Geometry Exploring Congruent Triangles

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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 together0005

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 triangle0220

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 parts0734

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 congruent1061

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