BAM! Mr. Tarrou! In this video, we are going to continue our discussion of transversals and the types of angles that are formed when the transveral comes through and cuts two or more lines. In this video, we are again, going to be talking about corresponding angles, alternate interior angles, alternate exterior angles, and same side interior angles. Those are the types of angles that are formed when you have two or more lines being cut by a transveral. Remember that transversal has to cut, or go through, those 2 lines at 2 distinct points. In the previous video, we did all these definitions and explained and talked about these types of angles. But, when those lines that are being cut by a transversal are parallel, as indicated by these two orange arrows, extra arrows I have put on my lines – then a lot of really cool things happen. And thus, we have, what we have going on here. Now, again, we have line L is parallel to line M. But, in a geometry diagram, you are going to be looking for a pair of dark triangles that are added to those lines. We always end lines with arrows, but there should be another pair of arrows highlighted on those lines somewhere to indicate that the lines are parallel. So if a transversal intersects parallel lines, then we have corresponding angles are going to be congruent. Corresponding angles: when you look at these two intersection areas here, let’s say in the upper left hand corner of the first intersection area and the upper lefthand corner of this second intersection area. I have angles of 4 and 2. Well, instead of just saying 4 and 2 are corresponding angles, we are now able to say they are congruent. Let me write all those pairs out for you. We have angle 4 is congruent to angle 2. We have angles 5 and 7, both in the upper right hand corner. We have angles 6 and 8. Finally, we have angles in the bottom lefthand corners, of 3 and one. Okay. I am writing all of the possible pairs of corresponding angles and that they are congruent in this particular diagram. By the way, we are going to do 4 algebraic examples. Fairly simple examples. And the last example in this video is going to be a proof involving 2 parallel lines being cut by a transversal. And learning something about those angles and proving some facts. Alternate interior angles are congruent. Now, alternate interior angles – alternate is, remember, when we defined these in another video, we want to go from the left side of the transerval to the right side of the transversal, or that alternate interior angles are on opposite sides of the transversal. And interior is the interior, inside, inbetween those two lines that are being cut by that transversal. So that area in here is going to be the interior, and outside these parallel lines that being cut by the transversal, that is the exterior. My alternate interior angles, which are congruent, are going to be the pairs 3 and 7 and 2 and 6. So angles 3 and 7 are congruent, and 2 and 6. Again, we have congruent angles, which means that their measures are equal. But, if I were talking about measures there would be no congruency mark there. It would be the measure of angle 3 is equal to the measure of angle 7. And that is what congruency means. Alternate exterior angles. Where we are still going to alternate from one side of the transversal to the other, or be on opposite sides. Only, we want to pick the pairs of angles that are outside of those parallel lines being cut by the transversal. Now we are talking about 4 and 8, and 5 and 1. So angle 4 is congruent to angle 8. And angle 5 is congruent to angle 1. Okay. Again, like the 5, right side of the transveral, the one is on the left. And they are outside those two parallel lines. And finally, same-side – so we want 2 angles which are on the same side of the transversal. Either both on the left, or both on the right. And we want them to be interior angles. Same-side interior angles are supplementary – let’s not forget that supplementary means that the measure of two angles add up to 180 degrees. Complementary is adding up to 90 degrees. We are talking about 3 and 2 – those are both in the left and both on the inside of our parallel lines. And we have 6 and 7. So, either one, we have the measure of angle 2 plus the measure of angle 3 is equal to 180 degrees. Or, if for some reason we needed to work on the other side, that is going to be the measure of angle 6 plus the measure of angle 7 is equal to 180 degrees. So! Those are our 4 types of angles and their properties when you are given intersecting parallel lines and not just regular, any old two individual lines being cut by a transversal. Again, we have 4 pretty simple, I hope, algebraic examples involving these types of angles, and then we are finishing with a proof. First 2 examples! Here, we’ve got actually 4 parallel lines. We have 2 parallel lines going horizontal, and we have two parallel lines going off at an oblique or slant angle here. And we are going to need to find in this question, we are going to need to solve for x and y. Now, we’ve got a lot going on here, and this is a more complicated diagram than what I started off with. So, I want to first focus my attention on solving for x. And to do that, what we are going to do is kind of ignore part of the diagram. The extra stuff in there that I do not need really to solve for x. If I cover up this portion of the diagram as best I can with my notebook paper, we have two parallel lines being cut by a transversal. What we look at here is if these are my two parallel lines, being cut by my tranversal, and I have 120 here and x over here. Well, you might want to call these same-side exterior angles. But that is not one of the specific 4 types of angles that I gave you. Now, you might already know, or think you know, a relationship that is going on between 120 and x, but I want to use those 4 definitions that we have to find right now. We are going to have to go through angle 1, because angle 1 and angle x…these are, you see how they are both in the upper right hand corner of our intersection areas? So angle 1 and angle x here are going to be corresponding angles. If I can figure out what the measure of this angle