Common Core Geoemtry Unit 4 Lesson 2 Constructing Angles and Parallel Lines

Hello and welcome to another common core geometry lesson by E math instruction. Today we're going to be doing unit two lesson number two on constructing angles and parallel lines. So unit four is all about constructions. And remember from our last lesson, that means that we're going to be using a compass and a straight edge to make these constructions and nothing else. So if you haven't already taken out your compass and straight edge, take a moment to do so now. All right, let's jump right into it and understand what we're going to be doing today. Now the title of the lesson, pretty much says it all. But one thing we're going to be doing is copying an angle. In other words, if I give you an angle, can you use a straight edge and a compass to create an angle that has the same measure as this one? Now the parallel line business, that's a little bit different, right? The idea behind the parallel lines is that we're going to have some line. And some point that's not on the line. And we're specifically going to want to create or construct somewhere. Where's my other line? There it is. We're going to want to construct a line that is parallel to this one and go through that point. Remember, that gets back to what we call Euclid's parallel postulate, which is the following. If we've got a line and a point not on the line, there's only one line that's parallel to this one that goes through that point. All right? And today, we'll learn how with only a compass and a straight edge we can create that line. But first, we need to learn how to copy an angle. All right, so let's get into it. A lot of these constructions now, I'm going to want to walk you through them step by step, and they'll be step by step directions in the actual lesson itself. Because constructions can be tough for students. In fact, a lot of schools and a lot of teachers will put them off until the absolute last thing that you do because they can be hard to remember for the exam. Luckily, you're going to have these videos to go back and reference right before you take that exam. And constructing an angle is one of the classics that will show up. So let's go through it. Here we go. Exercise number one, given angle a, shown below, do the following. Letter a, draw a line segment starting at R in any direction. All right, let me get rid of my compass. Let me bring my ruler over to point R now, I mean, I can draw this line segment in any direction I want, and it doesn't change the construction at all. But it probably makes sense to draw more or less horizontal line segment. So that's what I'm going to do. Okay? Now that's going to serve as one of the two rays to construct this angle. And again, it doesn't matter which rays which. Make sure you've got that. Now, let's do part B it says draw an arc about point a so that it intersects the two sides of a label the intersection points B and C all right, let's do that. Let's bring our compass out. Now notice I didn't specify how big the radius had to be. Just that it had to intersect the two sides. So that's good enough. And let's draw that arc. Now, don't change the radius of that arc. If you do change it or you accidentally changed it, you're going to have to go back and set it to be that exact radius. Now, it says to label these points B and C so I'm going to do that. And that's simply so that we can reference them later. Let's keep going. Letter C, draw an arc of the same radius. This is key. About point R so that it intersects the line segment you drew through R mark the intersection of this arc and segment as S all right? So without changing my arc, or the radius of my arc, I'm going to do this. Kind of go like that. And bring it back. Now keep in mind, a lot of times when we do constructions, we're essentially copying exactly what we did here over here. So the first thing I did is I kind of laid out a line segment here that is more or less like the line segment here. I drew an arc here with the center at a, I drew the same radius arc here with the center of R and it said label that intersection point as S okay, so there we have it. All right, let's keep going. Try to move this up without losing anything. We'll come back to you learn a bit. It says with S as its center. Draw an arc that has a radius that is the length of BC. Okay. With S as its center, we're going to draw an arc. But that arc has to have some radius to it. And the radius is going to be the length of segment BC, which I haven't drawn. I haven't drawn that segment in. And I don't need to draw that segment in. I just need to bring my compass over here. Okay. Maybe switch it around like this. I'm going to have to rotate the compass a bit. I'll do that up front. Easier for you than me. Bring it up here. Adjust the length. Now, I want to mark an arc right now that goes through C to prove. To prove that I measured the distance of BC. So I'm going to see if I can make that just a little bit smaller. There it is. Okay. That arc right there proves I'm measured that distance. Now I'm going to follow the directions and I think I'm going to do this just to make it a little bit easier on myself. That didn't work. And maybe it did. I got to come back up here, grab this, bring it down. I didn't quite work the way I wanted it. There it is. All right. So I just drew an arc with a center at S, whose radius was the same as BC. And notice, right? Let me get this out of the way. Notice how similar now that diagram is to this diagram. Truly, X marks the spot. So what did I want to do? Did I want to label that? Yeah, I wanted to label it as point T. Okay? Finally, draw RT, verify using tracing paper that angle a is congruent to angle R so in other words, I'm going to come back here. X marks the spot, bring my straight edge, not ruler, straight edge down here, line it up. Draw my ray, you could put arrows at the end of it. You don't have to put arrows at the end of it. The two angles look pretty congruent, but