Geometry Proofs Made Easy: Proving Parallel Lines Step-by-Step

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[00:00:00]Okay, let's talk about geometry proofs.

[00:00:02]And these are problems that you typically start to encounter at the high school level. So, if you're taking high school geometry and your teacher is giving you problems that say something like prove this or prove that. Matter of fact, let's go ahead and take a look at this example. Uh this will kind of um you know, clarify what I'm talking about. So, given that angle one is congruent to angle three, and I'm talking about uh what's going on in this figure right here, we want to prove that line L is parallel to line M. Now, these two lines appear to be parallel, but we can't write this down be like, "Well, line L looks to be parallel to line M, therefore uh L and M are parallel."

[00:00:43]Well, of course, we're going to need an actual concrete justification, some proof. And this is no different than how like a lawyer you know, or an attorney is talking in court, right? They're trying to convince a jury of something.

[00:00:56]They're saying, "You know, uh this, then this, and then therefore it must be this." Proof is proving something, okay?

[00:01:04]So, you need to kind of put that kind of thinking cap on it. All right, how do I justify something? Well, in mathematics, you have to follow very specific rules.

[00:01:13]And the rules or the kind of the um structure going to be using at the high school level in terms of high school level geometry proofs is something called deductive logic, okay? So, I'm going to be kind of going over what are very common type of proofs um that high school students um face when they take geometry. Now, I will say this, most students hate doing these type of problems. But just remember that there's not one way to prove something, okay? You could do a nice proof. It might be different than what your teacher would do, but it could still be technically correct. But you do have to follow very specific rules, and that's what this video is going to be about.

[00:01:49]So, what we're going to be doing here is what we call a two-column proof. It's the most typical type of format for um this level of mathematics for what you're going to be dealing with. Again, this is generally speaking most of you out there taking high school geometry.

[00:02:04]It's kind of what these videos targeted at, but maybe some of you need to just review geometry in general. Okay, so a two-column proof is basically looks like this. Okay, there's two columns.

[00:02:16]Here you're going to make a statement.

[00:02:17]You're going to say something. Then you're going to back that statement up.

[00:02:21]You're going to give a reason why you're saying that or justifying that. So you're going to make a statement and then you're going to give a reason. Then you're going to make another statement and then you're going to give another reason, but the statements have a certain flow. Okay, what you're trying to do is go and argue. Okay, you're trying to prove your final point. Okay, so you're trying to prove something and it's no different than saying you know, to your best friend, "Hey, you know, you're trying to let's say convince them of something and you're debating or arguing about something. You say, "Well, don't you agree that this happened?" And you'll say, "Oh, yeah.

[00:02:55]Okay, yeah, that happened. All right, boom." Right? So you say, "Okay, that happened. All right." Then you then you try to walk them to your next point. I'm pretty sure everyone, you know, understands what I'm saying. Maybe it's your brother or sister or your boyfriend or girlfriend, whatever the case might be. You're like, "Okay, what about this?

[00:03:11]Now this happened, right?" "Okay, yeah, yeah." "Or that happened, too. Boom."

[00:03:14]And so you're trying to step-by-step logically, you know, get to a certain conclusion.

[00:03:19]Okay, this is the idea in doing a proof.

[00:03:22]But again, where there's very specific rules because what we're dealing with geometric proofs. All right, so we're going to give a statement and then we're going to give a reason and we're going to just kind of uh continue to make as many statements and back those up with the with as many reasons until we reach our final conclusion, until we have proved what we want to prove. Now, here's the thing. You can see I have some stuff going on over here. The reasons you can list and here is the main kind of point or one of the main points of this video.

[00:03:56]The reasons that you can give aren't opinions. You're not going to say, uh uh don't you think that, you know, these two angles are uh equal because they look to be equal, you know, because you know, I measured them on my paper and they look and they are equal. No, you can't give a reason like that. The only reasons you can give, okay, in a geometric proof, and this is kind of classic geometry, is the following. One is given information, okay? Now, what am I talking about given information? Well, let's go back up here.

