Geometry Classifying Triangles

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In this next lesson, we are going to talk about triangles and some different ways we can classify them.0003

A triangle is a three-sided polygon; now, a polygon is any figure that has sides and is closed.0011

A polygon can look like that, as long as it is closed, and all of the sides are straight.0027

If I have that, it would not be considered a polygon; if I have just maybe something like this,0036

that is not a polygon, either, because it is not closed--that is not an example of a polygon.0047

A triangle is a polygon with three sides; now, the triangle is made up of sides, vertices, and angles.0056

The sides are these right here, side AB, side BC, and side CA (or AC).0066

The vertices are the points; the word "vertices" is the plural of "vertex," so if I just talk about one, then I would just say "vertex."0083

But since I have three, it is "vertices"; and that would just be point A, point B, and point C--those are my vertices.0098

The angles: now, for this angle right here, let's say, I can say "angle ABC," or I can just say "angle B,"0116

because as long as there is only one angle...if I have a line that is coming out through here,0127

then I will have several different angles, so I can't label it angle B;0134

but as long as there is only one single angle from that vertex, then you can label the angle by that point.0137

This angle right here can be called angle B; this angle can be called angle A, since there is no other angle there.0147

This angle can be called angle C, or you can just do it the other way: angle ABC, angle BCA, angle BAC.0157

But this is the easiest way: angle B, angle A, and angle C; that is the triangle.0170

Now, to classify triangles, we can classify them in two ways: by their angles and by their sides.0181

These are the ways that we can classify the triangles by their angles, meaning that, based on the angles of the triangle, we have different names for them.0191

The first one: an acute triangle...well, we know that an acute angle is an angle that measures less than 90, smaller than 90.0200

So, an acute triangle is a triangle where all of the angles are acute; all of the angles measure less than 90.0209

The next one: obtuse triangle: one angle is obtuse.0226

Now, all triangles have at least two acute angles.0231

The way we classify these other ones: for this one, an obtuse angle is an angle that measures greater than 90;0243

only one of them will be obtuse--there is no way that you can have a triangle with two obtuse angles.0253

That means two of the angles (since we have three) are going to be acute, and then one of them is going to be obtuse.0260

But all triangles must have at least two acute angles; so here is an obtuse triangle; here is your obtuse angle, and then your acute angles.0268

The same thing for the next one, a right triangle: a right triangle is when only one angle is a right angle, like this.0285

Again, the other two angles must be acute.0300

Now, if I try to draw two angles of a triangle to be right, see how there is no way that that could be a triangle,0304

because a triangle, remember, has to have three sides; "tri" in triangle means three.0322

Now, an equiangular triangle means that all of the angles are equal--"equal angle"--can you see the two words formed in there?0340

So, equiangular is when all of the angles are congruent.0352

Now, if all of the angles are congruent, then each angle is going to measure 60 degrees.0357

Now, we are going to go over this later on; but the three angles of a triangle have to add up to 180.0368

If all three angles are congruent, then I just do 180 divided by 3, because each angle has to have the same number of degrees.0378

So, 180 divided by 3 is going to be 60; so this will be 60, 60, and then 60.0390

And that is an equiangular triangle; and of course, an equiangular triangle is an acute triangle, because 60 degrees is an acute angle; all three angles are acute.0396

In an equiangular triangle, all angles are congruent; they all measure 60 degrees.0412

OK, so then, classifying triangles by size: we just went over by angles, depending on how the angles look--0419

if it is a right angle in the triangle, an obtuse angle, or an acute angle.0426

We can also classify triangles by the size.0433

If I have a triangle where no two sides are congruent--all three sides are different lengths (like this is 3, 4, 5), then this is a scalene triangle.0438

So, I can show this by making little slash marks; I can make this once; then this, I will do twice to show that these are not congruent;0456

and then I will do this one three times--that shows that none of these sides are congruent to each other.0465

And that is a scalene triangle.0473

The next one, an isosceles triangle, is when I have at least two sides being congruent.0477

And then, an equilateral triangle is when all of the sides are congruent.0493

So, an equilateral triangle will be considered an isosceles triangle, because any time you have at least0499

(meaning two or three) sides being congruent, then it is considered isosceles.0505

But this is the more specific name if all three are congruent.0510

So, you would call this an equilateral triangle; but it is also considered an isosceles triangle.0514

Now, remember this one: equilateral; equiangular is when all of the angles are congruent, and equilateral is when all of the sides are congruent.0523

These are some ways you can classify triangles by the sides.0532

Isosceles triangle: I see here that these two sides are congruent, which is an isosceles triangle.0540

If I label this as triangle ABC, then I can say that triangle ABC is isosceles.0549