let's test it. So for you, you should test it with tracing paper. For me, I've got my little angle already, right? Obviously that overlaps angle a and now if I bring it over here and actually let me see if I can bring it to the front so that it will overlap. Order, ring the front and take a look at that. All right. By the way, not perfect. Pretty good. I got to say pretty good given that I'm doing this electronically, but not perfect. You know, it's a little bit off up here. But still, we can see with the tracing paper that the two angles are the same size. The two angles are the same size. And remember, an angle being the same size as another angle. It doesn't have anything to do with how long the rays are. Raise our infinite in length. What it has to do with right is the amount of rotation that it would take to move one ray into the other. So that's why this thing doesn't have rays that are as long as these two. And likewise over here. But the angles do coincide. All right? Copying an angle. Pretty easy. All right? We're going to need that construction quite a bit in this lesson, but we're going to get a little bit more practice on it in a bit. The first thing we'd like to do, though, is maybe take a look at why it works. Okay? So this is essentially a reproduction of what you just had. And notice exercise two says, given informal proof of paragraph proof. For why this construction works. All right. So let's do this together. This is pretty easy. Remember, what we did, the first thing that we did is we drew this arc with a center and a through this angle to have points B and C now, honestly, what that meant was by construction. Right? By construction AB, this distance, right? Was congruent to AC. This distance. Now honestly, that's actually not so important. What is important is I then copied it over here, right? I made it exactly the same arc as over here. So I can really kind of extend this. This is also congruent to RS. N congruent to RT. So this is congruent to this. And this is congruent to this. Honestly, I could have just used one dash all over the place. But I think you'll see where I'm going here. Then I came back in and I marked the length of BC. And I came in and I marked the same length over here, which created ST. So also by construction. Also by construction, BC is congruent to ST. Just like that. Well, now we can now say that triangle ABC is congruent to triangle RST. By side side side, right? And because those two triangles are equal. This angle must be equal or congruent to that angle. Angle a is congruent to angle R by CP CT C now, one of the great ironies I think about copying an angle is that essentially copying an angle is the same as copying a triangle. It really is. What did we really do? We created two congruent triangles here. And because of the two triangles are congruent, it means that angle a and angle R must also be congruent. That's just kind of the deal. All right? CPC TC reasoning. So if you were able to construct a triangle in our last lesson, you really should be able to construct an angle. In this lesson. But let's get a little more practice, okay? Here we go. Nice obtuse angle. Create a copy of obtuse angle W shown below. All right, well let's see what you remember. Pause the video now and take a little bit of time and then we'll go through the construction. All right, let's go through it. So to create an angle, we really had to lay out two rays. The first ray is mildly unimportant in terms of the way that you lay it out, but I'll just kind of do it horizontal. And again, the idea is I'm essentially trying to reproduce this over here. All right, maybe I'll even put a little starting point here. All right, now I'm going to take my compass, bring it up here. Stretch it out so that it's got some kind of an arc. Perfectly good. This is one place where it differs from, let's say, creating or constructing or copying another triangle. There were another triangle here. We'd actually have a third side that we had to kind of deal with. Here we're almost creating the third side ourselves. So I'm going to put the point at W and I'm going to mark that arc, just like that. Okay? I'm going to bring this over to here. And I'm going to make the same arc. Okay. So far, so good. Again, notice how I'm more or less reproducing the two pictures. Now, what I really need to do is I need to locate this point, if you will, over here. It's somewhere on this arc. The question is, where is it? Well, in order to locate it, I have to create another arc specifically with this distance as its radius. Okay? So I'll bring this thing over to here, and I'll put the sharp end at either one of these. Actually, I need to rotate my compass sum. That should do it. Okay, I'll put it right on that intersection. Widen it out. Of course I don't have enough. All right, let me rotate it down a little bit more like that. It's not bad. A little more. A little bit more. Beautiful. Bring it back. So where you'll have an easier time than me. I've set my compass to be that length. I now make an arc like that. I bring it over here. Come on. There we go. Make the same arc. Oh, those two barely intersected. But I just needed a minute intersect. The X marks the spot. Right there, you don't even have to put a point. I now bring my ruler over. Rotate it around. Again, I shouldn't even call it a ruler. I should only call it a straight edge. And there we go. All right? You can then easily verify that these two angles are congruent by using tracing paper. And it's kind of cool because for me, I can just sort of draw it in. Like that. And notice how well they lie on top of each other. All right? I would suggest that you do the same when you're doing your constructions. Have a piece of that tracing paper out, use it to make sure that the two angles are congruent. That's it, copying an angle. Again, the steps in copying an angle, right? Are really very, very similar to the steps in copying a triangle. In a certain sense, though, you're sort of creating this triangle