Geometry Isosceles and Equilateral Triangles

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For this next lesson, we are going to be taking a look at isosceles and equilateral triangles.0002

OK, the isosceles triangle theorem: now, just to review, an isosceles triangle is a triangle with two or more congruent sides.0010

Here we have an isosceles triangle, because AC is congruent to BC.0022

So, a triangle where two sides are congruent (or three) is known as an isosceles triangle.0027

And the isosceles triangle theorem says that if two sides of a triangle are congruent, then the angles opposite those sides are also congruent,0037

meaning...remember: we learned that these two sides that are congruent are called legs;0049

this is called the base--the non-congruent side is called the base; this is called the vertex,0060

and these angles are called the base angles; so if the two sides are congruent, then the angles opposite those congruent sides--0069

that means if this side is congruent, then the angle opposite is this angle right here,0094

and this angle right here--these angles are also congruent.0102

That means that we will have this angle being congruent to this angle.0112

This isosceles triangle theorem is also called the base angles theorem, because it is saying that for an isosceles triangle, base angles are congruent.0116

See how the angles opposite the sides are actually the base angles; so, this is also called the base angles theorem.0136

OK, here we have an isosceles triangle; here are the legs; that means that this angle right here and this angle right here0148

are the base angles, and they are congruent; so if I want to find x, then I can just make them equal to each other.0158

5x + 20 = 8x - 10; if I subtract the 8x over, I am going to get -3x; if I subtract the +20 over, I am going to get -30; divide the -3; x is 10.0168

So, in this problem, they just want us to find x; that is my answer.0193

Again, when we have an isosceles triangle, the theorem says that the base angles are congruent.0199

Now, the converse of the isosceles triangle theorem, we know, is the opposite.0211

The original isosceles triangle theorem says that, if two sides of a triangle are congruent, then the angles opposite them are congruent.0220

The converse says that if the two angles of a triangle are congruent, then the sides opposite are congruent.0229

Here are the angles that are congruent; those are the opposite sides.0239

That means that those sides are congruent.0249

The isosceles triangle theorem works both ways: you can say that, if these sides are congruent, then the base angles are congruent;0256

or if the base angles are congruent, then the sides opposite are congruent.0262

Some corollaries: again, corollaries are kind of like theorems, where they are supposed to be proved;0272

but they are a little more on the common-sense side; it is like they are important, but not as important as theorems.0279

And you can prove these by using theorems.0296

A triangle is equilateral if and only if it is equiangular.0300

Remember "if and only if": this means that this conditional statement and its converse are both true.0304

So, I can say, "If a triangle is equilateral, then it is equiangular," but the "if and only if" says that the converse can also be true;0314

so, "If it is equiangular, then the triangle is equilateral."0323

That means that, if I have this, then I have this; or the other way around--if I have this, then I have this.0332

Equilateral and equiangular go hand-in-hand; if you have one, then you have both.0349

The next one: Each angle of an equilateral triangle measures 60 degrees.0356

Well, if I have an equilateral triangle, I know that I have an equiangular triangle, also.0361

And the angle sum theorem says that all of the angles of a triangle have to add up to 180;0368

so if it is equiangular, then if that is x, then this has to be x, and this has to be x, because they are all the same--equiangular.0377

So then, all three angles are going to add up to be 180; so x + x + x = 180; 3x = 180; divide by 3; x = 60 degrees.0391

That means that, if I have an equilateral triangle, or an equiangular triangle, then each angle measure is going to be 60 degrees.0406

It must, must, must be, because they have to be the same measure, and then they all have to add up to 180.0417

So, it is just 180 divided by 3.0424

So, let's go over our examples: Find the value of x.0430

The first one: let's look at this: we have an isosceles right triangle, meaning that our two legs are congruent, and the hypotenuse is our base.0435

Remember: I think two lessons ago, we talked about the angle sum theorem.0451

If we have a right triangle, since all three angles have to add up to 180, and one of the angles is a right angle (that is 90 degrees--0460

this is 90)--automatically, we know that the other two angles, the remaining two angles, have to add up to 90,0472

because since all three add up to 180, this angle already used up half of that; half of 180 is 90.0480

That means that the other two, the remaining two, are going to have to add up to the second half, which is 90.0488

So, this angle, angle A...the measure of angle A, plus the measure of angle B, is going to equal 90 degrees.0494

We know that the measure of angle A is x; what is the measure of angle B?0511

Well, look: it is an isosceles triangle still, so if these sides are congruent, then the base angles have to be congruent.0516

That means that this has to be congruent to this; so if this is x, then this has to be x, also; x + x = 90; 2x = 90; so x = 45.0524

