WEBVTT mathematics/geometry/pyo 00:00:00.000 --> 00:00:02.000 Welcome back to Educator.com. 00:00:02.000 --> 00:00:05.400 The next lesson, we are going to explore congruent triangles. 00:00:05.400 --> 00:00:11.900 We are going to be looking at triangle parts, and then we are going to compare two triangles together 00:00:11.900 --> 00:00:18.000 to see if we can see that they are congruent. 00:00:18.000 --> 00:00:30.600 In order for triangles to be congruent, they have to have the same size and shape, 00:00:30.600 --> 00:00:42.900 because if two triangles just have the same shape, then one can be this small, and another one can be this big. 00:00:42.900 --> 00:00:46.600 They have the same shape, but they are not the same size. 00:00:46.600 --> 00:00:52.300 They have to be the same size and shape. 00:00:52.300 --> 00:00:57.500 Here we have two triangles, triangle ABC and (it is congruent to) triangle DEF. 00:00:57.500 --> 00:01:02.600 Now, if you want to state triangles, remember: you can just write a little triangle symbol in front of that. 00:01:02.600 --> 00:01:06.400 And so, that is how you write "triangle ABC" in symbols. 00:01:06.400 --> 00:01:11.500 Triangle ABC is congruent to triangle DEF; these two triangles are congruent. 00:01:11.500 --> 00:01:24.200 Now, in order for two triangles to be congruent, they don't have to just look and be in the same upright position. 00:01:24.200 --> 00:01:32.000 Here there are three different ways that triangle DEF is shown. 00:01:32.000 --> 00:01:37.000 And even if you move it around, and you flip it, and you do all of these things to it, 00:01:37.000 --> 00:01:42.400 which are called congruence transformations, you are still going to get a congruent triangle. 00:01:42.400 --> 00:01:56.600 It is still going to be the same thing; so from this triangle, DEF, to this triangle, this is when it just moves. 00:01:56.600 --> 00:02:06.500 And when it just moves to a different position, that is called sliding; so this is sliding. 00:02:06.500 --> 00:02:11.800 Nothing changes about it, but it just moves. 00:02:11.800 --> 00:02:21.600 The next one, right here: see how it looks like it is turned a little bit. 00:02:21.600 --> 00:02:32.400 That is kind of doing one of these; and that is rotating; this one is rotated. 00:02:32.400 --> 00:02:44.700 And that is just when you take the triangle and just move it so that this top angle is no longer the top angle; you are just rotating it. 00:02:44.700 --> 00:02:54.900 And the next one, right here: this one is flipping; all we did was to take this, and we just flipped it from up to down. 00:02:54.900 --> 00:03:07.400 You just flip it; so this one is kind of like that--you are flipping it. 00:03:07.400 --> 00:03:11.400 There are three congruence transformations: slide, rotate, and flip. 00:03:11.400 --> 00:03:15.500 And no matter what you do to it, it is always going to be congruent to triangle ABC. 00:03:15.500 --> 00:03:20.400 This triangle, DEF, is still congruent to this triangle, ABC. 00:03:20.400 --> 00:03:30.900 So that means that, if I have triangle ABC like this, and triangle DEF like this, they are still congruent, 00:03:30.900 --> 00:03:34.900 even though they are not in the same upright position, like these two. 00:03:34.900 --> 00:03:40.500 These are congruence transformations. 00:03:40.500 --> 00:03:55.700 Now, corresponding parts of triangles have to do with their angles; you are comparing the angles of one triangle 00:03:55.700 --> 00:03:59.500 to the angles of another triangle, or the sides of one triangle to the sides of the other triangle. 00:03:59.500 --> 00:04:05.000 So, the parts of a triangle, we know, are angles and sides; and you are saying that, 00:04:05.000 --> 00:04:11.700 when it is corresponding...remember corresponding angles?--if we have two parallel lines and a transversal, 00:04:11.700 --> 00:04:17.000 remember how the corresponding angles were the angles in the same position? 00:04:17.000 --> 00:04:27.500 If we had an angle on the top right from the top part, or just one of those, then the bottom for the next part... 00:04:27.500 --> 00:04:40.900 let me just draw it out for you: remember: this angle right here (if these lines are parallel) and this angle were corresponding angles. 00:04:40.900 --> 00:04:48.500 This angle is the top right, and this angle is also the top right; so they have the same position. 