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### Similar Triangles

- AA (Angle-Angle) Similarity: If two angles are congruent to two angles of another triangle, then the triangles are similar
- SSS (Side-Side-Side) Similarity: If the measures of the corresponding sides of two triangles are proportional, then the two triangles are similar
- SAS (Side-Angle-Side) Similarity: If the measures of two sides of a triangle are proportional to the corresponding sides of another triangle and the included angles are congruent, then the triangles are similar

### Similar Triangles

m∠N = m∠A, m∠M = m∠B

Determine whether ∆NMO is similar to ∆ABC.

For two right triangles, if one acute angle of one triangle is congruent to one acute angle of the other triangle, then the two triangles are similar.

Trapezoid ABCD

Determine whether ∆BEC is similar to ∆DEA.

- trapezoid ABCD
- ―AD ||―BC
- ∠ADB ≅ ∠CBD, ∠CAD ≅ ∠ACB

Determine whether ∆MNO is similar to ∆ABO.

- [ON/OM] = [4/8] = [1/2]
- [OB/OA] = [4/5]
- [ON/OM] ≠ [OB/OA]

If the measures of two sides of a triangle are proportional to the corresponding sides of another triangle, and one angle of the first triangle is congruent to another angle of the other triangle,

then the triangles are similar.

If the measures of the corresponding sides of two triangles are proportional, then the two triangles are similar.

m∠ABC = 2x + 8, m∠DEF = 4x + 6, find x.

- m∠ABC = m∠DEF
- 2x + 8 = 4x + 6
- − 2x = − 2

∆ABC and ∆DEF are similar

AB = 2x + 5, AC = 14, DE = 3, DF = 6, find x.

- [AB/DE] = [AC/DF]
- [(2x + 5)/3] = [14/6]
- 2x + 5 = 7

Given: ―DE ||―BC

Prove: [AD/AB] = [AE/AC]

- Statements; Reasons
- ―DE ||―BC ; Given
- ∠ADE ≅ ∠ABC ; corresponding angles theorem
- ∠A ≅ ∠A ; reflexive prop of ( = )
- ∆ADE is similar to ∆ABC ; AA similarity
- [AD/AB] = [AE/AC]; corresponding sides of ∼ ∆'s are proportional.

―DE ||―BC ; Given

∠ADE ≅ ∠ABC ; corresponding angles theorem

∠A ≅ ∠A ; reflexive prop of ( = )

∆ADE is similar to ∆ABC ; AA similarity

[AD/AB] = [AE/AC]; corresponding sides of ∼ ∆'s are proportional

*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

### Similar 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
- AA Similarity 0:10
- Definition of AA Similarity
- Example of AA Similarity
- SSS Similarity 4:46
- Definition of SSS Similarity
- Example of SSS Similarity
- SAS Similarity 8:04
- Definition of SAS Similarity
- Example of SAS Similarity
- Extra Example 1: Determine Whether Each Pair of Triangles is Similar 10:59
- Extra Example 2: Determine Which Triangles are Similar 16:08
- Extra Example 3: Determine if the Statement is True or False 23:11
- Extra Example 4: Write Two-Column Proof 26:25

### Geometry Online Course

### Transcription: Similar Triangles

*Welcome back to Educator.com.*0000

*For this lesson, we are going to go over similar triangles.*0003

*We already discussed what it means to be similar and the whole concept of similarity.*0005

*We are going to talk about similar triangles now, and we are going to go over different theorems in order to prove that triangles are similar.*0011

*The first one is angle-angle similarity (AA stands for angle-angle).*0020

*And that just means that, if two angles are congruent to two angles of another triangle, then the two triangles are similar.*0027

*Now, in the previous lesson, we talked about what it means to be similar.*0034

*And remember the two things: it was that angles had to be congruent, and sides have to be proportional.*0041

*They can't be congruent also; if angles are congruent and sides are congruent, then that would just be congruency; that would make the triangles congruent.*0060

*We are talking about similarity: that two triangles are two polygons have the same shape, but different size.*0068

*If you were to maybe draw a map of the city, then you would be using the concept of similarity,*0079

*but only if it is to scale, because if it is to scale, then you would be drawing something that is kind of the same shape,*0088

*but then a different size--a lot smaller version of it.*0097

*We are talking about two triangles that are similar; again, triangles only have angles and sides,*0102

