### Exploring Congruent Triangles

- Two triangles are congruent if they have the same size and shape
- Corresponding angles and sides of triangles must be named in the same order
- Definition of Congruent Triangles: Two triangles are congruent if and only if their corresponding parts are congruent
- Congruence of triangles is reflexive, symmetric, and transitive

### Exploring Congruent Triangles

If three angles in a triangle are congruent with three angles in another triangle, then the two triangles are congruent.

If ∆ ABC ≅ ∆ DEF, then ∆ ABC ≅ ∆ EFD.

If ∆ ABC ≅ ∆ OPQ, and ∆ OPQ ≅ ∆ GEF, then VABC and VGEF are ____ congruent.

If ∆ JFK ≅ VONT, then ∆ ONT ≅ ∆ JFK.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Exploring Congruent Triangles

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Congruent Triangles 0:15
- Example of Congruent Triangles
- Corresponding Parts 3:39
- Corresponding Angles and Sides of Triangles
- Definition of Congruent Triangles 11:24
- Definition of Congruent Triangles
- Triangle Congruence 16:37
- Congruence of Triangles
- Extra Example 1: Congruence Statement 18:24
- Extra Example 2: Congruence Statement 21:26
- Extra Example 3: Draw and Label the Figure 23:09
- Extra Example 4: Drawing Triangles 24:04

### Geometry Online Course

### Transcription: Exploring Congruent Triangles

*Welcome back to Educator.com.*0000

*The next lesson, we are going to explore congruent triangles.*0002

*We are going to be looking at triangle parts, and then we are going to compare two triangles together*0005

*to see if we can see that they are congruent.*0012

*In order for triangles to be congruent, they have to have the same size and shape,*0018

*because if two triangles just have the same shape, then one can be this small, and another one can be this big.*0030

*They have the same shape, but they are not the same size.*0043

*They have to be the same size and shape.*0046

*Here we have two triangles, triangle ABC and (it is congruent to) triangle DEF.*0052

*Now, if you want to state triangles, remember: you can just write a little triangle symbol in front of that.*0057

*And so, that is how you write "triangle ABC" in symbols.*0062

*Triangle ABC is congruent to triangle DEF; these two triangles are congruent.*0066

*Now, in order for two triangles to be congruent, they don't have to just look and be in the same upright position.*0071

*Here there are three different ways that triangle DEF is shown.*0084

*And even if you move it around, and you flip it, and you do all of these things to it,*0092

*which are called congruence transformations, you are still going to get a congruent triangle.*0097

*It is still going to be the same thing; so from this triangle, DEF, to this triangle, this is when it just moves.*0102

*And when it just moves to a different position, that is called sliding; so this is sliding.*0116

*Nothing changes about it, but it just moves.*0126

*The next one, right here: see how it looks like it is turned a little bit.*0132

*That is kind of doing one of these; and that is rotating; this one is rotated.*0141

*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.*0152

*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.*0165

*You just flip it; so this one is kind of like that--you are flipping it.*0175

*There are three congruence transformations: slide, rotate, and flip.*0187

*And no matter what you do to it, it is always going to be congruent to triangle ABC.*0191

*This triangle, DEF, is still congruent to this triangle, ABC.*0195

*So that means that, if I have triangle ABC like this, and triangle DEF like this, they are still congruent,*0200

*even though they are not in the same upright position, like these two.*0211

*These are congruence transformations.*0215

*Now, corresponding parts of triangles have to do with their angles; you are comparing the angles of one triangle*0220

*to the angles of another triangle, or the sides of one triangle to the sides of the other triangle.*0236

*So, the parts of a triangle, we know, are angles and sides; and you are saying that,*0239

*when it is corresponding...remember corresponding angles?--if we have two parallel lines and a transversal,*0245

*remember how the corresponding angles were the angles in the same position?*0252

*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...*0257

*let me just draw it out for you: remember: this angle right here (if these lines are parallel) and this angle were corresponding angles.*0267

*This angle is the top right, and this angle is also the top right; so they have the same position.*0281

*It kind of means the same thing here, too.*0288

*Again, we have that triangle ABC is congruent to triangle DEF.*0292

*Now, remember: we have six parts total: three angles and three sides--six parts total.*0295

*That means that, if I have two congruent triangles, this triangle congruent to this triangle, then all of its parts...*0301

*see how, for this triangle, A is named first; it doesn't have to be named first, but it is.*0317

*Then, this D is named first for this triangle; then angle A is congruent to angle D.*0324

*This angle is corresponding to angle D.*0332

*So, to write "corresponding," I can draw an arrow like that, and that means "corresponding"; "corresponding" is like that.*0337

*Now, B is second, and E is second; that means automatically that this angle and this angle right here are corresponding.*0348

*B is corresponding to E; and then, the last angle that is mentioned, C, is corresponding to angle F; C is corresponding to F.*0357

*That means that angle C is congruent to angle F.*0371

*So then, this just shows that they are corresponding; I can also write the same thing, showing that they are congruent.*0379

*I know that, because we know that the triangles are congruent, all of their parts are congruent.*0386

