Geometry Proving Triangles Congruent

Welcome back to Educator.com.0000

This lesson, we are going to prove triangles congruent.0003

In the previous lesson, remember, we talked about the definition of congruent triangles,0007

and how, if we have two triangles, their corresponding parts are going to be congruent.0013

We are going to take a closer look at some of the methods we can use to prove that two triangles are congruent.0019

The first method to prove that two triangles are congruent...0028

Now, according to the definition of congruent triangles, if all of the corresponding parts of the two triangles are congruent, then the two triangles are congruent.0033

But that is kind of a lot to do; that is a lot of work, because then, you would have to prove0047

that six parts are going to be congruent in order to prove that the triangles are congruent.0052

Instead, we have postulates and theorems that make it easier.0058

The first one is the SSS Postulate, or you could just call it SSS, which stands for Side-Side-Side Postulate.0065

And that is just saying that, if all of the sides of one triangle are congruent to the sides of the second triangle, then the triangles are congruent.0077

This is one method: instead of having to show that all six corresponding parts are congruent,0086

in order to prove that the triangles are congruent, you only have to do the three.0095

So, if AC is congruent to DF, AB is congruent to DE, and BC is congruent to EF,0100

once you have shown that those three sides are congruent to the three sides of the other triangle,0110

then you have proven that the two triangles are congruent.0115

So, the Side-Side-Side Postulate is that, if three sides of one triangle are congruent0118

to three sides of the other triangle, then triangle ABC is congruent to triangle DEF.0123

The next one: now, there are going to be a few different methods--the next method is SAS, which is Side-Angle-Side.0146

Again, as long as you can prove that a side is congruent to a side of the other triangle,0159

an angle is congruent to the angle, and another side; then you can prove that those two triangles are congruent.0167

But there is a condition: the angle, this angle right here, has to be an included angle.0177

"Included" means that the angle has to be in between the two sides--0188

a side, and then the angle, and then the side; side, side, side, angle, angle, and then side, side.0198

It can't be an angle that is one of the two outside angles; it has to be the included angle, between the two sides that you are showing are congruent.0208

That is side-angle-side--very important.0220

If two sides of one triangle are congruent to two sides of the other triangle, and the included angle, then the triangles are congruent.0225

Let's do one proof on the SAS Postulate.0239

Now, this is not actually a proof to prove this postulate; it is just a proof to show that the two triangles are congruent, using the SAS Postulate.0245

Given that E, this point right here, is the midpoint of BD, and then E is the midpoint of AC,0259

I know that if E is the midpoint, then that means that the two segments are congruent.0273

And then, I want to prove that triangle AEB is congruent to triangle CED.0279

I want to prove that this triangle is congruent to this triangle.0288

Whenever you get a proof where you have to prove triangles congruent, you can use any of the methods,0299

depending on what information you have--what is congruent to what--what you can show to be congruent.0306

But this one is going to be SAS, because we are just going to practice using this one.0312

And so far, I have a side of one of the triangles congruent to a side of the other triangle; so I have an S.0318

Then, I have another side congruent to the side of another triangle; so I have the other S.0329

Now, I need an A; but the A, the angle, has to be the included angle, so it has to be in between the two sides.0337

That means that that angle is going to be this angle right here--this angle, and then this angle.0345

But do I have a reason--can I just say that they are congruent?0353

No, the reason would be that they are vertical, and we know that vertical angles are congruent.0357

So, now I have all three parts, and I can go ahead and write out my proof.0363

Let's do a two-column proof, where I have statements and reasons.0370

Statements/reasons: I am just going to write it out like this.0378

#1: The given is always number 1.0383

Now, sometimes, if you look at examples of proofs in your book, the first one might not always be the given.0391

You might have the given as #1, and then maybe #3, or later on in the proof.0397

Sometimes, if you have two different given statements, you don't have to write both for step 1.0403

You can write one of them that you are going to use at a time, that you are going to use first.0411

And then, if you don't need the other given statement until later on in the proof,0417

then you can just wait until that step where you need it and then write it in; and your reason will still be "Given."0420