is, then I know what the measure of angle x is. Okay. So, let’s see. Let’s cover up some more of the diagram. If I cover up even more of this diagram, we see that this angle here, 120 degrees, and this angle that I have put the one inside the corner, this is a linear pair. These are two adjacent angles who are forming a straight line. That is the definition of, that is what a linear pair is: two adjacent angles whose nonadjacent sides form a line, or opposite rays, if you will. So these are going to add up to 180 degrees. We have 120 degrees plus the measure of angle one is equal to 180. We will subtract both sides by 120 degrees And the measure of angle one is equal to 60. If I go back up here, and fill in some missing information…now I know that the measure of angle one is equal to 60 degrees. This angle and this angle…these are corresponding angles. Corresponding angles are congruent. If I know the measure of angle one is 60, then I can say that x is also equal to 60 degrees. Because corresponding angles are congruent. So that is what x is – 60 degrees. What we are going to do now, is find out what y is equal to. As I look down here at this angle, which has an expression which involves y, I see two parallel lines here being cut by…Now I am going to be focused on this part of the diagram. Let’s once again cover up a little bit of this. I know what x is now – 60 degrees. I can write that in there. Now, I can take out some of my distraction here and look at this! I’ve got two parallel lines, as marked by the orange triangles I have here. I have two parallel lines being cut by a transversal. In this intersection point, top righthand corner – This intersection point, top righthand corner, those are corresponding angles. And once again, if we are intersecting parallel lines, those corresponding angles are equal. If these two angles are supposed to be equal, I can say that 3y plus 6 is equal to 60 degrees. Subtract both sides by 6, moving away from the variable y, which we are trying to solve for. 3y is equal to 54. Divide both sides by 3 to undo that multiplication with the 3 and the y. You are trying to undo – do the opposite math operation. 3 divided by 3 is 1. We get y is equal to 8. No! Not 8, what am I thinking? 54…how many 3s are in 5? One. And if I take a 3 out of 5, that leaves me with 2, so 24. 24 divided by 3 is 8. I just remembered the last digit of the number instead of actually doing it in my head. So that is the end of my first example. And I have my value of x and my value of y. Moving over here… This one is going to take less work. It is just the matter of a different looking diagram and trying to again, try to ignore the extraneous information we have. Let’s see here. I see that we have two parallel lines and I see a transversal – this one coming up here. If I cover up this part of the diagram, I’ve got two parallel lines being cut by a transversal. If I have two parallel lines cut by a transversal, then my corresponding angles are congruent. Well, the 61 is sort of in the upper right or sort of just above this intersection. So is the y. So, y, this angle here, and this angle here with 61 degrees in it… those are corresponding angles. Y is equal to 61 degrees because corresponding angles are congruent. Now, we have the x and I still see my parallel lines, but angle x is not being formed by this line intersecting my two parallel lines. Angle x is being formed by this parallel line and this transversal. Since I have already figured out what y is equal to, let’s do this. Let me take this transversal out of here. And do you see what is forming? We have two parallel lines being cut by a transversal. Let me just extend this line for a second. You see this z shape? Right here? Whenever you have parallel lines being cut by a transversal, the alternate interior angles – if you can see that or many highlight it with a pen or something on your diagram you’ve already made with a pencil, that z shape is what is formed by your alternate interior angles. Kind of like you’re Zoro! So, the alternate interior angles – my parallel lines being cut by a transversal – This angle is above the transversal. This angle is below it. They are both inside the lines being cut by the transversal. So alternate interior angles are equal. We have x is equal to 60 degrees. Really no algebra there. Just a bit more of a complicated diagram. And needing to sort of ignore the parts of the diagram that are not making the angles that we are dealing with. And hopefully, being able to recognize whether we have corresponding angles, alternate interior angles, alternate exterior, or same-side interior. I’m trying to think if an old textbook of mine used to call those consecutive angles…but same-side interior angles Let’s get to the next two algebraic examples. Given: line L and line M are parallel, prove that angle 1 and angle 6 are supplementary. So we have two parallel lines, and the first line of the proof is almost always given. Yes, L and M are parallel, and that is given. We have two parallel lines being intersected by a transversal. And I have just numbered every single one of these angles. And we want to prove that angle 1 and angle 6 are supplementary. 1 and 6…those are not any of the special 4 types of angles that we set up. They are not corresponding angles. They are not alternate interior angles. They are not alternate exterior angles. They are not same-side interior angles. We need to prove that they are supplementary. And of course, I’ve got a proof set up to work with you, and help explain it to you. You can certainly come up with another proof and still have a perfectly nice sequence of logic and have a correct answer. So I am going to give you a version, or a proof…it is certainly not going to be the only one that you can come up with. This one is very, very flexible with how many ways you can work it out. So, I want to somehow link 1 to 6. And I need to link it through angles that we do know: corresponding, alternate interior, alternate exterior, or so on. The first thing I thought of when I looked at this, and again, there are many ways you can approach this… Is we are going to to ahead and match up 1 to 3.
