### Right Triangles

- LL (Leg-Leg) Theorem: If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent
- HA (Hypotenuse-Angle) Theorem: If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent
- LA (Leg-Angle) Theorem: If one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent
- HL (Hypotenuse-Leg) Postulate: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent

### Right Triangles

If two sides of one right triangle are congruent to the corresponding two sides of another right triangle, then the two triangles are congruent.

fill in the blank of the statement with always, sometimes or never.

Right ∆ABC and right ∆DEF, if―AB ≅ ―DE , ―AC ≅ ―DF , then ∆ABC is _____ congruent to ∆DEF.

- ―CD ≅ ―BD
- 3x + 5 = 2x + 3

If the two acute angles of one right triangle are congruent to the corresponding angles in another right triangle, then the two triangles are congruent.

right ∆ABC and right ∆DEF, ―AC = 2x + 6, ―DF = 12, ―DE = 8, ―AB = 3y + 2, find the ∆Alue of x and y so that the two triangles are congruent.

- ―AC ≅ ―DF
- 2x + 6 = 12
- x = 3
- ―DE ≅ ―AB
- 8 = 3y + 2

Given: Isosceles triangle ABC, ―AB ≅ ―AC , ∆ABD and ∆ACD are right triangles.

Prove: ∆ABD ≅ ∆ACD

- Statements; Reasons
- Isosceles triangle ABC, ―AB ≅ ―AC; Given
- ∠ABD ≅ ∠ACD; Isosceles triangle theorem
- ―AB ≅ ―AC; Given
- ∆ABD and ∆ACD are right triangles; Given
- ∆ABD ≅ ∆ACD; HA theorem

Isosceles triangle ABC, ―AB ≅ ―AC; Given

∠ABD ≅ ∠ACD; Isosceles triangle theorem

―AB ≅ ―AC; Given

∆ABD and ∆ACD are right triangles; Given

∆ABD ≅ ∆ACD; HA theorem

Given: Right triangles ABC and DCB, ―BE ≅ ―CE

Prove: ∆ABC ≅ ∆DCB

- Statements; Reasons
- ―BE ≅ ―CE; Given
- ∠ACB ≅ ∠DBC; isosceles triangle theorem
- ―BC ≅ ―BC; reflexive prop of eq.
- Right triangles ABC and DCB; Given
- ∆ABC ≅ ∆DCB; LA Theorem

―BE ≅ ―CE; Given

∠ACB ≅ ∠DBC; isosceles triangle theorem

―BC ≅ ―BC; reflexive prop of eq.

Right triangles ABC and DCB; Given

∆ABC ≅ ∆DCB; LA 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

### Right 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
- LL Theorem 0:21
- Leg-Leg Theorem
- HA Theorem 2:23
- Hypotenuse-Angle Theorem
- LA Theorem 4:49
- Leg-Angle Theorem
- LA Theorem 6:18
- Example: Find x and y
- HL Postulate 8:22
- Hypotenuse-Leg Postulate
- Extra Example 1: LA Theorem & HL Postulate 10:57
- Extra Example 2: Find x So That Each Pair of Triangles is Congruent 14:15
- Extra Example 3: Two-column Proof 17:02
- Extra Example 4: Two-column Proof 21:01

### Geometry Online Course

### Transcription: Right Triangles

*Welcome back to Educator.com.*0000

*This next lesson is on right triangles; we are actually going to prove right triangles congruent.*0002

*Just like we did a few sections ago, where we used SAS, ASA, AAS, and SSS theorems to prove that two triangles are congruent,*0007

*these theorems and postulates are going to be used specifically just for right triangles.*0019

*The first one is LL theorem: LL theorem is actually Leg-Leg.*0027

*Leg-Leg theorem is when the two legs of one triangle are congruent to two legs of another triangle.*0034

*Again, those other theorems and postulates--SAS, ASA...let me just write them down...AAS, and SSS...these were used*0042

*to prove that two triangles were congruent--just two non-right triangles.*0056

*These work the same way as these, but these theorems are just only for right triangles, proving right triangles congruent.*0062

*These are just to prove triangles congruent.*0071

*So again, the Leg-Leg theorem: If the legs of one triangle (that is AB and BC) are congruent*0074

*to the two legs of the other triangle (and of course, they have to be corresponding: DE and EF), then the triangles are congruent.*0083

*If I mark it like this, if BC is congruent to EF, and AB is congruent to DE, then these triangles are congruent by the LL theorem.*0093

*Now, even though we use the LL theorem because they are right triangles, if you look at this, this is the same thing*0105

*as side, and then angle (because we know that right angles are all congruent, so then angle), side.*0114