Or you can just write the whole thing under step 1.0428

E is the midpoint of BD, and E is the midpoint of AC; my reason is "Given."0434

OK, so then, from here, I have to prove each of these; and then I can prove that the triangles are congruent,0457

because the postulate says that if the side with the corresponding side and the angle with the corresponding angle0466

and the side with the side are congruent, then the triangles are congruent.0473

So then, I have to prove all of these first.0477

I can say that, since E is the midpoint of BD, BE is congruent to DE.0482

Make sure that, if you are going to say BE, then you have to say DE, because it has to be corresponding--the way you write it in order.0494

So, if I decide to write it as EB, that is fine; then you would have to write it as ED next.0502

So, BE is congruent to DE; what is the reason?0510

It is "definition of midpoint," because the definition of midpoint says that, if you have a midpoint,0515

then the midpoint will cut the segment into two equal parts.0527

So then, that is why these parts will be congruent--the definition of midpoint.0537

And then, I can say that angle AEB (I am looking at this angle right here for the second one) is congruent to angle CED.0544

And that is because they are vertical, and we know that vertical angles are congruent.0565

So, any time you have vertical angles, you can say that they are congruent; the reason is that vertical angles are congruent.0573

Now, my next step: I have to say that the other sides, these sides now, are congruent.0580

AE, this side of one of the triangles, is congruent to side CE; that is also because of the definition of midpoint.0585

So then, now can I say that the triangles are congruent?0603

This step right here, BE is congruent to DE: see how that is one of the triangle being congruent to one side of the triangle.0607

So, I have a side; then my next step was to show that the angles are congruent--I did that;0615

angle...and then another side of each of the triangles; did I meet all of the conditions?0623

Yes, I showed corresponding sides congruent, corresponding angles congruent, and corresponding sides congruent.0632

Now, I can say that the triangles are congruent: so triangle AEB, this triangle right here, is congruent to triangle CED.0642

And the reason is the SAS Postulate.0663

Again, these are just methods--SAS, SSS, and then we are going to go over a couple more.0674

Those are just methods to prove triangles congruent.0679

Otherwise, if we didn't have these methods, it would be a lot harder, because you would have to prove that each part, all six corresponding parts, are congruent.0686

These actually make it easier; we only have to prove three things congruent, three parts;0697

and then we can say that the triangles are congruent.0701

The next postulate is Angle-Side-Angle (ASA); here is one angle, angle A, congruent to angle F;0707

if angle A is congruent to angle F, and then AC is congruent to (remember, if I am going to say AC, then I have to say) FD0724

(because they are corresponding in that order), and angle C is congruent to angle D0743

(so then, I have: this is an angle; the corresponding angle is congruent; the corresponding side is congruent;0753

and the corresponding angle is congruent), then triangle ABC is congruent to triangle...0762

A is corresponding with F, so triangle F...B with E, and then C with D.0774

So, to prove that these two triangles are congruent, I can use this postulate.0784

Now again, the other thing to mention here is "included side"; if it is ASA, make sure that the side is in between the two angles.0789

So then, if I am going to use angle A and angle F and angle C and angle D, the side has to be in between.0801

So, if I have, let's say, two triangles, and I did it with the angle here and another angle, the angle here, and side--this side here,0809

this is not Angle-Side-Angle, even though I have two angles and I have one side.0821

This would not be Angle-Side-Angle, because the side is not included, meaning it is not in between the two angles.0827

This is actually going to be Angle-Angle-Side; this is not ASA.0837

This one is AAS, Angle-Angle-Side, when we have two angles congruent to the corresponding angles of the second triangle,0855

and then a side congruent to the other side, but not included (meaning it is not in between).0867

See how this is angle, and then angle, and then side; so it goes in that order: Angle-Angle-Side.0875

That is when you can state the Angle-Angle-Side theorem; this is a theorem.0882

Make sure that, if you are proving triangles this way, using this theorem, you state that it is Angle-Angle-Side, and not Angle-Side-Angle.0891

They both have two angles and one side, but if the side is in between the two angles, then it is Angle-Side-Angle;0904

and if the side is not in between--it is not included--then it would be Angle-Angle-Side.0913