I somehow need to…you know, I could match 6 to 2, I could go through the angle of 4…I could go through the angle of 3, which is how we are going to approach this proof. And regardless of whether you want to show a relationship between… or you could go through 5…I want to go with the 1 and 3. Now, if I have angle 1 and angle 3, and I cover up the rest of the diagram that is not specifically related to those two angles, I see that angles 1 and 3 are adjacent and forming a straight line. Which means that angle 1 and 3 are a linear pair.
And that means that not only 1 and 3 are going to be a linear pair, but what does that mean?
Linear pairs are supplementary. By definition. Let’s get that written for step number 2. Well, you might want to say linear pairs are supplementary. Maybe your teacher is happy with just definition of linear pair. Maybe they are just happy if you write linear pair. Talk to your teacher about what they want. If angle 1 and 3 are supplementary, what does that mean? Supplementary angles are two angles that add up to 180 degrees. Let’s get that written out. So perfect example of when you build a proof…I am already out of sequence from what I wrote in my notes. When you build a proof, all that is important is… you know, if I want to make a statement in line 6, it is probably going to need to be using information that I built on in the previous statements. Some statements you can make at any point and it is perfectly fine. And other statements you need to pull from facts that you have justified earlier in the proof.
At this point, I have made some statements and I didn’t need to pull on any kind of knowledge or statements that have been made earlier in the proof. At this point, I am just stating facts. I have not really seen a connection or string of information that I need to put together. I am just writing stuff down that I figured out. So I have a relationship between 1 and 3.
Now that I have some kind of relationship between 1 and 3, we might not know really where it is going yet…If I am making statements about 3, we do know how 3 is related to 6.
Ultimately, we want to prove something about 1 and 6. That they are supplementary. So I need to string those angles together through some other angle. And that now is going to be 3, because I have introduced it into my proof. In lines 2 and 3. We are going to say that angles 3 and 6 are congruent. And they are, because they are alternate interior angles. And when parallel lines are cut by a transversal, the alternate interior angles are congruent. And I do not want to talk about 4 and 5 because they have nothing to do with ultimately what we want to prove. So I am going to step out and say that angle 3 is congruent to angle 6. So alternate interior angles of parallel lines are congruent. And that means that angle 3 is congruent to angle 6. But ultimately, I’ve got this equation up here that has a statement about measures. And I do not have anything about measures of 3 and 6 together, so just a small step… but what are congruent angles? Angles that have equal measure. Now we are that the point where the next line is going to have to build on information that I have justified and stated earlier in my proof.
And if I look here, I’ve got the measure of angle 1 plus the measure of angle 3 equals 180. Again, the proof is about angles 1 and 6 being supplementary or 1 and 6 adding up to 180. But down here, I have justified now that the measure of angle 3 is equal to the measure of angle 6. So, I have take my 3 – the measure of angle 3, that symbol or notation that is in this equation… take it out and plug in what it is equal to. Which is 6.
Do you see what is happening? I am going to have one equation saying that with substitution, taking the 3 out and plugging in the measure of angle 6…I am going to have a statement that says the measure of angle 1 plug the measure of angle 6 is equal to 180.
And that means I am almost done with my proof! Very cool! Well… This is not what I am trying to prove. This does not say EXACTLY the same thing. This says the measure of angle 1 plus the measure of angle 6 is equal to 180. And I am trying to prove that they are supplementary. But…isn’t that kind of the same thing?
It’s not formally in the proof yet, in my sequence of logic, but it is what is coming up next.
We are just going to say that the measure of angle 1 and 6 are supplementary because that is the definition of supplementary. Whoo! BAM! There we go! Proof is done! Again, as we go through geometry, some people will get these proofs like boom, boom, boom! And some people are going to struggle with them. Maybe just always struggle with them. You need to make sure that you know your defintions, you postulates, your theorems right off the top of your head.
Make an organized…some kind of notes where the properties are on one side and what they say are on the other. And you can fold it and know them back and forth. Or flash cards… something where you know those statements right off the top of your head. And as we do these examples, I am going to build these proofs from scratch and hopefully help you see the logic string… you know, we have a complex problem, but there were at least a few steps to get from point A to point B.
The real point of proofs is to help you develop those logical thinking skills of, “hey! I’ve got this complex problem, how can I work it out and find a good, logical sequence of steps that will allow me to complete the problem or solve the issue?” You do not get that in a lot of classes. However, I do see a lot more with the newer books…they give you these proofs, and it is just like You are not given the reason, or you are given the reason and not the statement. That is more, unfortunately, just a memorization type problem. It does not strengthen your logic building skills. They are good examples, and you should pay attention to how the facts are stringed together, to strengthen your own logical problem-solving skills.
That is really what you need to take out of these kinds of problems, once you are out of geometry. I’m Mr. Tarrou! BAM! Go do your homework!