*So, even though this is LL, this can also be considered SAS; we just call it LL because they are right angles.*0122

*So again, this is the same as the SAS theorem, but again, it is just because they are right triangles; we just call them LL.*0131

*The LL is the first one; and the next one is HA...let me just write LL so that we keep track of the different theorems*0142

*that we are going to go over, and that we went over...the next one is Hypotenuse-Angle.*0151

*Now, we know that the hypotenuse is the side of the right triangle opposite the right angle.*0157

*This is the right angle; we know that these two are called the legs; and the other side, the long side, the side opposite the right angle, is the hypotenuse.*0165

*So, for the HA theorem, if you want to prove that two right triangles are congruent by the HA theorem,*0176

*we have to prove that the hypotenuses, which are AC and DF, are congruent, and an angle.*0184

*Now, an angle, of course, has to be an angle that is not the right angle.*0195

*So, it is either A and D, because they are corresponding, or angles C and F--it doesn't matter which one.*0200

*I can say that those two angles work; as long as we know that the hypotenuse and an angle are congruent, then these two right triangles are congruent.*0209

*Now, just like the LL, the HA is the same thing as...we have an angle, the next angle, and the next side.*0225

*So, this is the same thing as Angle-Angle-Side; but again, because these are right triangles, you use HA.*0239

*Now, you don't have to use HA, but the reason why we use HA instead of AAS for right triangles is because this is less work.*0251

*In this case, you only have to prove that these two are congruent, in order to prove that these two triangles are congruent,*0260

*whereas, with AAS, you have to prove all three; so it is less work--that is why HA would be the one that you would use for right triangles,*0267

*because you don't have to prove that these right angles are congruent.*0277

*That eliminates one of the angles; so again, it is the HA theorem.*0280

*LL, HA...that was the first one; that was the second one; the next one was the LA theorem.*0286

*L is for leg; A is for angle; again, this is only for right triangles, to prove that they are congruent.*0298

*A leg can be like this one or this; it could be these two legs or these two legs, as long as they are corresponding.*0307

*And there is an acute angle like this, and then an angle...so then LA would be the same thing as Angle-Side-Angle.*0318

*Now, it can be whichever angle; let's say I choose the other angle, or I only have the other angle to work with.*0341

*Then, this is also the same thing as Side-Angle-Angle, or AAS.*0352

*This LA can be the same thing as ASA or AAS; that is LA, a leg and an angle.*0364

*Let's use the Leg-Angle theorem to find the values of x and y.*0382

*Here we have that, let's see...here are the right angles; these two sides are the corresponding sides;*0388

*now, it is not good to assume which sides are corresponding.*0404

*Usually, there will be some kind of indicator to tell you which sides and which angles are corresponding.*0408

*But for this problem, I will tell you that this leg is corresponding to this leg.*0413

*Since they are corresponding, I want to use the LA theorem so that the triangles are congruent.*0421

*Find x and y: that means this 3y - 4 would be congruent to 20, so then you just make that 3y - 4 equal to 20.*0430

*3y...I am going to add 4...equals 24; so y equals 8.*0442

*And then, for x, again (let me just mark these, and then those angles), 2x + 30, that angle, is congruent to 62, that angle, so it has the same measure.*0450

*2x + 30 is equal to 62; so to solve it, subtract 30; 2x = 32, and then x = 16 (divide the 2).*0467

*y is 8, and x is 16; and if x is this value, and y is that value, then these sides will be congruent, and the angles will be congruent;*0486

*and therefore, that is using the Leg-Angle theorem.*0497

*So then, we have another one; first we did HA, then LA and LL, those three; and now we have HL.*0504

*HL, HA, LA, LL: two of them start with the hypotenuse, and two of them start with the leg.*0520

*If the hypotenuse and a leg (it could be either this leg or this leg; it doesn't matter)--any leg,*0530

*so let's just say that is the leg--are congruent, then the triangles are congruent.*0543

*Now, this postulate is the only postulate that is different than a few sections ago when we used SAS, ASA, and all of those theorems.*0552

*This is the only theorem that is different; all of the rest of the other ones were the same.*0563

*They were the same theorems; it is just that they eliminate one of the angles so that it is not as much work,*0571

*because the right angles, we know, are congruent; so then you don't have to prove that part.*0579

*So, they just gave new names for them; this one, however, HL, is the only one that does not apply to both regular triangles and right triangles.*0583

*If you look at this, this is the same thing as Side-Side-Angle.*0597

*Now, if you read this backwards, it is a bad word; so we know, as I mentioned before,*0605