If angle A is congruent to angle F, and angle B is congruent to angle E, and BC is congruent to ED, then I can say,0922

because again, this is an angle, and the next is an angle, and the next is a side--then I can say that triangle ABC0946

is congruent to triangle...what is corresponding with A? F; with B, E; and then D.0957

So, once you prove that those three parts are congruent, then you can prove that those two triangles are congruent.0967

So, just to go over the methods again: these four, again, are methods to prove that two triangles are congruent.0977

SSS means Side-Side-Side; if I have two triangles that I want to prove congruent, then this is one method, one way to do it,0994

by showing that all three sides are congruent; so if this, then the triangles are congruent.1012

This is Side-Angle-Side; and that is saying that a side with an included angle, and then a side, like that;1027

the angle has to be in between the two sides; if it is this angle or this angle, then that is not it.1048

And then, if this is true--if I prove that these parts are congruent--then the triangles are congruent.1060

This is Angle-Side-Angle; if you can prove that those three parts are congruent, then you can prove that the triangles are congruent.1072

Angle-Angle-Side: now, when you use these in your proofs, you don't have to actually write out "Side-Side-Side,"1099

"Side-Angle-Side"...you can just write SSS; and these three are postulates, and this one is a theorem.1113

The last one, Angle-Angle-Side, is a theorem; you can just write SSS, though; that should be OK.1121

Or you can just write these.1125

This is Angle-Angle-Side: see how there is a difference between this and this.1128

In this one, the side is in between the two angles; and in this one, the side is not in between the two angles.1142

So, if you prove that these parts are congruent, then you can say that the triangles are congruent.1146

Now, this last one is from the last section, the last lesson: CPCTC is actually not used to prove that triangles are congruent.1155

This one stands for Corresponding Parts of Congruent Triangles are Congruent.1170

That means that the triangles have to be congruent first.1189

So, if I want to prove that...let's say I have this, and I have to prove that maybe this side is congruent to this side;1193

now, if there is no simple way to just prove that those two sides are congruent, then what I can do is:1209

using one of these methods, first prove that the triangles are congruent.1215

So, with just Side-Side-Side, or whatever it is, prove that these two triangles are congruent.1222

Once those triangles are congruent, then I can say that any corresponding parts are congruent.1231

Let's say it is given to me; if it is given to me, whether it is given or whether you proved it, once you say1241

that triangle ABC is congruent to triangle DEF, once this statement is written, then I can say AB is congruent to DE.1250

And the reason would be CPCTC.1268

Corresponding parts of congruent triangles are congruent.1274

These four are used to actually prove that the triangles are congruent; this one is used after the triangles are congruent.1277

So, let's go over our examples: Determine which postulate or theorem can be used to prove that the triangles are congruent.1289

Now, for this one, I have a side; here is the first triangle, and here is the other triangle.1298

Side and side--that is one part that is congruent; and then you have another side.1313

But then, you only have two; you need three; so I know that vertical angles,1321

these angles (even if it is not given to me, just from the fact that they are vertical, I can say that they) are congruent.1325

So then, this one...which one is this going to be?--Side-Side-Angle.1333

This is not Side-Angle-Side, because the congruent angles are not included; it is not in between the two congruent sides.1347

This is not Side-Angle-Side; this would be Side-Side-Angle.1358

Now, there is no such thing as Side-Side-Angle; that doesn't exist.1361

If you spell this backwards, the other way around, then it spells out a bad word.1368

If it spells out a bad word, it doesn't exist; just think of it that way.1374

You cannot use this to prove that the triangles are congruent;1382

there is no postulate or theorem that says that you can prove these, and then the triangles are congruent.1386

In this case, for this problem, we can't prove that they are congruent, because this is not a rule.1392

Now, if I said that angle D is congruent to angle B, then you can say Side-Angle-Side, and that is a rule.1403

So then, you can prove that the triangles are congruent that way.1412

But for this one, this is all that is given--this side with this side, this side with this side, and then vertical angles; so that is not one.1417

The next one: Now, again, I only have a side; I am trying to prove that this triangle is congruent to this triangle.1426

I have a side congruent to a side; I have a side.1436

I have another side congruent to this side; but then I am missing one more thing.1440