[00:04:29]In this proof, I told you given that angle one is congruent to angle three, I gave you that information, so you can justify that in uh given information, your first statement is typically almost always going to be you're going to kind of repeat that given information, okay? So, that's one thing you can you can uh give, but oftentimes you can use this given information, you know, in other um statements throughout the proof if you need to. So, that's one allowable justification for a statement, okay, or reason uh that you can back it up. The second thing is definitions, okay? So, definition, so like a triangle has a specific definition of what it is, right? A circle or parallel lines, okay?

[00:05:14]These they have very specific mathematical definitions, and this is where your notes come in big time. If you're not taking like notes in sp- in any math course, but especially geometry, because you're going to be learning a ton of theorems, properties, postulates, axioms, corollaries uh there's even these things called lemmas. There are a ton and ton and ton, I mean, like it can just like add up hundreds and hundreds and hundreds of these little definitions, postulates.

[00:05:44]You need to be um highly organized and be able to call upon these, okay? And this is where I think where most students uh struggle with and in terms of geometry course cuz they're not as, you know, as good as note takers as they need to be.

[00:05:58]You need to have super organized notes, okay? So, that's definitions.

[00:06:02]Now, that goes along with postulates, okay? So, uh a postulate, right, is different than a theorem, okay? A postulate basically is something that we um are kind of given. We prove theorems, okay? Postulates are kind of given to us. They're like There's another thing in mathematics called axioms. We just assume them to be true, okay? Theorems we actually can prove. Now, just I'm kind of skipping over this here for a second just so you can um uh kind of get this idea of theorems. You probably know what this is, a squared plus b squared is equal to c squared.

[00:06:38]That's the Pythagorean theorem, okay?

[00:06:41]That can be proved. We can actually do a nice justification and prove that particular theorem, okay? But a postulate or just These are state These are things in mathematics that we just kind of assume to be uh true and they uh There's another kind of word that goes uh for them uh uh like axioms, okay? But postulates are typically this word is used pretty heavily in geometry. Okay, so let's go on to our next reason here and that's the properties of algebra and congruence. I'm going to show you some examples of uh what this is, but this is There's a lot of these as well, okay?

[00:07:16]There's a ton of these or a lot of these There's a lot of these and of course there's a ton of theorems which I explained. So, anytime you give a statement for a proof, the only thing you can say is that oh, that's given information, this is a definition, this is a postulate, maybe this is a theorem, that's uh a definition. So, you just can't have opinions, you got to back it up with these things right here, okay?

[00:07:39]So, therefore, you better have super great notes, okay? Or this is going to become very difficult, and that's why a lot of students don't like doing these problems cuz, you know, there's no way you're just going to be able to be able to remember all this stuff. Okay, so let's going to take a look at some examples of properties of algebra and congruence. So, what am I talking about?

[00:08:01]Well, here's just a couple of quick examples. There's actually quite a few of these things, but uh let's go ahead and take a look at a few. Okay, so you're probably saying, "Oh, I remember learning this kind of stuff."

[00:08:13]Well, let's take a look at the addition property of algebra, okay? Now, you can just kind of abbreviate this the addition property, but let's take a look at this. If A equals B and C equals D, then A + C is equal to B + D. Now, this seems like very obvious, right? Look, if this is equal to this, okay?

[00:08:37]A is equal to B.

[00:08:40]Well, then if um C is equal to D, well, if I add C over here, A + C, that's the same thing as B + D. Okay, if you kind of study this for a second, it would seem like logical, right? You'll be like, "Yeah, you know, yeah, if A equals B and C equals D, then of course A + C would equal B + D." But we need this right here has to be justified. Okay, you have to back this up formally, okay? So, you have you'd have to say, "Yes, this is true because of the addition property of algebra."

[00:09:11]Okay, you just can't write this down.

[00:09:13]You have to actually know the name, okay? So, it is important that you can identify these properties, definitions, etc. by name. And there's no way you're going to be able to remember all remember all this stuff. You got to reference your notes, okay?

[00:09:26]All right, so let's take a look at another example of uh another property example in algebra, and this is the transitive property. Whoop, I'm missing a little uh N here, transitive property.

