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### Isosceles and Equilateral Triangles

- Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent
- Isosceles Triangle Theorem Converse: If two angles of a triangle are congruent, then the sides opposite those angles are congruent
- Isosceles Triangle Corollaries:
- A triangle is equilateral if and only if it is equiangular
- Each angle of an equilateral triangle measures 60 degrees

### Isosceles and Equilateral Triangles

- ∆ DEF is an isosceles triangle.
- ∠DEF = m ∠DFE
- 6x + 4 = 3x + 2

If two sides of a triangle are congruent, then there must be a pair of congruent angles in this triangle.

All the angles in an isosceles triangle are 60 degree.

- ∆ ABC is an isosceles triangle.
- ―AC = ―AB
- 16 − x = 3 + 4x

Isosceles angles are _____ equilateral angles.

- ∠ABC = 60
- ∠ADB = 90
- ∠ABC + m ∠ABD + m ∠BAD = 180
- ∠BAD = 180 − m ∠ABC − m ∠ABD

If all the three sides of one triangle are congruent, then the triangle is an isosceles triangle.

Given: AD||BC, E is the midpoint of AD, ―BE ≅ ―CE

Prove: ∆ ABE ≅ ∆ DCE

- Statements ; Reasons
- AD||BC ; Given
- ∠AEB ≅ ∠CBE, ∠DEC ≅ ∠BCE ; alternate interior angles
- ―BE ≅ ―CE ; Given
- ∠CBE ≅ ∠BCE ; Isosceles ∆ theorem thearom
- ∠AEB ≅ ∠DEC ; Transitive prop of ≅∠s
- E is the midpoint of AD ; Given
- ―AE ≅ ―DE ; Definition of midpoint
- ―BE ≅ ―CE ; Given
- ∆ ABE ≅ ∆ DCE ; SAS post

AD||BC ; Given

∠AEB ≅ ∠CBE, ∠DEC ≅ ∠BCE; alternate interior angles

―BE ≅ ―CE ; Given

∠CBE ≅ ∠BCE ; Isosceles ∆ theorem

∠AEB ≅ ∠DEC ; Transitive prop of ≅∠s

E is the midpoint of AD ; Given

―AE ≅ ―DE ; Definition of midpoint

―BE ≅ ―CE ; Given

∆ ABE ≅ ∆ DCE ; SAS post

Prove: BAC ≅ CBD

- Statements ; Reasons
- ―AC ≅ ―AB ; Given
- ∠ABC ≅ ∠C ; Isosceles ∆ theorem thearom
- ∠ABC = m∠C ; definition of ≅ angles
- ∠BAC + m ∠ABC + m∠C = 180 ; triangle angles sum theorem
- ∠BAC + m∠C + m∠C = 180 ; transitive prop ( = )
- ∠BAC = 180 - 2 m∠C ; substraction prop ( = )
- ―BC ≅ ―BD ; Given
- BDC ≅ C ; Isosceles ∆ theorem
- ∠BDC = m∠C ; definition of ≅ angles
- ∠CBD + m ∠BDC + m∠C = 180 ; triangle angles sum theorem
- ∠CBD + m∠C + m∠C = 180 ; transitive prop ( = )
- ∠CBD = 180 - 2 m∠C ; substraction prop ( = )
- ∠BAC = m∠CBD ; transitive prop ( = )
- BAC ≅ CBD ; definition of congruent angles.

―AC ≅ ―AB ; Given

∠ABC ≅ ∠C ; Isosceles ∆ theorem

∠ABC = m∠C ; definition of ≅ angles

∠BAC + m ∠ABC + m∠C = 180 ; triangle angles sum theorem

∠BAC + m∠C + m∠C = 180 ; transitive prop ( = )

∠BAC = 180 - 2 m∠C ; substraction prop ( = )

―BC ≅ ―BD ; Given

BDC ≅ C ; Isosceles ∆ theorem

∠BDC = m∠C ; definition of ≅ angles

∠CBD + m ∠BDC + m∠C = 180 ; triangle angles sum theorem

∠CBD + m∠C + m∠C = 180 ; transitive prop ( = )

∠CBD = 180 - 2 m∠C ; substraction prop ( = )

∠BAC = m∠CBD ; transitive prop ( = )

BAC ≅ CBD ; definition of congruent angles.

^{o}, find m ∠BAD.

- ∠B ≅ ∠C
- ―AD ⊥―BC
- ∠ADB = m ∠ADC = 90
- ―AD ≅ ―AD
- ∆ ABD ≅ ∆ ACD
- ∠BAD ≅ ∠CAD
- ∠BAD = m∠CAD

^{o}

*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

### Isosceles and Equilateral 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
- Isosceles Triangle Theorem 0:07
- Isosceles Triangle Theorem
- Isosceles Triangle Theorem 2:26
- Example: Using the Isosceles Triangle Theorem
- Isosceles Triangle Theorem Converse 3:29
- Isosceles Triangle Theorem Converse
- Equilateral Triangle Theorem Corollaries 4:30
- Equilateral Triangle Theorem Corollary 1
- Equilateral Triangle Theorem Corollary 2
- Extra Example 1: Find the Value of x 7:08
- Extra Example 2: Find the Value of x 10:04
- Extra Example 3: Proof 14:04
- Extra Example 4: Proof 22:41

### Geometry Online Course

### Transcription: Isosceles and Equilateral Triangles

*Welcome back to Educator.com.*0000

*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 here*0148

*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

0 answers

Post by Sayaka Carpenter on August 28, 2011

the equilateral triangle corollaries page says isosceles triangle corollaries. ;P 4:30