If this is 90, then this has to be 45 and 45.0538

The next one: we have the measure of angle A being 71, the measure of angle B being 71, AC as x, and BC as 22.0545

I don't have any markings to show any congruence, so I have to just look at this.0557

Angle A and angle B have the same measure of 71, so I know that they are congruent.0567

If these are congruent, then the converse of the base angles theorem, or the isosceles triangle theorem,0579

says that the sides opposite them (this side and this side) must also be congruent.0584

So then, in this case, if this is 22, then x has to be 22.0594

OK, for this next problem, I have a triangle here with base angles congruent, which means that this is congruent to this side.0604

And then, I have another triangle here with base angles congruent, which makes this side congruent to this side.0614

So then, if I look at this top triangle again, this measures 60, and these two are the same.0625

So, if I make this y, this has to also be y, because they are congruent.0635

And the angle sum theorem says that all three angles have to add up to 180.0641

That is 60 + 2y = 180; so 2y = 120, and then y = 60.0652

That means that each of these angles measures 60 degrees.0663

And that just means...if all three are 60 degrees, that means that I have an equilateral triangle, or an equiangular triangle.0669

That means...if this is 3x + 2, this is also 3x + 2, and this is 3x + 2.0679

And then, this right here is 5x - 6, and this is 5x - 6, because of this triangle here.0690

Since this side right here is 3x + 2, and it is also 5x - 6, I can just make them equal to each other.0700

So, 3x + 2 = 5x - 6; if I subtract the 5x over, I get -2x; if I subtract the 2 over, I get -8; x = 4.0708

For the next one: see how we have two triangles; this is an isosceles triangle, because these sides are congruent.0733

That means that these base angles have to be congruent.0753

Then, for this triangle, the same thing: these are congruent; that means that these base angles have to be congruent.0756

And then, if you look here, we have parallel lines.0765

Now, parallel lines, with this transversal, mean that we have some congruent angles.0769

Since for parallel lines (parallel line, parallel line, transversal), alternate interior angles are congruent,0782

that means that this angle right here is also 7x - 6, which is also 6x.0795

I can say 6x = 7x - 6; if I subtract 7x over, I get -x = -6, so x is 6.0809

You just have to look at it: I have isosceles triangles; if you have parallel lines, that will definitely help you with angles.0826

You will need to see those parallel lines to show that these alternate interior angles are congruent.0835

The next example: for examples 3 and 4, we are going to be working on a couple of proofs.0845

Let's see what is given to us: AB, this side right here, is congruent to DC;0855

angle 1 is congruent to angle 4; and I want to prove that these two angles are congruent.0862

In order to do this, let's see: Now, what do I have to work with here?0876

Well, I know that I am dealing with triangles; so here, I see a lot of triangles.0888

Now, if I make these base angles, these congruent angles, then since these base angles are congruent...0896

Now, these base angles are from the big triangle, triangle ABD; so that means that this side...0906

Now, you have to ignore these two segments right here, because we are just looking at the big one.0915

The big triangle with these base angles...I have to make it twice, since there is already one right here...we know that those two are congruent.0923

Now, if I want to prove that these two angles are congruent...these are the base angles of this triangle right here;0936

So, in order for me to say that these two base angles are congruent, these two sides have to be congruent.0945

How can I show that these two sides are congruent?0952

Well, let's see: look at how I have Side-Angle-Side.0956

From the previous lesson, we can prove that these two triangles (this one right here and this one right here) are congruent.0979

Why?--because of Side-Angle-Side: that is one of the rules, one of the methods to proving triangles congruent.0990

Then, I can say that, since this is a side (this is a part of this triangle), and this is a part of this triangle--0998

once I have proved that these triangles are congruent, then I can say that corresponding parts are congruent.1008

So then, I can just say that this is congruent to this.1013

And then, once those sides are congruent, then these base angles will be congruent, because of the isosceles triangle theorem.1017

So, that is how I am going to go about my proof.1025

So again, step 1 is to prove that these two triangles are congruent.1027

Then, I am going to say that these two sides are congruent, because of CPCTC.1035

And then, I can say that angles 2 and 3 are congruent, because of the isosceles triangle theorem.1043

So now, I just have to write everything out: statements and reasons.1052

Statements: #1: AB is congruent to DC, and angle 1 is congruent to angle 4; what is the reason?--"Given."1065

Again, just to explain what I am doing here: I am going to prove that these red triangles are congruent,1082

so that these sides will be congruent, so that these angles will be congruent.1088

My first focus is to prove that those two triangles are congruent.1102

And I am going to do that by one of the methods that we went over in the previous lessons.1106

So, I have AB--I have a side; there is a side; I have an angle; I need one more thing, which would be AE being congruent to DE.1110