So, I have to try to see if there is a given--if there is something here that, even if they didn't give it to me, I know automatically that it is congruent.1448

And what that is, is this side right here.1458

So, if I split up these triangles, it is going to be like this and like that.1461

So, I am trying to prove that these two triangles are congruent; this side is congruent, and this side is congruent.1472

This right here is the same as this right here; they are sharing that side.1480

Since they are sharing that side, it has to be the same; it has to be congruent, automatically.1490

And that is the reflexive property, because AC is going to be congruent to itself.1499

So, AC is the same for this triangle, and it is the same for this triangle.1507

So, in this case, this is Side-Side-Side; even though this side is not given to you, we know that it is still congruent,1510

because it is reflexive; it is equaling itself; this is the same side for this, and it is the same side for this.1520

Automatically, it is congruent; it would be the Side-Side-Side Postulate.1526

OK, we are going to do a few proofs now.1541

So then, let's take a look at our given: angle A is congruent to angle E.1546

I am just going to do that; C is the midpoint of AE; that means that, if this is the midpoint, then these two parts are congruent.1555

That is given; and then I have to prove that this triangle, triangle ABC, is congruent to triangle EDC.1570

Since I am proving that two triangles are congruent, I have to use one of the methods: SSS, SAS, ASA, or AAS--one of those four methods.1577

Since I only have two parts--I have an angle, and I have a side--I need to look at this and see if anything else is given.1590

And I see that this angle right here is congruent to this angle right here, automatically, because they are vertical.1599

So, statements and reasons, right here: the #1 statement is going to be that angle A is congruent to angle E,1608

and that C is the midpoint of AE; the reason is "Given."1631

Now, since I know that my destination, my last step, my point B, is going to be that these two triangles are congruent,1644

do I know already what method I am going to be using to prove that those two triangles are congruent?1655

I have an angle, a side, and an angle, so I know that I am going to be using the ASA Postulate.1660

Now, I just have to state out each of these; I have to state out the angle; I have to state out the sides; and then I have to state the angles.1669

And then, once all three of these are stated, then I can say that these two triangles are congruent.1678

The next step: now, see how angle A is congruent to angle E--that angle is already stated, so that here, this is an angle.1688

Since C is the midpoint of AE, I am going to say that AC is congruent to EC.1700

The reason for that is "definition of midpoint."1709

And then, that is my side; and then, I am missing an angle, which is this right here.1722

Now, I can't say angle C, because angle C represents so many different angles.1736

So, I have to say angle ACB, or BCA; so ACB is congruent to...and if I am going to say angle ACB, then I have to say angle ECD,1742

because A and D are corresponding, because they are congruent.1755

ECD--see how that is another angle; what is my reason for that?--that vertical angles are congruent.1761

Now, see how I stated all of them now: I stated my angle; I stated my sides; and I stated my angle.1780

So then, I can say that triangle ABC is congruent to triangle EDC.1786

And what is my reason--what method did I use?--Angle-Side-Angle Postulate.1799

You are going to do a few of these kinds of proofs, where you are proving the triangles using one of the methods.1813

Always look at what is given; look at your diagram.1820

Your diagram is going to be very valuable when it comes to proofs, because you want to mark it up1824

and see what you have, what you have to work with, what more you have to do, and how you are going to get to this step right here.1828

It is like your map.1839

BD bisects AE; now, remember "bisector": when something bisects something else, it is cutting it in half.1843

So, if BD is bisecting AE, that means that BD is cutting AE in half--think of it that way--it is cutting it in half.1858

Don't get confused by what is cut in half, which one is going to be cut in half.1869

Is BD cut in half, or is AE cut in half?1879

Whichever one is doing the bisecting is the one that is doing the cutting.1884

Since BD is bisecting AE, BD is cutting AE in half; if BD is bisecting AE, that means that BD cut AE in half, so AE is cut in half.1888

And then, what else is given? Angle B is congruent to angle D.1907

I know that angle 1 is congruent to angle 2, because they are vertical angles; I am just going to mark that.1919