*so all angles must be congruent, and sides must be proportional.*0108

*For this one, the AA similarity theorem, we know that A and A are both angles.*0117

*And so, since angles have to be congruent, we are saying, "OK, well, then, the two angles that we are talking about*0124

*have to be congruent to two angles of the other triangle."*0130

*We are not talking about any sides--just purely, if two angles of one triangle*0134

*are congruent to two angles of another triangle, then they are similar.*0138

*Now, they could be congruent; but this is the bare minimum to prove that they are similar.*0143

*Let's see: we have this angle right here at 80 degrees, and let's say that this angle right here is 55 degrees.*0155

*Now, that doesn't really tell us much; but when we look at the other triangle, if I tell you*0166

*that this angle right here is corresponding to this angle right here, and this is B, and this is E;*0170

*here is C, and this is F; the triangle ABC with triangle DEF...that means A is corresponding to D; B is with E; and C with F.*0183

*That means that AB is the corresponding side to DE; AC is corresponding with DF; and so on.*0199

*If A is corresponding to D, if I tell you that D is 80, and then B is 55, but then I give you*0210

*that F is 45, now you can assume that, since this angle right here and this angle right here are congruent,*0220

*so we have one of the A's (we have that one), and then here this is 55 and this is 45,*0230

*but they are not corresponding, then B is corresponding with E.*0238

*So then, I would have to subtract it from 180, and then I would get 55.*0242

*That means I show that angle E has a measure of 55.*0248

*Then, I know that A is congruent to D, and B is congruent to E; so automatically, I can say that,*0253

*since I have the second angle, that these two triangles are now similar.*0266

*So, I can draw that little symbol right there that means "similar."*0273

*Triangle ABC is similar to triangle DEF by AA similarity.*0278

*The next one is SSS similarity; now, don't get this confused with the SSS congruence theorem.*0287

*SSS similarity is a little bit different; you use it the same way; it is the same concept, but for two different reasons.*0298

*If you are trying to prove that two triangles are congruent, then you would use the SSS congruence theorem.*0308

*If you are trying to prove that they are similar (they have the same shape, but different size) then it would be the SSS similarity theorem.*0314

*Again, angles are congruent; sides are proportional; here we are talking about sides, so then it would have to be sides being proportional.*0325

*The measures of corresponding sides (all three sides) have to be proportional to the corresponding sides of the other triangle.*0346

*Then, the two triangles are similar by this SSS similarity.*0353

*Here, if I have three sides, say ABC, DEF, we know that this side and this side are corresponding; this side with this side; and this side with the last side.*0363

*Here, if I say that that is, let's say, 6, and this is 12; this is 5, and this is 10; this is 8, and this is 16;*0381

*then I know that these two would be similar, because each corresponding pair of sides have the same scale factor; they are proportional.*0402

*Then, this side with this side is 6:12; that is the ratio, which is equal to this side to this side, which is 5:10.*0414

*And then, the last two...the pair is 8:16; see how they are all equal to 1/2.*0425

*That means that they all have the same scale factor, which means that they are proportional.*0432

*And therefore, if all three sides are proportional to the three sides up here (and it has to be three;*0437

*for the sides, it has to be all three of them), then these two are similar.*0444

*If we had this ratio with this ratio (so then this pair of sides and this pair of sides being congruent,*0455

*having the same ratio), but then, let's say, the third pair of sides wasn't the same;*0464

*it was maybe 8/15; then it would not work, because it has to be all three that have the same ratio--they all have to be proportional.*0470

*And the third one is the Side-Angle-Side similarity theorem.*0485

*We have an SAS congruence theorem; you remember that one; but it is different.*0492

*Remember again: that is the congruence theorem; that SAS congruence theorem is to prove that two triangles are congruent.*0499

*Angles are congruent; sides are congruent; but in this one, again, sides (and I am writing this*0505

*over and over again for each slide, so that, that way, you remember this) are proportional.*0515

*Two things make it similar: angles are congruent, and sides proportional.*0523

*When we talk about sides here, this S and this S, the sides, have to be proportional.*0527

*And then, for the angle, it has to be congruent.*0533

*So, the Side-Angle-Side similarity theorem is just saying that two sides are proportional to the corresponding sides of the other triangle.*0537

*And then, the included angle, if you remember, is the angle between the two sides.*0545

*Here are two triangles, ABC and DEF (just so you know that the corresponding angles will be A, B, C, and then D, E, F).*0555