Any time you have a triangle, you can label it like this, triangle ABC, just like when you have an angle, you can say angle A.0560

So, if you have a triangle, you are going to say triangle ABC.0566

This right here, this side that is not congruent to the other two sides, is called your base.0573

These two sides that are congruent are called your legs.0587

Now, the two angles that are formed from the legs to the base are called base angles.0598

Base angles would be this angle right here and this angle right here.0609

And then, this angle right here, or that point, is called your vertex.0618

This right here, that is formed from the two congruent legs, is called your vertex.0627

So, we have legs; we have a vertex; we have the base; and then, you have base angles.0634

And this makes up your isosceles triangle.0641

And then, just to go over the right triangle: these are also called your legs, but this one is not called the base; it is called a hypotenuse.0644

That is just to review over that.0668

In the isosceles triangle, these two are congruent, so they are called your legs; this one is your base.0672

Now, don't think that the base is the side always on the bottom; no.0679

It depends on the triangle: I can move this triangle around and make it look like this, and I can say that these two sides are congruent.0684

Then, this would be my base, and these are my legs.0694

Find x, AB, BC, and AC; let me just write this to show that they are my segments.0701

To solve this, if I want to solve for x, I know that these two are congruent.0714

Since they are congruent, I can just make these equal to each other.0722

2x + 3 = 3x - 2; then, if I subtract the 3x, I get -x =...I subtract 3 over, so I get -5; x is 5.0726

There is my x; AB is (and I am not going to put a line over this one, because I am finding the lengths;0742

if I am finding the measure, then I don't put the line over it) 2 times 5, plus 3; so that is 10 + 3, is 13.0753

And then, for BC, since I know x, even though this side is the base, and it doesn't have anything to do with these sides;0772

since I know x, I can solve for BC; that is 5 + 1, so BC = 6.0783

And then, AC is 3(5) - 2, so that is 15, minus 2 is 13; and that is everything.0794

So again, if you have an isosceles triangle, and they want you to solve for x, then you can just say that,0811

since these two sides are congruent, you can make these two congruent.0818

All right, let's go over a few examples now: Classify each triangle with the given angle measures by its angles and sides.0826

We have to classify each of these triangles with the given measures in two ways: by its angles and by its sides.0835

The first one: the angles are here, and the sides here.0845

To classify this triangle by its angles, look: I have one that is really big--that is an obtuse angle; that means that this would be an obtuse triangle.0859

I am just going to draw a little triangle right there.0875

And then, by its sides, remember: sides are scalene, isosceles, and equilateral.0877

No two sides are congruent (the same), so this would be a scalene triangle.0886

The next one: 50, 50, and 80: by its angles--look at the angle measures: they are all acute angles.0897

So then, this triangle would be an acute triangle.0905

And then, with its sides, are any of them the same?0912

We have two that are the same; these two are the same; then, I have an isosceles triangle.0915

The next one: this is an acute triangle, because they are all the same; but more specifically, this would be an equiangular triangle.0926

And then, since they are all the same, even though it could be isosceles (because isosceles is two or more), a more specific name would be equilateral.0946

And the last one: 30, 60, 90: well, I have one angle that is a right angle, so this is a right triangle.0964

And then, my sides: this would be scalene, because they are all different.0975

The next example: We are going to fill in the blanks with "always," "sometimes," or "never."0989

Scalene triangles are [always/sometimes/never] isosceles.0996

Well, we know that a scalene triangle is when they are all different.1001

An isosceles triangle is when we have two or more sides that are the same.1006

So, a scalene triangle would never be isosceles, because in order for the triangle to be scalene, all of the sides have to be different.1011

In order for the triangle to be isosceles, you have to have two or more the same, so this would be "never" isosceles.1022

The next one: Obtuse triangles [always/sometimes/never] have two obtuse angles.1034

Well, if I draw an obtuse angle here, and then I draw an obtuse angle here, this has to be greater than 90,1042

and this has to be greater than 90; a triangle has to have three sides only, so there is no way for that to be a triangle.1050

So, this is "never."1058

Equilateral triangles are [always/sometimes/never] acute triangles.1066

If they are equilateral, that means that it has to be like this.1075

Can we ever have an obtuse triangle where they are the same?--no, because you know that this can only be extra long.1082

And then, if we have a right triangle, no, because we can't have the hypotenuse be the same length as one of the legs.1092

So, equilateral triangles are "always" acute.1099

Isosceles triangles are [always/sometimes/never] equilateral triangles.1110

An isosceles triangle is like this or like this; "isosceles" can be two or three sides being congruent.1115

