angles 1 and 2 are adjacent and form linear pair, so they add up to 180 degrees. The consecutive interior angles — angles 1 and 3 — are not adjacent. But hold on, isn't angle 2 congruent to angle 3? Yes, definitely! They are alternate interior angles. And since angles 2 and 3 are congruent (or equal), we can substitute here and... Hey! Now we can see that our consecutive interior angles (angles 1 and 3) are supplementary. This is exactly what the consecutive interior angles theorem tells us! In this specific situation, where the transversal crosses parallel lines, the consecutive interior angles are supplementary. To recap: Alternate exterior angles, alternate interior angles, corresponding angles, and vertical angles are all congruent. Consecutive interior angles are supplementary. Make sure you feel comfortable with these relationships before moving on. If you need to pause or rewind here, feel free to do so. Let's give it try! Our first problem states: Given that the measure of angle 1 is 48 degrees, find the rest of the angle measurements. Since we know from those little arrows in our diagram that we have two parallel lines, we know that our angle pairs are either congruent or supplementary. Use those relationships to angle 1 to start moving through the rest of the diagram. See how far you can get! There are many ways that you could use angle relationships to find the rest of the angle measurements in this diagram. I'm going to go through one way with you here. If you took different steps than me, that is OK — as long as our final answers match! You'll find an activity in your Notes Template PDF to help you explore some alternate strategies for this problem. Let's start by working out from angle 1. First, we can use the alternate exterior angle from angle 1; that would be angle 8, and they're congruent. So we now know that the measure of angle 8 is also 48 degrees. Angle 1 doesn't have an alternate interior angle because it's not interior. The corresponding angle would be angle 5, and we know those are congruent, so the measure of angle 5 is also 48 degrees. The vertical angle would be angle 4, and those are congruent, so there's another angle that's 48 degrees. Angle 1 still doesn't have consecutive interior angle, so have run out of relationships with angle 1. But now, have three new angles to work with: 4, 5, and 8. You can use any of these next, but I'm gonna go with angle 4. Are there any angles that we still need to find that have relationship with angle 4? Yeah, there's one! The consecutive interior angle with angle 4 is angle 6, and those are supplementary, which means they add up to 180 degrees. That means that angle 6 is 132 degrees. OK, now we're getting somewhere! Let's take angle 6 now that we know its measure. Its alternate interior angle is angle 3. Alternate interior angles are congruent, so that's also 132 degrees. The corresponding angle with 6 is 2, so that's also 132 degrees. The vertical angle with 6 is 7, so that's also 132 degrees. one, two, three, four— hey, that's all of them! So that tells us that if we have just one angle measurement from two parallel lines cut by transversal, we can find any other angle that we want. That's going to help lot! Let's do one more together. Given the diagram below, if the measure of angle 3 is 100 degrees and the measure of angle 9 is 4x, find Let's start by trying to find relationship between angle 3 and angle 9 so that we can create an equation. Hold on second... angle 3 and angle 9 don't have relationship! They're not alternate exterior; they're not alternate interior; they're not corresponding; they're not vertical, they're not consecutive interior. Well, if can't create an equation straight from angles 3 and 9, maybe can use another congruence relationship to help. Like our last practice problem, there are many ways to do this. I'm going to do it one way with you now, and you can explore some alternative options in your Notes Template PDF. Angles 3 and 4 are supplementary because they're consecutive interior angles. And angles 4 and 9 are congruent because they're vertical angles. Can you guess what we're going to do next? We can substitute the measure of angle 9 in for the measure of angle 4 since those are congruent. Take it from here and finish this problem off! From here, you should have substituted 100 for the measure of angle 3 and 4x for the measure of angle 9. Subtract 100 from both sides and divide by 4 to get = 20. Is this our final answer? Yes, it is! Keep in mind that all of the angle relationships we talked about throughout this lesson only apply if the transversal is crossing parallel lines. Move ahead to the practice game next to continue building your skill with using transversal congruence theorems. When you finish the practice game, you'll find challenge activity in your Notes Template PDF. This will challenge you to use the transversal congruent theorems while also bringing back some solving skills from Algebra 1 to help you stay strong. See you next time!