yourself. And that triangle always being an isosceles triangle that you're then kind of copping over here. This would be that third, that third side. All right. So copying an angle. Now, let's talk about parallel lines. Okay, constructing a line parallel to another through a point not on the line. Again, real simple idea. I've got this line, I've got this point, right? And I want to construct a line that goes through that point, but is parallel to this line. Okay? Now, the key to this is really understanding how to prove that two lines are parallel. Remember, we can prove that two lines are parallel by showing that either alternate interior angles are congruent. Or by showing that corresponding angles are congruent. And it's that corresponding angle piece, right? That's going to be the key to this construction. Because what we're really going to do strangely enough is we're going to copy an angle. And by copying an angle, we're going to get parallel lines. Let's take a look at how that works. Okay, so we get this down here. Let's take a look at this exercise. Given line M shown below and point a we would like to construct a line parallel to M, passing through a all right, so it's the scenario we have. Now, one of the problems with this construction is that there's a lot of freedom to do a lot of different things. Including, let's take a look at the first part. Draw a line segment passing through point a and intersecting line M so I need to draw a line that goes through a and intersects line M and it can really do it in any way you want. Now I've located a little red point down here. I'm going to ensure that the line passes through a and through the red point. So that we can do something a little bit later on. But honestly, honestly, you can draw a line any which way you want as long as it passes through the point that you're interested in. That's point a I mean. And intersects the line that you're supposed to draw this parallel line through. All right? So something like that. I really kind of wanted that in blue and I see what happens if I trace over it in blue. Wonderful. I thought maybe I'd get purple, but I just got blue. All right. What do I get the feeling that's going to come back to haunt me in the end? Whatever. All right, so there we have it. Now, why did we do that? Let me give you a little preview. We did that to create an angle. And now what we're going to do is we're going to copy that angle. We're going to copy that angle, but we're going to use a as the vertex. Okay? So part B using the line segment through a as one of the sides, copy the angle, the line segment, makes with M at a, right? So that corresponding angles are congruent. In other words, I want to take this angle and I want to create it up here, okay? With a as its vertex. So let's go through that. All right, I'm going to do actually the exact construction I just went through. All right, I'm going to put my compass at that red point. Now again, I'm going to put the compass at the point where that line that I had to draw intersects this line that I'm trying to get the parallel line to. All right, so now how did that work? Well, let's see. Actually, I didn't quite get it through the red. So we're going to do that. Remember, I draw an arc. All right, that's my first step in copying an angle. Then what happens is I take this thing and I put it at point a and I draw the same arc with the same radius. Here we go. All right. Make sure that the arc is big enough so that it intersects so that it intersects this line. Now again, take a look. I got this arc. I got that arc. Now what I'm really trying to do is find this point. I want to find that point on this arc. And I'm going to do it exactly the way I did it before. I'm going to take this thing. And I'm going to put it like this. I'm going to move this. I'm going to try to move this. I'm going to rotate this and try to set the radius to be this length. All right, the radius to be the length between this point and that point down there. I'm going to prove to the person grading my paper that I made that length. By putting a little arc mark through it, and now no great surprise. I will do exactly the same arc up here like that. Get rid of my compass. And now I will simply draw the line segment. That goes through it. And look at that. Look at that. Parallel lines. Parallel lines. It's one of the most interesting constructions I think. But it very much relies on you being able to copy an angle. Being able to copy an angle. Now let me just go down a little bit. It says use tracing paper to verify that the two angles you created are congruent. Explain why the side of the newly constructed angle that does not intersect M must be parallel to it. So again, why is this parallel to this? Well, first, this is why I had to have that red dot down here. Let me try to bring this out front. Order, bring the front, wonderful, just verify. There it sits. You know, I had to have that angle kind of already predetermined in order to show you that yes, in fact, I copied the same angle, okay? So tracing paper can tell us that this angle is congruent to that angle. Now the question is, why do those two angles being congruent tell me that these two lines or line segments are parallel? Think about that for a moment. All right. These two angles, let me extend it a little bit. The two congruent angles. Angles are corresponding. Am I really not going to be able to get that in? Oh, I did. Our corresponding angles. Remember corresponding angles? These two angles, if I kind of extended this, let's say, right? These two angles are in the same place on the T, right? A T down here, and a T up here, that angle and that angle are both in the, I guess, upper right hand corner of that T and because the two corresponding angles are congruent, these two lines then have to be parallel. It's really kind of cool. It's the way that we always will create parallel lines. In fact, when in the old days, when people were creating architectural drawings and things