00:04:48.500 --> 00:04:52.300 It kind of means the same thing here, too. 00:04:52.300 --> 00:04:55.400 Again, we have that triangle ABC is congruent to triangle DEF. 00:04:55.400 --> 00:05:01.600 Now, remember: we have six parts total: three angles and three sides--six parts total. 00:05:01.600 --> 00:05:17.100 That means that, if I have two congruent triangles, this triangle congruent to this triangle, then all of its parts... 00:05:17.100 --> 00:05:24.100 see how, for this triangle, A is named first; it doesn't have to be named first, but it is. 00:05:24.100 --> 00:05:32.400 Then, this D is named first for this triangle; then angle A is congruent to angle D. 00:05:32.400 --> 00:05:37.200 This angle is corresponding to angle D. 00:05:37.200 --> 00:05:48.600 So, to write "corresponding," I can draw an arrow like that, and that means "corresponding"; "corresponding" is like that. 00:05:48.600 --> 00:05:57.300 Now, B is second, and E is second; that means automatically that this angle and this angle right here are corresponding. 00:05:57.300 --> 00:06:10.700 B is corresponding to E; and then, the last angle that is mentioned, C, is corresponding to angle F; C is corresponding to F. 00:06:10.700 --> 00:06:19.500 That means that angle C is congruent to angle F. 00:06:19.500 --> 00:06:25.800 So then, this just shows that they are corresponding; I can also write the same thing, showing that they are congruent. 00:06:25.800 --> 00:06:31.200 I know that, because we know that the triangles are congruent, all of their parts are congruent. 00:06:31.200 --> 00:06:48.300 So, angle A is congruent to angle D; angle B is congruent to angle E; and angle C is congruent to angle F. 00:06:48.300 --> 00:06:56.500 Now, each one of these letters represents an angle. 00:06:56.500 --> 00:07:02.600 And if you put two of them together, they represent sides. 00:07:02.600 --> 00:07:07.900 Whatever is written first is going to be congruent to this one that is written first. 00:07:07.900 --> 00:07:16.000 So then, I can't say that angle A is going to be congruent to angle F, because they have to be named in the same order. 00:07:16.000 --> 00:07:20.200 Congruent parts have to be named by that same order. 00:07:20.200 --> 00:07:32.200 Now, if I were to say triangle BCA, that is OK; I can name this triangle however I want. 00:07:32.200 --> 00:07:43.600 I can name it triangle ABC; I can name it triangle BCA; I can name it triangle BAC, triangle CAB, CBA...whatever. 00:07:43.600 --> 00:07:48.400 I can name this triangle however I want, but if I have two congruent triangles, 00:07:48.400 --> 00:08:04.400 and I am going to write a congruence statement, then whatever I write next is going to depend on what I wrote first. 00:08:04.400 --> 00:08:18.700 If I am going to write it like this, then I have to write what is congruent to angle B (angle E) first; C is F; A is D. 00:08:18.700 --> 00:08:23.100 You can write it like this, or you can write it like this. 00:08:23.100 --> 00:08:31.800 If you decide to name this triangle CAB, then you would have to say that it is congruent to triangle FDE. 00:08:31.800 --> 00:08:43.000 So again, it doesn't matter how you name the first triangle; but the second triangle will be dependent on how you write that first triangle. 00:08:43.000 --> 00:09:00.600 So, again, here are the corresponding angles; and then, for the sides, AB is congruent to DE. 00:09:00.600 --> 00:09:15.600 And you can also look at that in this way, too: AB is congruent to DE; BC is congruent to EF; 00:09:15.600 --> 00:09:27.300 now, if I said BC, then I can't say FE, because again, remember: B is congruent to E; 00:09:27.300 --> 00:09:32.500 so if you are going to say BC, then the same thing applies for this. 00:09:32.500 --> 00:09:37.800 You can't say FE; you have to say EF, because B and E are corresponding. 00:09:37.800 --> 00:09:41.300 It has to be in the same order. 00:09:41.300 --> 00:09:53.800 And then, the next one: AC is congruent to DF; if you said CA, then you would have to say FD, 00:09:53.800 --> 00:10:00.300 because C and F are congruent; so CA would be congruent to FD, and so on. 00:10:00.300 --> 00:10:07.800 So then, those are my six congruence statements for my corresponding parts. 00:10:07.800 --> 00:10:15.100 Here is the congruence statement for the congruent triangles. 