Now, sometimes they are like this, and sometimes they are like that.1127

That means that sometimes they are going to be equilateral.1131

Because isosceles triangles can be considered equilateral triangles (but not always; if it is only two, then it is not; if it is three, then it is), it is "sometimes."1137

Acute triangles are [always/sometimes/never] equiangular.1151

If I have an acute triangle, yes, it could be equiangular; but can I draw an acute triangle that is not equiangular?1158

How about like this? This is not equiangular, but they are all acute angles.1170

Sometimes it could be like this, or sometimes it could be like this; "sometimes"--acute triangles are sometimes equiangular.1184

If I have, let's say, 50, 50, and 80; see how these are all acute angles.1200

They are acute angles, but then it is not equiangular; only if they are 60, 60, 60 are they equiangular.1215

I can have other acute triangles that are not equiangular.1223

OK, the next example: Find x and all of the sides of the isosceles triangle.1231

Here, it is this side and this side that are congruent.1238

Now again, this is the base (even though it is not at the bottom, it is still called the base).1244

These are my legs, the vertex, and the base angles.1250

I am going to make 9x + 12 equal to 11x - 4, because those sides are congruent.1256

Then I am going to solve this out; I am going to subtract 11x over there, so I get -2x = -16; x = +8--there is my x.1267

Then, I have to look for all of my sides: AB is going to be 9 times 8 plus 12; that is 72 + 12, so AB is equal to 84.1281

AC is 11 times 8, minus 4; that is 88 - 4, which is 84; that is AC.1305

And notice how they are the same; they have to be the same, because that is the whole point.1320

They are isosceles; these are congruent; we make them equal to each other so that they will be the same.1323

And then, BC is 2 times 8 plus 10, so this is 16 + 10, so BC is 26.1329

And that is it for this problem.1347

Use the distance formula to classify the triangle by its sides.1351

Here is my triangle, ABC; and then, you are going to use the distance formula to find what kind of sides there are.1355

So, if I find the distance of A to C, then I will find the length of this side.1364

Then I can use the distance formula to find the length of that side, and again do the same thing here.1372

And then, you are going to compare those distances of the three sides.1376

That means I will have to use the distance formula three times, for each of the sides.1382

There is A, and there is B, and there is C.1386

Now, A (let me just write it out) is at point...here is -2...1, 2, 3; B is at (-1,3); and C is at...here is 1, 2, 3...-1.1389

Let's find AB first; it doesn't matter which one you find first.1426

AB: I am going to use points A and B, these two; this is going to be x₂, and then I will just write out the distance formula again right here.1431

All you do is subtract the x's, square that number, and add it to that number.1441

AB is (-2 - -1)² + (-3 - 3)²; -2...minus a negative is the same thing as plus,1453

so that is going to be -1 squared, is 1, plus -3 - 3, is -6, squared is +36; so this is going to be √37; there is AB.1482

And then, BC is B and C: it is (-1 - 3)² + (3 - -1)².1503

And then, -1 - 3 is -4, squared is 16; plus...this is going to be 3 + 1; that is 4; this is 16; and then, that is the square root of 32.1523

This can be simplified; remember from the last lesson: if you want to simplify radicals,1544

square roots, then you have to write this down; you can do a factor tree.1551

This is going to be 16, and this is going to be 2; or you can just do 8 and 4; it doesn't matter.1557

Circle it if it is prime; now, 16 here is a perfect square, so I can just go ahead and do that.1563

But just to show you: it is going to be 8 and 2 (or 4 and 4; it doesn't matter), 4 and 2, 2 and 2.1568

Whenever you have two of the same number, it is going to come out of the radical a single time.1578

So then, as long as there are two of the same number, it comes out as one.1589

So then, that comes out as a 2; these come out as a 2; and then, this one is still left.1595

Whatever is left has to stay inside, and then, whatever came out (2 came out here, and 2 came out here)...that becomes 4√2.1604

And then again, you know that is 16 times 2; we know that because it is 16 + 16; and then, just do that.1616

That is BC; and then, AC is going to be (-2 - 3)² + (-3 - -1)².1626

So, this is going to be -5, squared is 25, plus...this is going to be -2, because it is going to be plus; -2 squared is 4; so this is...1647

OK, let me just write it here: AC = √29.1666

And you can't simplify that any more.1675

Now that I finished my distance formula, applying the distance formula to each of the sides,1680

AC was √29; BC is 4√2; and then, AB is √37.1687

Now, see how all three sides are different: the whole point is to classify the triangle by its sides.1696

That means that it is either going to be a scalene, an isosceles, or an equilateral triangle.1704

Since all three are different, we know that this is a scalene triangle; and that is your answer.1709

That is it for this lesson; thank you for watching Educator.com.1721