*It is triangle ABC with triangle DEF.*0574

*And then, if this side right here, let's say, is 5 (AB), DE is 7, let's say BC is also 5, FE is 7, and let's say that the measure of angle B--*0582

*this has to be the included angle; that means that if these are the two sides that are the sides that we are talking about,*0603

*then the included angle would be angle B--so that angle, let's say, is 120, and this is 120*0610

*(because, remember, angles have to be congruent), then these two would be similar by SAS similarity.*0617

*Side is proportional to side, side to side, and then the angles.*0627

*And then again, these are proportional because 5/7 is the ratio, the scale factor, and that is the same thing as the other one.*0632

*So, 5/7 is this side, and then 5/7 again for this side.*0641

*There are three of them: angle-angle similarity, SSS (side-side-side) similarity, and side-angle-side similarity.*0651

*With those three, let's use them to solve our examples.*0661

*Determine whether each pair of triangles is similar.*0668

*Here, I don't have any angles, so I am probably going to use the side-side-side similarity to see if these two triangles are similar.*0673

*And so, here are my triangles that I can base corresponding parts to.*0686

*And then, I know that, let's say, side AB is corresponding with side DE; that means that the ratio would be 6/9, or 6:9.*0694

*And then, BC to EF...BC is 7; 7 to...where is EF?...EF is 10.5, so here, let's just solve these out first, or simplify them.*0709

*6/9 is 2/3, and 7/10.5...if you just want to check those, what you can do...*0727

*OK, let's do this a different way, because you have that decimal, so it is not like you can easily simplify that.*0739

*So, what you can do: see how I have two pairs of the sides--so then, I am going to make them into a proportion.*0745

*Remember: a proportion is when you have two equal ratios.*0753

*I am just going to make them into a proportion, just to see if they are equal ratios.*0759

*I am going to solve them out and see if I get the correct answer.*0764

*Remember: with cross-products, I have to multiply my extremes with my means, so 6 times 10.5.*0768

*And let's just do them right here: 10.5 times 6 is 63; that equals...9 times 7 is 63, so see that that works, so it is true.*0776

*That means that this is a correct proportion, meaning that this ratio equals this ratio; they are equal to each other.*0796

*Then, those two work (so far, so good); and then, we have to try our last pair of corresponding sides.*0805

*That is AC, which is 10, to DF; that is 15.*0813

*Here, this is going to be 2/3; remember how this one was 2/3; so then, if this comes out to 2/3,*0823

*then this also has to come out to 2/3, because they are equal; and this comes out to 2/3.*0833

*So then, for this one, this one is "yes"; they work.*0837

*And normally you can just simplify; but the reason why we had to multiply this out is that, if you have decimals,*0844

*or you have fractions that make it hard for you to just look at it and simplify, then you can just solve it*0852

*as a proportion to see if those two are the correct ratio.*0857

*The next one: now, here I see that I have 95 and 95; so automatically, I know that these are congruent angles.*0863

*Now, I look at the next one; they are not the same--do you automatically assume that they are not similar now?*0872

*No, because we have to check to see if they are even corresponding angles.*0880

*They kind of look like they would be, but you would have to check these triangles.*0885

*Angle A is corresponding to angle D; see, it is not corresponding--this angle is corresponding to this angle.*0893

*So, just by looking at this, you can't assume that, just because they are different angle measures, this is automatically "no, they are not similar."*0900

*You have to check to see if they are supposed to be the same, first of all.*0910

*I am going to find the missing measure here: 95 + 53 is 148.*0918

*And then, I take 180, and I subtract that, and I get 32 degrees.*0929

*So then, here I have that this is 32; and since angle A and angle D are corresponding, they are congruent, and so this is "yes, by angle-angle similarity."*0937

*And for this one, what was the rule there? It was SSS similarity.*0958

*The next one: Determine which triangles are similar.*0968

*Here, we have a couple different shapes; we have three triangles--I have triangle ABF;*0973

*I have the bigger one, triangle ACD; and then, I have this triangle right here, FED.*0983

*And then, I have a parallelogram, parallelogram BEDC.*0992

*Now, of course, I am not going to use a parallelogram to prove anything; I am not going to use that to show similarity.*0999

*But I do need it...I am probably going to need it to determine which triangles are similar.*1007

*Let's see...let's look at our parallelogram: now, we have parallel lines here...*1016

*Or, no; back to the parallelogram: you know that opposite angles are congruent.*1025

