Geometry 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 used0042

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 congruent0074

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 thing0105

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 theorems0142

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 option1317

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