So, how would I be able to prove that these two triangles are congruent?1925

Angle-Angle-Side: it is not Angle-Side-Angle; it is Angle-Angle-Side.1929

Now, but then, look at my "prove" statement; it is not asking me to prove that the triangles are congruent.1939

It is asking me to prove that AB, this side, is congruent to this side.1943

But is there any way for me to be able to prove that those two sides are congruent?1951

I don't see a way to say that these two sides are congruent; there is nothing here that allows me to prove that AB is congruent to ED, except for CPCTC.1956

Remember: it is a random side that I have to prove congruent; whenever it is just a random side,1973

a random part, that you have to prove congruent, then you have to first prove that the triangles are congruent,1983

which we can do by Angle-Angle-Side; and then, once the triangles are congruent, you can say that corresponding parts are congruent.1991

So then, once the triangles are congruent, I can say that AB is congruent to ED,2002

because corresponding parts of congruent triangles are congruent.2008

I know (wrong one!)...statements/reasons...1: BD bisects AE; angle B is congruent to angle D; that is given.2013

OK, so one of the parts is already stated out: angle B is congruent to angle D--that is an angle.2048

Now, if you are going to do this like how I am doing it, how you are writing out what you are showing on the side,2058

that is good; but just be careful when it comes to your included angle or your included side,2066

because if you are not doing the right order--let's say I mention the angle--see how this angle is mentioned first,2072

and then maybe the next step--what if I mention the sides?2079

Then, be careful so that it is not going to be in that order.2084

I will show you when we get to that.2089

Here, the next step: I am going to say that AC is congruent to...now, if I am going to say AC,2094

I can't say CE; I have to say EC; remember corresponding parts--with AC, what is congruent to A?2103

E is, so then, if I say AC, then I have to say EC.2112

What is the reason--why are those two sides congruent?--because this is "definition of segment bisector."2118

If it was an angle that was bisected, then it would be "angle bisector."2133

But since this is a segment, it is "segment bisector."2139

OK, so then, this one is the side that is mentioned; and then, angle 1 is congruent to angle 2.2144

What is the reason for that?--"vertical angles are congruent," so that is my angle.2157

I have three parts mentioned, so now I can say that my triangles are congruent; triangle ABC is congruent to...2169

A is corresponding to E, so it has to be triangle E...what is corresponding with B?--D; C.2183

What is the reason? Now, ASA is one method, but we didn't use ASA--we used Angle-Angle-Side.2194

Now, that is what I was talking about earlier.2210

Be careful, because I didn't mention it in the order of AAS; I mentioned an angle, and then I mentioned the side, and then I mentioned the angle.2213

It is OK if this is out of order, but just be careful if you are going to write it out on the side like this, like how I am doing.2226

Then, you don't put it in that order, ASA; you have to look at the diagram and see what the order is.2233

It is Angle-Angle-Side--it is not Angle-Side-Angle.2239

I just mentioned it in this order, but it is not the actual order of the diagram.2242

Just be careful with that; it is OK to list them out like this, but then, when it comes to the order,2249

look back and say, "Is it ASA? No, it is AAS."2254

And then, I am done with the proof, right?2262

The whole point of proving these two triangles congruent is so that I can prove that parts of the triangles are congruent.2267

So then, now that it is stated that the triangles are congruent, I can now state any of the corresponding parts congruent.2279

Now, I can say that AB is congruent to ED, because these are parts of these congruent triangles.2291

What is my reason?--"corresponding parts of congruent triangles are congruent."2301

The corresponding parts are congruent, as long as they are from congruent triangles.2310

OK, we are going to do one more proof on this.2316

Let's see what we have: AB is parallel to DC, and then, AD is parallel to BC.2328

Now, just like those slash marks, I have to write this out twice.2340

And then, I want to prove that angle A is congruent to angle C.2349

Is there any way that I can prove that those two angles are congruent?2354

No, I don't think that there is anything; how would you prove that those two angles are congruent?2363

Well, then, can I do it in two steps, where I can prove that these two triangles are congruent,2371

and then use CPCTC to say that these parts are congruent?2379

Let's say if I can prove the triangles congruent: well, if these two are parallel (remember parallel lines?),2385

here is my transversal; see, extending it out makes it easier to see.2393

Then, alternate interior angles, that angle with this angle of this triangle, are going to be congruent.2400