like that, they would slide one triangle along another triangle to maintain those angles as being equal to create other parallel lines. But that's it. The key to drawing one line parallel to another line through a point, strangely enough, is making sure to keep angles the same. But parallelism is all about direction, and direction is really all about angles. So let's do some more of these. This one really takes some practice. Okay? I've broken up the next problem onto two different sheets. All right? And I don't have my ruler on this one. That's okay, we'll break out the line tool. But here it says construct lines parallel to the ones shown below through the point given. All right? So let's do, let's do one more of these together. And I'm going to actually break out of this just for a moment so that I can grab my ruler tool. I really need a ruler on this. Then we'll go right back in. Hopefully, there we go. I need that ruler. Because my first step is to draw a line that goes through the point and also intersects this line. So that's my first step. And again, the reason that I do this is that I'm creating an angle that I can then copy, right? I want to copy this angle up here at R, okay? But now I'm back to my copy the angle construction. I'm going to bring this thing down here. All right. Widen it out. Maybe even a little bit farther. Make sure that that arc intersects both sides of the angle. Bring it up to where I want the other angles vertex to be. Draw the arc, make sure that that arc intersects this side of the angle that's going to have its vertex at R bring this here now. Rotate down. Wow, that's going to be exactly the same. That's kind of weird. Anyway, if it weren't exactly the same, set that distance like that. Bring it up to here. Make the arc again. Again, take a look at how identical this is. Two arcs, one here, one here. The same two arcs produced up here. And now. Maybe even kind of slide my ruler back a little bit. And I've got my two parallel lines. All right. Again, to make a line, parallel to another line. Through a point not on the line. We draw this sort of like weird extra line in. That creates an angle down here. We then reproduce the same angle using this first line that we drew through the point that we care about, we copy that angle, making then these two angles congruent, and because those two angles are corresponding angles, it means these two lines are now parallel. All right? Interestingly enough, there's no real way to test or check the parallelism. The only thing you can really test is to see whether or not you created congruent angles, but you can do that with tracing paper. All right, let's go on to the last construction, which is the same one. But it's just in a different place in a different orientation. And what I would like you to do is all on your own now. Attempt to construct a line parallel to F that passes through B take a few minutes to do that. Oh, I have my ruler on this one. Excellent. All right. Let's go through it. Same steps, nothing new. I need to create an angle. Or create a line that goes through B let's see, how do I want to do it? Let me swing it up a little bit like this. Maybe like this. It's kind of debating in my head which way I wanted to intersect the line with. We'll kind of go like that. Great. I now have this angle. I want to reproduce it with its vertex at B so I'll come in right here. Stretch my compass out. And you get rid of my ruler. Not like that. I won't. Oh, man. There we go. All right. Make that arc. Right? Bring it up to point B. The other arc, okay, keep in mind now what I'm looking for is I'm really trying to locate the equivalent point here up here. I'll do that. By bringing this down, I can't keep making this thing nearly equilateral triangle, which I don't mean to. Measure that arc off. Right? Bring it over here for me, I've really got to switch this thing around. Up to here. Whoops. Move this away. And now X marks the spot. Keep rotating, rotating rotating. Bring this thing up a bit. And draw my line. Pretty good. Actually, that's my worst one so far today. But what are you going to do? All right. Yeah. Not overly happy with that. But it'll do. It probably looks a little bit different because I have the cross here versus the cross there. The only difference there, and I just want to emphasize that it really wouldn't make a difference. If I had brought this over here and sort of measured this distance like this, right? And then brought it over here. But it really should be the same distance. That is why I'm a little bit off, though. Is that this arc isn't quite the right length. And I bet those of you with a very good eye can tell that this line and this line aren't quite parallel. All right, but again, this construction would still receive full credit on any kind of test or standardized exam because all the correct arc marks that are there. And again, keep in mind, you only need one of these. That's not going to erase it all. That's okay. Apparently I can't erase arc marks once they're there, but I can just delete them. There we go. That's a little bit better. I like that one. All right. Let's wrap the lesson up. So today we saw two constructions, one of them hinged on the other. The first construction was copying an angle, which was very, very similar to copying a triangle from the last lesson. Using then our ability to copy an angle, we then were able to create a line that was parallel to another one that went through a given point not on that line. Euclid's postulate, all right? You quotes parallel postulate. So one of many different constructions, copying that angle, that then allows us to then do another construction, creating a parallel line. We'll see a lot more constructions that involve perpendicular lines, not parallel ones in the next lesson. For now, though, I'd like to thank you for joining me for another common core geometry lesson by E math instruction. My name is Kirk Weiler. And until next time, keep thinking. And keep solving problems.