*So, angle A is congruent to angle D; angle B is congruent to angle E; and angle C is congruent to angle F.*0391

*Now, each one of these letters represents an angle.*0408

*And if you put two of them together, they represent sides.*0416

*Whatever is written first is going to be congruent to this one that is written first.*0422

*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.*0428

*Congruent parts have to be named by that same order.*0436

*Now, if I were to say triangle BCA, that is OK; I can name this triangle however I want.*0440

*I can name it triangle ABC; I can name it triangle BCA; I can name it triangle BAC, triangle CAB, CBA...whatever.*0452

*I can name this triangle however I want, but if I have two congruent triangles,*0463

*and I am going to write a congruence statement, then whatever I write next is going to depend on what I wrote first.*0468

*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.*0484

*You can write it like this, or you can write it like this.*0499

*If you decide to name this triangle CAB, then you would have to say that it is congruent to triangle FDE.*0503

*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.*0512

*So, again, here are the corresponding angles; and then, for the sides, AB is congruent to DE.*0523

*And you can also look at that in this way, too: AB is congruent to DE; BC is congruent to EF;*0540

*now, if I said BC, then I can't say FE, because again, remember: B is congruent to E;*0555

*so if you are going to say BC, then the same thing applies for this.*0567

*You can't say FE; you have to say EF, because B and E are corresponding.*0572

*It has to be in the same order.*0578

*And then, the next one: AC is congruent to DF; if you said CA, then you would have to say FD,*0581

*because C and F are congruent; so CA would be congruent to FD, and so on.*0594

*So then, those are my six congruence statements for my corresponding parts.*0600

*Here is the congruence statement for the congruent triangles.*0608

*And then, here is the symbol to write that the angles are corresponding, just to show correspondence.*0615

*So then, these are corresponding parts; again, you have to make sure that you name them in the order of its congruence.*0627

*If you don't have a diagram--let's say you only have this congruence statement,*0639

*and you don't have a diagram, but you have to name all of its congruent parts;*0645

*then your angles, remember: the first angle, angle A, is going to be congruent to this angle, D;*0650

*angle B is congruent to angle E; angle C is congruent to angle F.*0657

*But then, for the sides, if I say side AC, then that would be congruent to the side DF.*0660

*So it is first/third, then first/third on this one.*0669

*If I said CB (that is third/second), then it has to be third/second on this one, FE.*0672

*So, that is how you can just name the parts of it.*0679

*Definition of congruent triangles: Two triangles are congruent if and only if their corresponding parts are congruent.*0686

*So, here I have two triangles: this can go both ways--"if and only if" means that you could have two conditionals;*0696

*if two triangles are congruent, then their corresponding parts are congruent; that is one way;*0708

*and then the converse would be that, if the corresponding parts are congruent, then the two triangles are congruent.*0713

*So, "if and only if" just means that this statement, as a conditional, is true, and its converse is also true.*0720

*When you switch the "if" and "then," that is also true; that is "if and only if."*0728

*So, if all of these corresponding parts, meaning all six parts of the corresponding parts*0734

*from the one triangle to the other triangle, are congruent, then the two triangles are congruent.*0741

*If you say that the two triangles are congruent, then all of its corresponding parts, all of the six parts, will be congruent.*0746

*So, if triangle GHL is congruent to this (and again, I have to write it so that it is corresponding with this:*0754

*G with M, H with P, and L with Q), then all of its corresponding parts are congruent.*0767

*Or you could say, "If all of the corresponding parts are congruent," if angle G is congruent to angle M,*0792

*and angle H is congruent to angle P, and so on and so on; then this triangle is congruent to this triangle.*0797

*And that is by the definition of congruent triangles.*0807

*Now, another name for this is CPCTC; this is very, very important for you understand in geometry.*0810

*CPCTC means Corresponding Parts of Congruent Triangles are Congruent.*0824

*Try to say that a few times, just so that it will sound familiar.*0842

*Corresponding parts of congruent triangles are congruent; so this is saying this right here, what we just said.*0849

*If the triangles are congruent, if somewhere it is stated--if it is given to you, or you proved it,*0862

*or whatever way you figured out that those two triangles are congruent--from there, then all of the corresponding parts are congruent.*0869

*They are congruent to each other--the corresponding parts of each of these triangles.*0881

*Again, CPCTC says that, if those two triangles are congruent, then corresponding parts are congruent.*0885

*Corresponding parts of congruent triangles--"of congruent triangles" means that it has to be congruent first.*0896

*Triangles have to be congruent first, and then all of the corresponding parts are congruent.*0900

*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,"*0909

*what proof do I have--what is my reason behind angle L being congruent to angle Q?*0921

*Well, I can just say "CPCTC," because since these triangles are congruent, then all of their corresponding parts are congruent.*0925

*So, once it is stated that those two triangles are congruent, from there I can say that any of its parts are congruent.*0934

*And then, my reason will be "CPCTC."*0941

*Again, if the triangles are congruent, then all of their corresponding parts are congruent.*0946