*So then, I don't want to say that this whole thing is equal to this whole thing.*1033

*I could, but then here, see how I can say that this angle E is congruent to angle C, because opposite angles are congruent.*1039

*Now, I wouldn't want to say that of angles B and D, because this is cutting into two different triangles.*1051

*It is cutting into this triangle, and it is cutting into the big triangle, so there is no point in me saying that they are congruent.*1062

*I have an angle; since I have an A, what two things could that be?*1069

*That can be...remember: I can use the three different similarity theorems.*1075

*The two that use an angle are angle-angle similarity and side-angle-side similarity.*1081

*So, we know that we are going to use one of these two.*1093

*And then, the other one: let's see: we can say that these two angles are congruent, because they are vertical angles.*1097

*But then, remember: I need two pairs, so since my first pair, this one right here that I just marked,*1108

*has to do with this big triangle right here (that is an angle from the big triangle and an angle from this triangle),*1121

*let's see if we can find something from those two triangles--another angle from this big one, and another one from this one.*1132

*This one...even though I can say that these are vertical angles, I have two angles here, but then I only have one here.*1142

*So, I can't really use this triangle if only one of the angles is congruent to this triangle.*1150

*That is why I want to try to see about this one, because I need two angles.*1159

*If you look at this very closely, if I extend this out, we have a line...I have two parallel lines.*1165

*And then, try to ignore this line right there, BE, so that all you see is CA and DE.*1184

*And then, right here, you see a transversal.*1195

*Let me just draw it out for you on the side: here is CA; here is DE; and here is a transversal.*1202

*Now, these lines are parallel; that means that we can say that this angle right here is congruent to this angle,*1218

*because alternate interior angles are congruent when the lines are parallel.*1228

*Those are alternate interior angles; so what angles are those now, in here?*1233

*That would be this angle and this angle; see, now we have two angles from this big triangle,*1240

*and they are congruent to the corresponding angles of this triangle right here.*1252

*So, you can say that triangle ACD is similar (be careful that you don't put "congruent") to triangle...*1261

*what is corresponding with A?...D...there is our D...where is C?...E, and F.*1288

*Now, another pair of angles...back to this angle: now, because I solved this angle first (that angle was this angle right here),*1303

*I wanted to use that first, because, since we found that, I wanted to use the big triangle with that other triangle that it involved.*1313

*Now, since we have another pair of congruent angles, I can also say that this triangle right here,*1323

*because it has the two angles congruent to two angles of this triangle (this angle is congruent to this angle,*1334

*and then these are congruent to each other)--that means that these two triangles are also similar.*1343

*I can also say that triangle AFB is similar to this triangle: it is DFE.*1353

*So, because of this theorem, where both of these pairs are, you can say that those are similar--*1371

*this big one with this one, and then the second would be this one with this one.*1383

*The third example: Determine if the statement is true or false; if false, show a counterexample.*1393

*Remember: a counterexample was an example of the opposite; you are showing an example of the statement that is false.*1399

*And here is the statement: If the measures of the sides of a triangle are x, y, and z,*1413

*and the measures of the sides of a second triangle are (x + 1), (y + 1), and (z + 1), the two triangles are similar.*1421

*Here is triangle 1, and say this is triangle 2.*1435

*That means that, if this is x, y, and z, this is (x + 1), (y + 1), and (z + 1); are they similar?*1442

*What you can do is just start plugging in numbers for x, y, and z, and seeing if they are going to have the same ratio.*1456

*Let's say that we are going to use the numbers 4, 5, and 6.*1463

*Well, x + 1 is 5; y + 1 is 6; and z + 1 is 7; so let's see if the ratios are going to be the same.*1472

*This one to this one is 4:5; this one to this one is 5:6; and I am putting a question mark over the equals sign, because I am trying to see if they are equal.*1483

*I don't know if they are equal yet...6:7.*1501

*Now, I don't think that they are the same; if you want to double-check, well, let's work with this first.*1506

*We can use cross-products to see if this ratio is equal to this ratio, because proportions mean that they have to be the same.*1514

*A correct proportion would be that this ratio is equal to this ratio.*1523

*We are going to see if that works; so then, the cross-product: 4 times 6 is 24...equal to 25.*1528

*No, they are not equal; so I don't have to check the second one or the last one, because these two, I know, are not the same.*1536

*Here is my counter-example; it is an example of the statement that shows that it is false.*1544