00:10:15.100 --> 00:10:26.900 And then, here is the symbol to write that the angles are corresponding, just to show correspondence. 00:10:26.900 --> 00:10:38.900 So then, these are corresponding parts; again, you have to make sure that you name them in the order of its congruence. 00:10:38.900 --> 00:10:45.200 If you don't have a diagram--let's say you only have this congruence statement, 00:10:45.200 --> 00:10:50.400 and you don't have a diagram, but you have to name all of its congruent parts; 00:10:50.400 --> 00:10:57.000 then your angles, remember: the first angle, angle A, is going to be congruent to this angle, D; 00:10:57.000 --> 00:11:00.200 angle B is congruent to angle E; angle C is congruent to angle F. 00:11:00.200 --> 00:11:09.000 But then, for the sides, if I say side AC, then that would be congruent to the side DF. 00:11:09.000 --> 00:11:12.000 So it is first/third, then first/third on this one. 00:11:12.000 --> 00:11:18.700 If I said CB (that is third/second), then it has to be third/second on this one, FE. 00:11:18.700 --> 00:11:26.000 So, that is how you can just name the parts of it. 00:11:26.000 --> 00:11:35.900 Definition of congruent triangles: Two triangles are congruent if and only if their corresponding parts are congruent. 00:11:35.900 --> 00:11:47.800 So, here I have two triangles: this can go both ways--"if and only if" means that you could have two conditionals; 00:11:47.800 --> 00:11:53.500 if two triangles are congruent, then their corresponding parts are congruent; that is one way; 00:11:53.500 --> 00:12:00.600 and then the converse would be that, if the corresponding parts are congruent, then the two triangles are congruent. 00:12:00.600 --> 00:12:07.800 So, "if and only if" just means that this statement, as a conditional, is true, and its converse is also true. 00:12:07.800 --> 00:12:14.200 When you switch the "if" and "then," that is also true; that is "if and only if." 00:12:14.200 --> 00:12:20.900 So, if all of these corresponding parts, meaning all six parts of the corresponding parts 00:12:20.900 --> 00:12:26.100 from the one triangle to the other triangle, are congruent, then the two triangles are congruent. 00:12:26.100 --> 00:12:34.000 If you say that the two triangles are congruent, then all of its corresponding parts, all of the six parts, will be congruent. 00:12:34.000 --> 00:12:46.700 So, if triangle GHL is congruent to this (and again, I have to write it so that it is corresponding with this: 00:12:46.700 --> 00:13:12.200 G with M, H with P, and L with Q), then all of its corresponding parts are congruent. 00:13:12.200 --> 00:13:17.500 Or you could say, "If all of the corresponding parts are congruent," if angle G is congruent to angle M, 00:13:17.500 --> 00:13:27.500 and angle H is congruent to angle P, and so on and so on; then this triangle is congruent to this triangle. 00:13:27.500 --> 00:13:30.300 And that is by the definition of congruent triangles. 00:13:30.300 --> 00:13:43.900 Now, another name for this is CPCTC; this is very, very important for you understand in geometry. 00:13:43.900 --> 00:14:02.000 CPCTC means Corresponding Parts of Congruent Triangles are Congruent. 00:14:02.000 --> 00:14:09.400 Try to say that a few times, just so that it will sound familiar. 00:14:09.400 --> 00:14:22.100 Corresponding parts of congruent triangles are congruent; so this is saying this right here, what we just said. 00:14:22.100 --> 00:14:29.600 If the triangles are congruent, if somewhere it is stated--if it is given to you, or you proved it, 00:14:29.600 --> 00:14:41.400 or whatever way you figured out that those two triangles are congruent--from there, then all of the corresponding parts are congruent. 00:14:41.400 --> 00:14:44.800 They are congruent to each other--the corresponding parts of each of these triangles. 00:14:44.800 --> 00:14:56.100 Again, CPCTC says that, if those two triangles are congruent, then corresponding parts are congruent. 00:14:56.100 --> 00:15:00.600 Corresponding parts of congruent triangles--"of congruent triangles" means that it has to be congruent first. 00:15:00.600 --> 00:15:08.800 Triangles have to be congruent first, and then all of the corresponding parts are congruent. 