And then, since these two lines are parallel, the same thing here: angle 1 is going to be congruent to angle 4.2407

If you want to see that again, these are the two parallel lines; this is AB, and this is DC.2417

Here is the transversal; this is 3, and this is 2; see if they are parallel--then the alternate interior angles are going to be congruent.2425

The same thing is going this way: my transversal...here is angle 4; here is angle 1;2438

as long as they are parallel, then these two alternate interior angles are congruent.2447

So then, I have two angles; I have Angle-Angle, but then I need one more; I need three.2453

Remember: earlier, we looked at a diagram similar to this, where we have two triangles, and they share a side.2461

If they share a side, then automatically, I can say that this side to this triangle is congruent to this side to this triangle.2471

That is another one of those automatic things: you have vertical angles that are automatically congruent, and you have a shared side that is automatically congruent.2479

Now I have three parts: I have an angle; I have a side; and I have angles.2491

Now, in order to prove that this angle is congruent to this angle, I can first say that this whole triangle is congruent to this whole triangle.2501

And then, these parts of those congruent triangles are going to be congruent.2512

Statements/reasons: 1: AB is parallel to DC, and AD is parallel to BC; "Given."2521

Step 2: Angle 1 is congruent to angle 4; and then, my reason for angle 1 being congruent to angle 4,2544

and angle 2 being congruent to angle 3, is going to be the same.2562

My reason is going to be the same, so I can just include it in the same step.2566

I don't have to separate it: angle 2 is congruent to angle 3.2571

And then, both of those are going to be...you can say "alternate interior angles theorem,"2577

or you can just write it out: "If lines are parallel, then alternate interior angles are congruent"--you could just write it like that.2589

Step 3: What do I have so far? I have my angle listed, an angle stated, and another angle stated; and now I have to state my side.2607

DB is congruent to BD; now, notice how I didn't write BD and BD; I wrote DB, and then I wrote BD.2620

If I draw this out again, if I separate out the two triangles, this is D, and this B; this is D, and this is B.2638

This angle right here is actually corresponding with this angle right here.2656

See how, if I flip it around, this angle and this angle are congruent; this angle and this angle are congruent,2661

because this is angle 1, and then, this is angle 4; and then, we know that angle 1 and angle 4 are congruent.2678

So then, this and this are corresponding; so then, I have to say B and D.2685

So, if it is DB, then I have to say BD; does that make sense?2692

Here is my side that I am sharing; that is this side right here.2698

Since angle 1 is congruent to angle 4 here, this angle and this angle are corresponding parts.2705

So, if I say DB, then I have to say BD, because this is corresponding to this, and this one is corresponding to this one.2715

This one is congruent to this.2723

Even though the letters are the same, DB and DB here, look at the angles: this one is corresponding to this angle,2727

so then, if you mention D here first, you have to mention B first for the next one.2736

Step 3: This is the reflexive property--any time something equals itself, this side equals the same side, then it is the reflexive property.2744

Then, did I say all that I needed to say?--yes, so now I can say, since I have all three parts, that the triangles are congruent.2761

Triangle ABD is congruent to triangle...what is corresponding with A? C; what is corresponding to B?--remember the angle, D; B.2775

What is my reason? Is my reason Angle-Angle-Side?2793

No, I have to look at the diagram: I used the Angle-Side-Angle Postulate.2796

I used the angle, an angle, and a side, but not in that order; it is in this order.2806

But that is not it; the whole point wasn't to just prove the triangles congruent.2812

The whole point was to prove them congruent so that these parts would be congruent.2819

So, angle A is congruent to angle C, and the reason is CPCTC.2826

Now, remember again: if I want to use this CPCTC rule, first the triangles must be congruent.2836

So, here this has to be stated somewhere before you use CPCTC.2844

And once it is stated, then you can use it, saying that any corresponding parts will be congruent.2851

That is it for this lesson; we will do a little more of this.2860

We are going to go over more triangle stuff in the next lesson, so we will see you then.2865

Thank you for watching Educator.com.2870