[00:09:38]Um so, the transitive property, big property. So, here is the transitive property. If A equals B and B equals C, then A equals C. So, makes sense, right?

[00:09:48]If A equals B and then B equals C. All right, so I'm like, okay, AB. So, you can almost think of it this way. A equals B and then B equals C. Therefore, A is equal to C.

[00:10:02]It's totally logical. Well, you got to back this up. This right here is true because of something called the transitive property of algebra, okay?

[00:10:12]Now, let's go over here and take a look at congruence. There's also properties of congruence. Now, properties of algebra, um you're typically going to have like the equal sign. Over here, with properties of congruence, we're um talking about this little congruent um symbol right there. So, what is congruence? Well, the congruence here, if I said like the number two is equal to two, okay? That is just like a property of algebra. Okay, that would be like the reflexive property.

[00:10:42]Now, we have a similar um property. Okay, it's basically the same property, but we call it the property of congruence.

[00:10:50]But, congruence has to deal with measure the measurement of something that's given us a measure, okay? I.e., let's let's actually take a look at an example. So, let's look at the reflective property of congruence. So, line segment AB is congruent to line segment AB. In other words, here's AB.

[00:11:09]It's basically the same measure of itself, okay? Well, we can state that because of the reflexive property of of congruence. Because we're dealing with line segments, okay? We're actually measuring something. So, when you're measuring line segments or angles, like here's another example of the reflexive property of congruence.

[00:11:29]Uh angle A is congruent to angle A. So, this would not be the reflexive property of algebra. To be more technical about it, you want to have the reflexive property of congruence. But, if I just said two is equal to two, that would just be the reflexive property. That's a property of algebra. So, going up here, okay? There are quite a few of properties of algebra and congruence.

[00:11:53]They should be somewhere in your notes.

[00:11:56]Now, a lot of you are probably kind of flabbergasted at this level. You might be going like this here just like, "This is why I hate doing these problems. You know, there's so much to know. Just like And then, I got to like figure out how to actually do these things." Well, listen. Don't give up the ship you just yet, okay? I'm trying to tell you what you need to work on in order to be organized, okay? And in geometry, when you're dealing with proofs, you're not going to, you know, be effective at this unless you can reference all this stuff. But, listen. If your notes aren't where they need to be, just improve them. You can definitely go back and get yourself uh to where you need to be, okay? It's just going to require a little bit of work. All right. Now, here comes the fun part.

[00:12:40]Are you struggling in math because of confusing lessons? Maybe the teacher's not showing you all the steps you need or things are happening too fast. Well, there is a better way. So, come on over to my math help program at tcmathacademy.com.

[00:12:57]There, you'll find clear step-by-step instruction by me that will definitely make a huge difference in your math success. So, make sure to check out all my courses by following the links in the description.

[00:13:11]So, after we have a sense of that we're going to be doing a two-column proof, we know our reasons that we can give, okay?

[00:13:18]We kind of went over some examples of properties of algebra and properties of congruence. Well, all of that, okay?

[00:13:26]It's just the tools. Okay, now we need to get creative and really our first move here those are kind of the rules of the game, but the game itself, in order to prove something, right? If you're going to like, you know, here's you and here's your brother or sister and you guys are having an argument over something. Now, the game is, "Hey, I got to convince this person that I'm right and they're wrong." Okay, that's the game. So, now that we know the rules, we need a game plan to prove our point. Okay, so this where it be- this is where it becomes fun.

[00:14:02]Okay, so let's take a look at the situation here and we're saying, "Okay, given angle one is congruent to angle three." We got to understand these words. So, here's angle one and we're saying it's congruent to angle three. In other words, this angle and this angle are equal. We're like, "Okay, all right, that's given to me.

[00:14:21]I want to prove that these two lines are parallel, okay?"