00:15:08.800 --> 00:15:21.200 If I wanted to tell you that these two triangles are congruent, and then I said, "OK, well, angle L is congruent to angle Q," 00:15:21.200 --> 00:15:25.600 what proof do I have--what is my reason behind angle L being congruent to angle Q? 00:15:25.600 --> 00:15:33.700 Well, I can just say "CPCTC," because since these triangles are congruent, then all of their corresponding parts are congruent. 00:15:33.700 --> 00:15:40.800 So, once it is stated that those two triangles are congruent, from there I can say that any of its parts are congruent. 00:15:40.800 --> 00:15:46.300 And then, my reason will be "CPCTC." 00:15:46.300 --> 00:15:55.400 Again, if the triangles are congruent, then all of their corresponding parts are congruent. 00:15:55.400 --> 00:16:03.400 From there, you can say that any part...I can say that HG is congruent to PM, or HL is congruent to PQ, and so on. 00:16:03.400 --> 00:16:10.600 And the reason why I am emphasizing this is because this is something that students make mistakes on all the time. 00:16:10.600 --> 00:16:21.800 Using CPCTC is a little bit hard to understand, especially when you have to use it for proofs. 00:16:21.800 --> 00:16:28.400 Just remember: with CPCTC, triangles have to be congruent first; 00:16:28.400 --> 00:16:39.800 and once they are congruent, then you can say that any of their corresponding parts are congruent. 00:16:39.800 --> 00:16:45.400 OK, triangle congruence: now, we know the reflexive property, symmetric property, and transitive property. 00:16:45.400 --> 00:16:52.100 Now, this is a theorem, so this is actually supposed to be proved. 00:16:52.100 --> 00:17:07.700 But I am just going to explain to you that we can now apply the reflexive property, symmetric property, and transitive property to congruent triangles. 00:17:07.700 --> 00:17:17.500 You know how the reflexive property is when you have triangle ABC congruent to triangle ABC. 00:17:17.500 --> 00:17:30.500 And that is the reflexive property; the symmetric property is that, if triangle ABC is congruent to triangle DEF, 00:17:30.500 --> 00:17:40.900 then triangle DEF is congruent to triangle ABC. 00:17:40.900 --> 00:17:54.300 And then, the transitive property: if triangle ABC is congruent to triangle DEF, and triangle DEF is congruent 00:17:54.300 --> 00:18:14.500 to triangle GHI, then triangle ABC is congruent to triangle GHI. 00:18:14.500 --> 00:18:25.600 So, you can apply the reflexive property, symmetric property, and transitive property to congruent triangles. 00:18:25.600 --> 00:18:33.800 OK, let's go over our examples: Write a congruence statement for each of the corresponding parts of the two triangles. 00:18:33.800 --> 00:18:42.900 "For each of the corresponding parts" means that we have six total. 00:18:42.900 --> 00:18:48.300 Now, I don't have diagrams, so I can't see which sides are congruent to the other sides, 00:18:48.300 --> 00:18:56.000 and so on; so I have to just base it all on this congruence statement right here. 00:18:56.000 --> 00:19:13.100 I know that angle D is congruent to angle X; angle E is congruent to angle Y; and angle F is congruent to angle Z. 00:19:13.100 --> 00:19:26.700 There are three parts; then, DE is congruent to XY; how do I know that?--because DE is here, and XY is there; 00:19:26.700 --> 00:19:45.900 and then, let's see: EF is congruent to...EF is this, so YZ; and DF (that is DF right there-- 00:19:45.900 --> 00:19:59.700 don't think that that is the symbol for a segment) is congruent to (that is D and F right there, so it has to be) X and Z. 00:19:59.700 --> 00:20:08.100 Now, if it is DF, can I say ZX? No, if I am going to say D first, then it has to be "DF is congruent to XZ." 00:20:08.100 --> 00:20:15.300 Now, if this seems like it is really picky, and it is just too much...it is not too bad. 00:20:15.300 --> 00:20:20.900 It is just that you have to write this congruence statement, because you are stating that they are congruent. 00:20:20.900 --> 00:20:28.300 And if you are going to say that they are congruent, then you have to make sure that you are stating that the right parts are congruent. 00:20:28.300 --> 00:20:32.000 It is something that is just necessary. 00:20:32.000 --> 00:20:42.500 DF is congruent to XZ; and that is it--we have all six corresponding parts. 