*that if it spells a bad word, it is not one of the ways you prove triangles congruent; it doesn't exist.*0610

*Now, in that case, it would be SSA; but because they are right triangles, this is a special case where we can use HL.*0617

*So then, this one is a little bit different.*0632

*If you are going to use the other theorems and not these, make sure that this one, you know to use for right triangles.*0634

*Again, one pair of corresponding legs, and the hypotenuse, is HL.*0644

*We have HA, HL, LA, and LL.*0652

*For the first example, we are going to state the additional information we need to prove each by the given theorem or postulate.*0658

*We want to prove that these two triangles are congruent by one of the theorems or the postulate.*0666

*But we are missing information; so we have to state what additional information we need.*0674

*Now, before, we went over four--the theorems and the postulate--and they are LL, LA, HL, and HA.*0679

*Think of them as...there are four of them; we can remember that two of them start with an L, a leg; and two of them start with an H.*0693

*They never start with an angle: LL, HA...and there is always a leg or an angle; leg, angle, leg, angle.*0707

*You can only have legs and angles if it is not the hypotenuse: Leg-Leg, Leg-Angle, Hypotenuse-Leg, Hypotenuse-Angle.*0720

*The first one is the LA theorem; we want to prove that these two triangles are congruent by the LA theorem.*0733

*Right now, all we have is that they are both right triangles (obviously--that is the only way that we can use these theorems and postulate)...*0742

*We know that these angles are congruent, because they are vertical angles.*0750

*These are vertical angles, so we have the A; now we need an L.*0756

*So then, the additional information that we need would be the leg.*0766

*It could be either this leg with this leg or this leg with this leg; it doesn't matter.*0775

*In the next one, the HL Postulate is what they want us to use to prove that these two triangles are congruent.*0780

*Now, so far, all we have is this segment right there, because it is the same; they are sharing the same hypotenuse.*0788

*So, that is the hypotenuse for this one, and that is the hypotenuse for this one.*0801

*So, the H is covered; and on a proof, you would say that that is congruent by the reflexive property,*0806

*because the reflexive property says that anything equals itself; so that is the H.*0813

*And then, we are missing the L; we are missing the leg, so either this leg has to be congruent to this leg, or this leg has to be congruent to this leg.*0823

*And then, we can prove that those two triangles are congruent by the HL Postulate.*0831

*And again, this is the only postulate; this is a theorem, theorem...HL, the one that is different,*0835

*the one that is not like SAS, ASA, and all of the others, is a postulate.*0842

*So again, we are missing the leg; that is what we are missing.*0849

*The next example: Find the value of x so that each pair of triangles is congruent.*0857

*For this one, let's see what we are working with.*0862

*We have triangle ABC; we have triangle FED; BC is congruent to DE, because it says 10, and it says 10 here.*0866

*Now, notice how these two triangles don't look exactly the same, but make sure that you don't base it on what it looks like.*0879

*You base it on the facts; so even if you think, "Oh, this side looks like that side," don't assume that they are corresponding or they are congruent.*0891

*You have to look at the numbers; look at what is given to you.*0901

*So, in this case, it is given that BC is 10, and then DE is 10.*0906

*Even if they don't look like they are the same size, just by them having the same measure, they are congruent.*0910

*And so, then we know that we are going to assume that the triangles are congruent, because that way, we can find x.*0918

*So, if these two are congruent, that means that AB has to be congruent,*0930

*because if these two triangles are congruent, then we know that all of the corresponding parts are congruent.*0934

*And that is CPCTC, "Corresponding Parts of Congruent Triangles are Congruent."*0940

*So, we can say that AB is congruent to EF; then, to solve it, I just make 25 - 2x equal to 21.*0945

*I am going to subtract the 25; -2x = -4; divide the -2; x is 2.*0959

*And the next one: we have that the measure of angle A is 60; the measure of angle F is 60;*0975

*so then, those angles are corresponding/congruent.*0985

*And then, the hypotenuse: I make 3x - 7 equal to 2x + 4.*0990

*I am going to subtract the 2x here; that is going to give me 1x.*1002

*I am going to add the 7 here; and then, I am going to get 11; so x = 11.*1007

*OK, we are going to do a two-column proof: this one says that triangle ABC and triangle CDE are both right triangles.*1024

*AB is congruent to DE; whatever is given--just mark it on your diagram; that way, it is easier to see what you are working with.*1038

*And we want to prove that the two triangles are congruent.*1049

*Now, since they are both right triangles, we are going to use one of the four theorems and postulate that we just went over.*1052

*If they were non-right triangles, then we would use SAS, ASA, AAS, SSS...*1059

*The four that we went over are LL, LA, HL, and HA; here is the postulate.*1067

*We need one more corresponding part to be congruent; we have a leg; that means we are going to use either this one or this one.*1084