[00:14:27]So, it's probably going to help us to review a lot of theorems and postulates about parallel lines. So, this is the key, okay? This proof has to do with proving two lines are parallel. So, what you want to be thinking in your little thinking cap there is you're going to be saying to yourself, "Okay, what do I know about parallel lines?" So, you need to be thinking about all the theorems you know, all the postulates you know, and like the definition of parallel lines. You got to be thinking, "Okay, what do I have?" Okay, this is all this stuff should be in your notes, okay?

[00:15:02]Now, we want to be thinking about this particular problem, a certain theorem that can help us out, but I'm kind of jumping the gun here.

[00:15:10]So, you're like, "All right, I know a lot about parallel lines and I'm seeing that two and angle one, these two angles here, okay? Cuz I didn't in the given information, angle two wasn't mentioned, but these two angles are what kind of angles? They're corresponding angles. Okay?

[00:15:30]And there's a theorem you're saying, "Oh, corresponding angle. Wait a minute.

[00:15:33]I remember saying a theorem that said if two core corresponding angles formed by a transversal and two parallel lines, okay? If two corresponding angles formed by a transversal and uh two lines, okay? If these two corresponding angles are congruent, then these lines are parallel." Now, I kind of just paraphrased that, but that's effectively what a theorem would look like. Now, in your book or in your course, you might be like theorem you know, your teacher might have that as theorem one or whatever. It's going to be categorized, but there you're going to have some sort of theorem that you'll learn that says, "Hey, if you have a transversal and two lines and these corresponding angles here, okay? Or or corresponding angles are congruent, well, that is telling you that the that the lines here are indeed parallel. So, that's what we want to shoot for. Our game plan is like, "Oh, okay. So, why don't we construct a plan to argue that angle two and angle one are corresponding?" Now, can we do that?

[00:16:35]Absolutely. Okay? So, we can say, "Hmm, angle three and angle two, all right?

[00:16:41]These two angles right here are what we call vertical angles." Okay? You're like, "Oh, yeah, vertical angles. These two angles are congruent." All right? In other words, they're the same. So, this is the same as this. Okay? Three is the same as two, and three is the same as one. So, therefore, two is the same as one. Okay? So, I can say, "Okay, two is the same as one." And that's what I mean. Okay? Two is the same as one, meaning that these corresponding angles are congruent. Therefore, I can make a um final statement that these two lines are parallel. So, this This where the interesting part comes in. You need to kind of a game plan. Don't start doing a proof until you construct this. Now, if you saw another strategy to get to this and maybe you're using another theorem or whatever the case is, there are other ways you can do this, okay? But you want to kind of look at the most elegant most effective efficient way to prove your point. But that's okay. I mean, as long as you get it right, that's really the main thing here, okay?

[00:17:38]All right. So, with that being said, if you want to try this on your own before you see my final answer, what you want to do now is make a statement, okay?

[00:17:49]Which is typically going to be the given. And then you're going to use a definition, postulate, property of algebraic congruence or theorem to kind of walk through, okay? And your final statement will be that is your your final statement is always going to be line L is parallel to line M in this particular case because of your final reason, okay? So, this is what you're going to think through. We have a game plan. So, now we have to kind of tell that story. All right. So, let's go down here and actually do this now. If you want to, you know, play around with this for a second, go to pause the video and construct a proof.

[00:18:26]But let's go ahead and see my work.

[00:18:28]Again, you could have done this maybe a different way, but this is going to be the most direct route to take. All right. So, two-column proof for this.

[00:18:37]So, given angle one is congruent to angle three, prove that line L is parallel to line M. Of course, that's the problem. Let's go ahead and get into it now.

[00:18:49]Okay. So, angle one is congruent to angle three. That is given, all right?

[00:18:54]This is typically going to be your first move. And I kind of think of this as playing a chess game or checkers game.

[00:19:01]You're like, okay, you're going to make one move and typically repeating the given information is almost always your first statement. So, just get in the habit of just putting that down. Boom.

[00:19:10]So, now I'm going to say angle three is congruent to angle two. So, remember over here my I'm already being told that three and one are the same angle. That's given information. But, in order for me to make my little case here, I need to justify that three and two are the same.

[00:19:29]And three and two are the same because of vertical angles. So, that's my next reason. I can't do anything until I make establish establish that fact. So, angle three is congruent to angle two. Why?