00:20:42.500 --> 00:20:47.200 Let's say that I want to just do one more thing: if I want to rewrite this congruence statement to how I like it-- 00:20:47.200 --> 00:21:00.000 let's say that I don't like that it is in the order DEF--then I can just...let's say I want to do triangle FDE; 00:21:00.000 --> 00:21:08.000 I can rewrite this triangle congruence statement the way I want it, as long as I change the second part, too. 00:21:08.000 --> 00:21:23.100 I can't keep it XYZ: so if I say FDE, then I have to write...what is corresponding with F? Z; triangle Z...what is corresponding to D? X; with E, Y. 00:21:23.100 --> 00:21:27.400 So, I can write it like that, too. 00:21:27.400 --> 00:21:32.300 The next example: Write a congruence statement for the two triangles. 00:21:32.300 --> 00:21:41.800 I am going to say that this is congruent to this, is congruent to this; this is congruent to this. 00:21:41.800 --> 00:21:48.900 And these slash marks: if I have one, all of the segments with one slash mark are congruent; 00:21:48.900 --> 00:21:57.500 all of the segments with two slash marks are congruent; and then, all of the segments with three are going to be congruent. 00:21:57.500 --> 00:22:03.200 Let's say the same thing with this: if I have one mark there, all of the ones with one are congruent; 00:22:03.200 --> 00:22:09.800 all of the ones with the two marks are congruent; and then, all of the ones with the three are congruent. 00:22:09.800 --> 00:22:19.000 So now, I have all six parts of this triangle being congruent to the corresponding parts of the other triangle. 00:22:19.000 --> 00:22:33.000 Now, I can write a congruence statement: Triangle SPT is congruent to triangle...what is corresponding with S? R. 00:22:33.000 --> 00:22:43.500 OK, R is congruent to S; what is corresponding with P?--M; and then, with T is T. 00:22:43.500 --> 00:22:52.700 Now, if you did this same problem on your own, then your congruence statement might be different than mine. 00:22:52.700 --> 00:23:00.900 It is OK, as long as you make sure that these three parts are corresponding to these three parts. 00:23:00.900 --> 00:23:10.000 If you have triangle PTS, then it has to be triangle MTR. 00:23:10.000 --> 00:23:16.000 Draw and label a figure showing the congruent triangles. 00:23:16.000 --> 00:23:35.500 I can just draw triangle...I can just do like that...here is MPO, TSR, and then I want to label and show the congruent triangles. 00:23:35.500 --> 00:23:48.300 MP is congruent to TS; and then, PO is congruent to SR; MO is congruent to TR. 00:23:48.300 --> 00:23:57.300 And then, M is congruent to angle T; P is congruent to angle S; and O is congruent to angle R. 00:23:57.300 --> 00:24:06.100 That would be showing a diagram of this congruence statement. 00:24:06.100 --> 00:24:21.300 Draw two triangles with equal perimeters that are congruent. 00:24:21.300 --> 00:24:34.000 If I said that this is 4, 5, 6; then this also has to be 4, 5, 6, because they are going to be congruent. 00:24:34.000 --> 00:24:40.900 And then, the corresponding angles would be congruent. 00:24:40.900 --> 00:24:43.200 So, in this case, they are going to have the same perimeter, because with perimeter, 00:24:43.200 --> 00:24:51.200 remember: we add up all of the sides, so 4 + 5 + 6 is going to be the same as 4 + 5 + 6. 00:24:51.200 --> 00:24:57.700 So, this would be an example of two triangles with equal perimeters that are congruent, 00:24:57.700 --> 00:25:06.000 because I showed that all three corresponding sides are congruent, and the corresponding angles are congruent. 00:25:06.000 --> 00:25:16.000 Draw two triangles that have equal areas, but are not congruent. 00:25:16.000 --> 00:25:28.200 Let's say I have a triangle here, and say that this is 8, and this is 5. 00:25:28.200 --> 00:25:47.400 Then the area here is going to be 20 units squared, because 8 times 5 is 40, divided by 2 is 20. 00:25:47.400 --> 00:26:00.000 So, that is one triangle; and then, another triangle--let's say the same area; that means, since these add up to 40, 00:26:00.000 --> 00:26:04.100 that this also has to add up to 40, the base and the height. 00:26:04.100 --> 00:26:14.400 Let's say this is 10, and this is 4; and that means that the area is...40 divided by 2 is 20 units squared. 00:26:14.400 --> 00:26:24.000 So again, they have the same area, but they are not congruent, because...see how this side and this side are different. 00:26:24.000 --> 00:26:40.500 And then, I can just say that this is, let's say, 7 and 9; this would be 6 and 5. 00:26:40.500 --> 00:26:46.000 That is it for this lesson; thank you for watching Educator.com.