*We need one more thing, and that is all of the information that they give me.*1094

*So then, I have to think, "OK, what other information do I have--what can I tell from this diagram?"*1098

*And I know that vertical angles are congruent; that means that here, even if they don't tell me that they are congruent,*1107

*I can assume that they are congruent, because they are vertical, and we know that vertical angles are congruent.*1113

*Now, we have two corresponding parts that are congruent; and that is a leg and an angle.*1121

*We are going to use this one: #1: here are the statements and the reasons.*1126

*#1: Triangle ABC and triangle CDE are right triangles, and then that AB is congruent (I don't know why these won't work) DE.*1143

*And that is all "given."*1167

*The next one: Here is my L, leg; then, angle ACB is congruent to...and if I am going to say ACB,*1172

*then remember: the other angle that I name has to be corresponding, so then I am going to say angle DCE,*1192

*because D and A are corresponding; they are right angles--angle DCE.*1200

*Again, I didn't have to say angle ACB; I could have said angle BCA.*1205

*If I say angle BCA here, then for the next one, I have to say angle ECB; whatever is corresponding with B, I have to state in that same order.*1211

*Then, that would be the reason "vertical angles are congruent."*1223

*Here is my A; I have both L and A; that means I can go ahead and say that the triangles are congruent.*1232

*Triangle ABC is congruent to triangle CDE, and the reason for that is "LA theorem."*1243

*The next example is also a proof: let's see what we have.*1263

*The given is that this angle and this angle are congruent; they want us to prove that AD is congruent to CB, that these two legs are congruent.*1269

*Now, there is no way to just prove directly that those two legs are congruent.*1287

*So then, I have to say, "OK, can I first prove that the triangles are congruent, and then say that corresponding parts of congruent triangles are congruent?"*1295

*If you are given two triangles, and then they want you to prove that a pair of corresponding parts are congruent,*1307

*then if you can't see a way to do it directly, then you have the other option*1317

*of first proving the triangles congruent, and then saying that corresponding parts of congruent triangles are congruent.*1322

*That is what we are going to do here: so then, we have to think, "How do we prove the triangles congruent?"*1332

*Well, if the two triangles share a side, then that side is automatically congruent, because of the reflexive property.*1338

*So again, my four are LA, LL, HA, and HL.*1348

*Now, this is going to be a leg of this triangle and a leg of that triangle.*1358

*So, I have a leg (that means that I am going to use one of these two) and an angle; so I am going use the LA theorem again.*1363

*And then, once I have proved that those two triangles are congruent by the LA theorem,*1374

*then I can say that AD is congruent to CB, because they are corresponding parts.*1378

*And if you have congruent triangles, then all corresponding parts are congruent.*1384

*Statements, and reasons on this side: Angle A is congruent to angle C; that is my angle, and then "given."*1394

*The next one: BD is congruent to...this one is a little bit...the way you say it...BD is actually going to be congruent to DB.*1416

*BD is going to be congruent to DB; and the reason why it is not BD to BD:*1439

*if you flip these around, then the B in this triangle is actually corresponding to the D in this triangle.*1448

*So, if you are going to say BD, then you have to say DB.*1456

*And then, the reason, again, is the reflexive property, and then that is your leg.*1463

*So then, my triangles are now congruent: triangle ABD is congruent to triangle...corresponding to A is C; with B is D; and then B again.*1480

*So then, again, B and D are corresponding; B of one triangle is congruent to D of the other triangle.*1505

*And then, that would be the LA theorem.*1513

*And then lastly, remember: we have to start here as our first step, and then we have to end here as our last step.*1522

*Then, our last step is going to be that AD is congruent to CB; the reason for that is CPCTC.*1531

*Again, the four theorems that we are going to use are LA, LL, HA, and...the last one is actually not a theorem; HL is a postulate.*1551

*So again, these theorems and postulate can be used to prove that two right triangles are congruent.*1561

*You could use the other theorems, but that is just going to be more work,*1567

*because then you have to prove that an additional angle would be congruent,*1572

*whereas with these, you only have two, because the angle is already covered, because they are right.*1576

*We have LA, LL, HA, and the HL postulate.*1583

*That is it for this lesson; we will see you for the next lesson, too--thank you.*1590

1 answer

Last reply by: Mary Pyo

Sat Oct 29, 2011 10:32 PM

Post by shawn poag on August 23, 2011

Shouldn't the triangles in example 3 be labelled in corresponding order? ABC congruent to DEC or does it only matter when talking about specific angles.