[00:19:42]Because vertical angles. Okay, vertical angles are congruent. All right, three and two are vertical angles by definition. And vertical angles are congruent. So, that is a definition of vertical angles.

[00:19:55]Okay? All right, so there is somewhere along the lines you learned about vertical angles. That is a definition.

[00:20:01]Okay, so now let's move on to our next statement. I'm going to say angle one is congruent to angle two. All right, so now that I made that point, three is equal to two. Three is also equal to one. So, therefore two is equal to one.

[00:20:17]Remember, this is my next strategy. I want to show that these corresponding angles are equal. So, let's go ahead and state that, but I got to back it up. So, angle one is congruent to angle two.

[00:20:27]Why? Because of the transitive property.

[00:20:30]Okay? And to be more technically correct, you could put the transitive property of congruence. Recall, let me go up here real quick.

[00:20:39]Remember what the transitive property is, and that is this. If A is equal to B, okay, so here we had if angle one is equal to angle three, which we already know, and angle three is congruent to angle two, okay?

[00:20:57]Therefore, A is equal to C. So, in other words, angle one is congruent to angle two, and that's what we want. But, that is a illustration of the transitive property.

[00:21:08]Now, because we're not using equal signs, we're using congruent signs, transitive property of congruence.

[00:21:14]Ooh, boy. Okay, so, you know, if you're struggling with proofs, you know, there are, you know, they're they're like a little puzzle you got to figure out, like a little Rubik's Cube. Gosh, I remember those things came out in the early '80s, they were driving people crazy. Uh typically, people would just take them apart and put them together the right way. You can't necessarily do that with a problem like this, but you can get better at math, and that's the whole idea here. Okay, so, transitive property, so, we're pretty good to go.

[00:21:40]We've um proved um you know, our up to this point angle one and angle two are congruent, and this is really kind of the secret to our kind of game plan here. As long as we could prove that these corresponding angles are congruent, well, that can only happen if the lines here, these lines formed with this transversal are parallel. What basically establishes that? Well, we have a thing called a theorem that backs this up. And just to use a paraphrase, so now we can finally make our justification, hey, angle uh I'm sorry, line L is parallel to line M.

[00:22:17]If I've been saying angle, I apologize.

[00:22:19]I am speaking a lot a lot of I'm trying to stay focused here, so even myself doing, you know, teaching this is, you know, can be challenging sometimes. So, line L is parallel to line M, okay?

[00:22:33]That's my final conclusion, but I got to back it up with a theorem it says, if two lines cut by a transversal, okay, and the corresponding angles are congruent, then those lines are parallel, okay, that those lines are parallel.

[00:22:50]And um this is an example of constructing a geometric proof, okay? It's pretty involved, all right? No doubt about it.

[00:22:59]And if you're having, you know, uh problems doing proofs, that's pretty normal, okay? Even for very good students. But, hopefully this little video here helped you kind of go from this little face to let's kind of give it a little bit better face here, maybe to like this face be like, "Okay, I think I can do this. I can assure you if you continue to practice, you'll end up with a nice little happy face. You'll be like, I am so smart." Uh I'll always show my teacher that I could do these problems every single time. And you can.

[00:23:30]So, don't get stuck in this situation, all right? What you have to do is, you know, uh learn the fundamentals, get your bearings, improve your notes, get yourself organized, and then you're like, "Okay, now I'm ready to kind of go." And as you continue to do problems successfully, you gain confidence, and eventually you'll be able to handle these problems. But, these problems are going to come up over and over again, okay? Especially in a geometry course.

[00:23:55]And if you continue on in uh more advanced mathematics, you're absolutely going to be having to think in terms of proving stuff.

[00:24:03]Okay, so hopefully this video helps you out. If that is the case, don't forget to like and subscribe. If you need more help with geometry, anything geometry, definitely check out my geometry course.

[00:24:11]You can find it on my math help program.

[00:24:13]But, with that being said, I definitely wish you all the best in your mathematics adventures. Thank you for your time, and have a great day.