### Proving Triangles Congruent

- SSS (Side-Side-Side) Postulate: If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent
- SAS (Side-Angle-Side) Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent
- ASA (Angle-Side-Angle) Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent
- AAS (Angle-Angle-Side) Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.
- CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

### Proving Triangles Congruent

If two sides and an angle of one triangle are congruent to two sides and an angle of another triangle, then the two triangles are congruent.

VABC and ∆ DEF, if ―AB ≅ ―DE , ―BC ≅ ―EF , ―AC ≅ ―DF , then ∆ ABC ≅ ∆ DEF

Given: ABC and DBC are right angles, ―AB ≅ ―BD

Prove: ∆ ABC ≅ ∆ DBC

- Statements ; Reasons
- ABC and DBC are right angles ; Given
- mABC = mDBC = 90
^{o}; Definition of right angles - ABC ≅ DBC ; Definition of congruent angles
- ―BC ≅ ―BC ; Reflexive prop ( = )
- ―AB ≅ ―BD ; Given
- ∆ ABC ≅ ∆ DBC ; SAS post.

ABC and DBC are right angles ; Given

mABC = mDBC = 90

^{o}; Definition of right angles

ABC ≅ DBC ; Definition of congruent angles

―BC ≅ ―BC ; Reflexive prop ( = )

―AB ≅ ―BD ; Given

∆ ABC ≅ ∆ DBC ; SAS post.

If two angles and a side of one triangle are congruent to the conrresponding two angles and a side of another triangle, then the two triangles are ____ congruent.

If two sides and an angle of one angle are congruent to the conrresponding two sides and an angle of another triangle, then the two triangles are ____ congruent.

If a triangle has a right angle, then its congruent triangle can have an obtuse angle.

Given: ―AD ||―BC , ―AB ||―CD .

Prove: ∆ ABC ≅ VCDA

- Statements ; Reasons
- ―AD ||―BC ; Given
- ∠CAD ≅ ∠ACB ; alternate interior angles
- ―AB ||―CD ; Given
- ∠DCA ≅ ∠BAC ; alternate interior angles
- ―AC ≅ ―AC; reflexive prop ( = )
- ∆ ABC ≅ VCDA ; ASA post.

―AD ||―BC ; Given

∠CAD ≅ ∠ACB ; alternate interior angles

―AB ||―CD ; Given

∠DCA ≅ ∠BAC ; alternate interior angles

―AC ≅ ―AC; reflexive prop ( = )

∆ ABC ≅ VCDA ; ASA post.

Given: ∠ABC ≅ ∠DCB, ∠BAC ≅ ∠CDB Prove: ∆ ABC ≅ ∆ DCB

- Statements; Reasons
- ∠ABC ≅ ∠DCB, ∠BAC ≅ ∠CDB ; Given
- ―BC ≅ ―BC ; reflexive prop ( = )
- ∆ ABC ≅ ∆ DCB; AAS theorem

∠ABC ≅ ∠DCB, ∠BAC ≅ ∠CDB ; Given

―BC ≅ ―BC ; reflexive prop ( = )

∆ ABC ≅ ∆ DCB; AAS theorem

*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

### Proving Triangles Congruent

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
- SSS Postulate 0:18
- Side-Side-Side Postulate
- SAS Postulate 2:26
- Side-Angle-Side Postulate
- SAS Postulate 3:57
- Proof Example
- ASA Postulate 11:47
- Angle-Side-Angle Postulate
- AAS Theorem 14:13
- Angle-Angle-Side Theorem
- Methods Overview 16:16
- Methods Overview
- SSS
- SAS
- ASA
- AAS
- CPCTC
- Extra Example 1:Proving Triangles are Congruent 21:29
- Extra Example 2: Proof 25:40
- Extra Example 3: Proof 30:41
- Extra Example 4: Proof 38:41

### Geometry Online Course

### Transcription: 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 prove*0047

*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 congruent*0118

*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 angle*0466

*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) FD*0724

*(because they are corresponding in that order), and angle C is congruent to angle D*0743

*(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 ABC*0946

*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 say*1241

*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 up*1824

*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

0 answers

Post by Kristie Ornellas on November 16 at 10:06:33 PM

Can we show a proof example of SSS postulate so I can understand that postulate? With the SAS proof I understood that postulate perfectly. So do you think you can show a proof example somehow please.

0 answers

Post by Mohammed Jaweed on August 12, 2015

0 answers

Post by Shahram Ahmadi N. Emran on July 1, 2013

Why the lectures kept stopping in the middle of the slide which is being taught?

0 answers

Post by jeeyeon lim on January 16, 2013

love your examples!!!!!!

1 answer

Last reply by: Shahram Ahmadi N. Emran

Mon Jul 1, 2013 1:13 PM

Post by Nadarajah Vigneswaran on November 17, 2012

Do the vertical angles have to be in the triangle or can they be exterior. For example on your example of the sas postulate if triangles BEC and AED were non existent could you use the exterior angles formed at point E as proof that angle AEB and angle CED.

0 answers

Post by reid brian on February 15, 2012

hey man! no...^

2 answers

Last reply by: Mary Pyo

Sun Sep 11, 2011 9:14 PM

Post by Sayaka Carpenter on August 22, 2011

for the SAS postulate, in the example you drew, the angle that you used as an example was the top angle, but can it be any of the 3 angles of the triangle? i dont really get the inside triangle part...