*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.*0955

*And the reason why I am emphasizing this is because this is something that students make mistakes on all the time.*0963

*Using CPCTC is a little bit hard to understand, especially when you have to use it for proofs.*0970

*Just remember: with CPCTC, triangles have to be congruent first;*0982

*and once they are congruent, then you can say that any of their corresponding parts are congruent.*0988

*OK, triangle congruence: now, we know the reflexive property, symmetric property, and transitive property.*1000

*Now, this is a theorem, so this is actually supposed to be proved.*1005

*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.*1012

*You know how the reflexive property is when you have triangle ABC congruent to triangle ABC.*1028

*And that is the reflexive property; the symmetric property is that, if triangle ABC is congruent to triangle DEF,*1037

*then triangle DEF is congruent to triangle ABC.*1050

*And then, the transitive property: if triangle ABC is congruent to triangle DEF, and triangle DEF is congruent*1061

*to triangle GHI, then triangle ABC is congruent to triangle GHI.*1074

*So, you can apply the reflexive property, symmetric property, and transitive property to congruent triangles.*1094

*OK, let's go over our examples: Write a congruence statement for each of the corresponding parts of the two triangles.*1105

*"For each of the corresponding parts" means that we have six total.*1114

*Now, I don't have diagrams, so I can't see which sides are congruent to the other sides,*1123

*and so on; so I have to just base it all on this congruence statement right here.*1128

*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.*1136

*There are three parts; then, DE is congruent to XY; how do I know that?--because DE is here, and XY is there;*1153

*and then, let's see: EF is congruent to...EF is this, so YZ; and DF (that is DF right there--*1167

*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.*1186

*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."*1200

*Now, if this seems like it is really picky, and it is just too much...it is not too bad.*1208

*It is just that you have to write this congruence statement, because you are stating that they are congruent.*1215

*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.*1221

*It is something that is just necessary.*1228

*DF is congruent to XZ; and that is it--we have all six corresponding parts.*1232

*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--*1242

*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;*1247

*I can rewrite this triangle congruence statement the way I want it, as long as I change the second part, too.*1260

*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.*1268

*So, I can write it like that, too.*1283

*The next example: Write a congruence statement for the two triangles.*1287

*I am going to say that this is congruent to this, is congruent to this; this is congruent to this.*1292

*And these slash marks: if I have one, all of the segments with one slash mark are congruent;*1302

*all of the segments with two slash marks are congruent; and then, all of the segments with three are going to be congruent.*1309

*Let's say the same thing with this: if I have one mark there, all of the ones with one are congruent;*1317

*all of the ones with the two marks are congruent; and then, all of the ones with the three are congruent.*1323

*So now, I have all six parts of this triangle being congruent to the corresponding parts of the other triangle.*1330

*Now, I can write a congruence statement: Triangle SPT is congruent to triangle...what is corresponding with S? R.*1339

*OK, R is congruent to S; what is corresponding with P?--M; and then, with T is T.*1353

*Now, if you did this same problem on your own, then your congruence statement might be different than mine.*1363

*It is OK, as long as you make sure that these three parts are corresponding to these three parts.*1373

*If you have triangle PTS, then it has to be triangle MTR.*1381

*Draw and label a figure showing the congruent triangles.*1390

*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.*1396

*MP is congruent to TS; and then, PO is congruent to SR; MO is congruent to TR.*1415

*And then, M is congruent to angle T; P is congruent to angle S; and O is congruent to angle R.*1428

*That would be showing a diagram of this congruence statement.*1437

*Draw two triangles with equal perimeters that are congruent.*1446

*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.*1461

*And then, the corresponding angles would be congruent.*1474

*So, in this case, they are going to have the same perimeter, because with perimeter,*1481

*remember: we add up all of the sides, so 4 + 5 + 6 is going to be the same as 4 + 5 + 6.*1483

*So, this would be an example of two triangles with equal perimeters that are congruent,*1491

*because I showed that all three corresponding sides are congruent, and the corresponding angles are congruent.*1498

*Draw two triangles that have equal areas, but are not congruent.*1506

*Let's say I have a triangle here, and say that this is 8, and this is 5.*1516

*Then the area here is going to be 20 units squared, because 8 times 5 is 40, divided by 2 is 20.*1528

*So, that is one triangle; and then, another triangle--let's say the same area; that means, since these add up to 40,*1547

*that this also has to add up to 40, the base and the height.*1560

*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.*1564

*So again, they have the same area, but they are not congruent, because...see how this side and this side are different.*1574

*And then, I can just say that this is, let's say, 7 and 9; this would be 6 and 5.*1584

*That is it for this lesson; thank you for watching Educator.com.*1600

0 answers

Post by Delores Sapp on June 12, 2014

WXYZ is congruent to JKLM. Name the congruent sides from the congruent statement. Will you explain ?

1 answer

Last reply by: Mary Pyo

Sat Feb 4, 2012 12:37 AM

Post by cinzia zullian on December 28, 2011

I love the way that you teach us, i am foreign exchange student, and my English is not very good, but i understand everything and now i like geometry. Thank you