What is the reason for that? Well, remember how we said that these sides are congruent, because the base angles are congruent.1134

Then, we can say "isosceles triangle theorem--converse"; "converse" is because it was given to us that the angles are congruent,1143

and then from there we concluded that the sides were congruent.1158

And then, now I have all of the parts that I need to prove that the triangles are congruent.1162

I can say now that triangle ABE (it doesn't matter how I label it for the first triangle,1172

and then the second triangle depends on that)...so what is corresponding with A? D; with B, the C, and then E and E;1181

so again, I don't want to just say Side-Angle-Side in this order until I look at it and make sure.1194

It is Side-Angle-Side, because the angle is included; it is in between the two sides.1203

So then, this is the Side-Angle-Side postulate.1208

I prove that the triangles are congruent; once the triangles are congruent, I can now say that any of the parts are congruent.1218

Now, I can't automatically just write that angles 2 and 3 are congruent, because they are not angles of the triangles that we proved.1228

So, we can't say that these angles are congruent.1237

Instead, we can say that EB is congruent to EC, because they are parts of the triangles that we just proved congruent.1240

So, they are corresponding parts of congruent triangles that are congruent.1259

So again, if you want to use this as a reason for corresponding parts to be congruent, this has to first be stated--two triangles being congruent.1267

And then, I can say that angle 2 is congruent to angle 3, my final step, because,1281

now that this is congruent because of CPCTC, I can say that the angles opposite, the base angles, are congruent now.1292

So then, here it is going to be "isosceles triangle theorem," and that wouldn't be the converse; that would just be the regular theorem.1303

That is it for this one; I know this was a little bit more difficult, but sometimes you have to work backwards.1320

I looked at what I had; that was the first thing I did--I looked at my given.1332

I made little markings if it is not already there.1336

And then, I have to see what I have to prove--what is going to be my last step.1341

So, from there, I can say, "OK, well, I can use this theorem if I prove that, and then I can prove that by proving something else."1347

Sometimes you have to work backwards, and just have to kind of look at it and think about it before you actually begin the proof.1354

We are going to do one more for this lesson: again, the first step is to look at my given;1362

angle 3 is congruent to angle 4; those angles are congruent; AB is congruent to DC--that is already marked.1371

And then, I want to prove that angle 1 is congruent to angle 2.1384

Again, if I want to say that these two angles are congruent, there is no way, just by what is given to me;1394

I can't just say that these two angles are congruent.1407

But then, I know that these two angles are the base angles of this triangle right here.1410

So, as long as I can say that the sides opposite, this side and this side, are congruent, then angles 1 and 2 can be congruent.1418

Is there any way that I can prove that these sides are congruent, then, instead?1429

Since I can't prove that the angles are congruent, can I prove that the sides are congruent?1434

Well, let's see: these sides belong to these triangles.1438

I can't directly say that these sides are congruent, but if I prove that the triangles are congruent,1447

then I can say that these sides that belong to those triangles are going to be congruent, because of CPCTC.1453

Then, I can say that these angles are congruent; that is the reasoning behind it.1463

Now, am I able to prove that these two triangles are congruent?1468

I have a side; I have corresponding angles; and then, I need one more--I only have two.1472

So then, I need one more; now, look: I have that angle 5 is congruent to angle 6--that is automatic, because they are vertical angles.1479

Now, I have Angle-Angle-Side; is that a valid method?1490

Yes, it is; Angle-Angle-Side is valid, so that is what I am going to do.1497

I am going to prove that these two triangles are congruent, and then say that these two sides are congruent, and then say that the angles are congruent.1503

So, it is kind of similar to the example that we just did.1511

Statements and reasons here: Angle 3 is congruent to angle 4, and AB is congruent to DC; "Given."1521

What do I have here that pertains to my triangles?1542

I have a side, my side; and I have my angles.1545

Step 2: I need another side or angle--something; so then, that would be my vertical angles.1552

Angle 5 is congruent to angle 6; "Vertical angles are congruent."1560

There is my other angle; now that I have everything I need, I have proven that the triangles are congruent.1572

Triangle ABE is congruent to triangle...what is corresponding with A?--D; C, because C is corresponding with B; E.1581

And is my reason ASA? No, because it is not the right order; it is Angle-Angle-Side, so it is AAS.1600

And this is actually the theorem, not the postulate.1612

And then, now that I have proven that the triangles are congruent, I can now say that the sides, one of which is AE...that is congruent to DE.1617

What is my reason? CPCTC.1633

Now that these sides are congruent, I can now say that these angles are congruent, which is my last step.1639

And then, the reason is "isosceles triangle theorem," because we just proved that these sides are congruent.1651

That is it for this lesson; thank you for watching Educator.com.1670