### Classifying Triangles

- Triangle: A three-sided polygon. A triangle has three sides, three vertices, and three angles.
- Acute triangle: All angles are acute
- Obtuse triangle: One angle is obtuse
- Right triangle: One angle is right
- Equiangular triangle: An acute triangle with all congruent angles
- Scalene triangle: No two sides are congruent
- Isosceles Triangle: At least two sides are congruent
- Equilateral Triangle: All sides are congruent

### Classifying Triangles

A triangle ____has 3 vertices.

A triangle with two acute angles and one obtuse angle.

- ―AB = ―BC
- 4x + 3 = 3x + 7
- x = 4

^{o}, m∠ACB = 60

^{o}, m∠BAC = 90

^{o}, determin what kind of triangle it is.

Isosceles triangle _____has three congruent angles.

Equilateral triangles are _____ acute triangles.

Scalene triangles ____have two sides are congruent.

- A(3, 4), B( − 3, − 1), C(4, − 3)
- ―AB = √{( − 3 − 3)
^{2}+ ( − 1 − 4)^{2}} = √{36 + 25} = √{61} - ―BC = √{(4 − ( − 3))
^{2}+ ( − 3 − ( − 1))^{2}} = √{49 + 4} = √{53} - ―AC = √{(4 − 3)
^{2}+ ( − 3 − 4)^{2}} = √{1 + 49} = √{50} . - No two sides are congruent.

*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

### Classifying 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
- Triangles
- Classifying Triangles by Angles
- Equiangular Triangle
- Classifying Triangles by Sides
- Isosceles Triangle
- Isosceles Triangle
- Extra Example 1: Classify Each Triangle
- Extra Example 2: Always, Sometimes, or Never
- Extra Example 3: Find All the Sides of the Isosceles Triangle
- Extra Example 4: Distance Formula and Triangle

- Intro 0:00
- Triangles 0:09
- Triangle: A Three-Sided Polygon
- Sides
- Vertices
- Angles
- Classifying Triangles by Angles 2:59
- Acute Triangle
- Obtuse Triangle
- Right Triangle
- Equiangular Triangle 5:38
- Definition and Example of an Equiangular Triangle
- Classifying Triangles by Sides 6:57
- Scalene Triangle
- Isosceles Triangle
- Equilateral Triangle
- Isosceles Triangle 8:58
- Labeling Isosceles Triangle
- Labeling Right Triangle
- Isosceles Triangle 11:10
- Example: Find x, AB, BC, and AC
- Extra Example 1: Classify Each Triangle 13:45
- Extra Example 2: Always, Sometimes, or Never 16:28
- Extra Example 3: Find All the Sides of the Isosceles Triangle 20:29
- Extra Example 4: Distance Formula and Triangle 22:29

### Geometry Online Course

### Transcription: Classifying Triangles

*Welcome back to Educator.com.*0000

*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 least*0499

*(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 x _{2}, 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

0 answers

Post by Delores Sapp on February 9, 2015

Correction to post : then all its exterior angles are obtuse.

0 answers

Post by Delores Sapp on February 9, 2015

Is this a true statement ? If a triangle is acute, then all i its exterior angles are obtuse.