*And this shows that it is false; so that is my answer, and that is my counterexample; it is false.*1552

*Now, for this to be true, if it was x, y, and z, then it would have to be multiplied; it can't be x + 1--you can't add 1.*1562

*If I multiplied each one of these by 2, then it would have the same scale factor.*1571

*But again, if you add a number, then it is not going to be the same.*1580

*The fourth example: Write a two-column proof: If you have BC parallel to AD, let's use that to prove that BE to ED is equal to CE to EA.*1586

*This is a proportion, and we have to prove that this proportion is correct;*1612

*so BE to ED, the ratio of this to this, is going to be equal to the ratio of this to this.*1621

*The only thing that I am given is that these two lines are parallel.*1634

*From there, if I have parallel lines, I can say a lot.*1640

*Here, I have parallel lines, and then my transversal; so then, I can say that this angle is congruent to this angle.*1652

*I can also say that this angle is congruent to this angle.*1671

*Now, can I say that this angle is congruent to this angle?--no, because these angles are with these lines, and those lines are not parallel.*1678

*Now, from here, I can also say that these two are vertical angles, and they are congruent.*1690

*But I don't need to, because all I need to prove that these two triangles are similar is angle-angle, our two angles.*1701

*So, I don't have to say that; that would be an extra step that is not necessary.*1711

*And why do I want to say that these two triangles are similar?*1716

*It is because, see, look at these parts: this BE is a side from this triangle; that is a part of this triangle;*1723

*ED is a part of this triangle; CE is from this triangle, and EA from this triangle.*1735

*These are the parts of these two triangles, and then these are scale factors; they are ratios.*1742

*So, I need to say that these two triangles are similar first; that way, I can say that the scale factor between the corresponding parts is going to be equal.*1752

*Step 1: Remember: for a two-column proof, you write your statements on one side, and then your reasons.*1762

*What are your reasons for that statement?*1776

*The first statement...*1780

*And if you haven't really understood proofs, just remember that your given statement is your starting step.*1783

*You are starting here, and then this statement right here, the "prove" statement--that is your ending; that is your last step.*1793

*First step, last step: you are going to go from here in a series of steps to end up there: this is your starting point, and that is your ending point.*1799

*Then, our first statement is going to be BC parallel to AD; what is the reason for that?--the reason is always "given."*1810

*And then, from there, what did I say about these angles--what angles are congruent?*1825

*So then, here you can't just say angle B, because angle B has all of these different angles here.*1831

*You have to say angle CBE; you can say angle CBD; you can say EBC; just make sure that you name this angle with these points.*1838

*Angle CBE is congruent to angle ADE; what is the reason for that?--"alternate interior angles theorem."*1853

*You might have to write it out; you might have to say, "If lines are parallel, then alternate interior angles are congruent."*1878

*I am just going to leave it like that, but if you are told that you need to write it out, then just make sure that you write it out.*1888

*If lines are parallel, then alternate interior angles are congruent.*1892

*Now, since the next pair of angles has the same reason, I can just write it under the same step; I don't have to rewrite the whole thing.*1900

*So then, just keep it under step 2: Angle BCA is congruent to angle DAE--and again, it is the same reason.*1907

*So then, see how I already have the angle here and the angle here.*1924

*Automatically, my third step is going to be that the triangles are similar.*1929

*So, triangle BCE is similar to triangle DAE; now, the reason for that would be angle-angle similarity.*1937

*And then, my last step (because it doesn't end there; this has to be my last step):*1962

*now that these two triangles are similar, I can say that BE/ED, this side to the corresponding side of the other triangle, which is ED,*1969

*is going to have the same ratio as CE, that side, to the side of the other triangle.*1984

*So then, the reason for that: well, corresponding sides of similar triangles (that is a triangle; it looks like an A)*1997

*are...not congruent...remember, what do we know about the sides of similar triangles?...they are proportional.*2017

*So then, that is it; here is our given statement: we start there, and then, given that,*2032

*we are going to take all of the steps that we need to end up here, and that should be our last step.*2039

*That is it for this lesson; thank you for watching Educator.com.*2047

0 answers

Post by Taylor Wright on June 12, 2013

Can you also say that Triangle AFB is similar to Triangle ADC? Since Angle AFB is congruent to Angle ADC due to the parallel lines and the transversal that intersects them. Therefore, all the triangles are congruent to one another.

0 answers

Post by bo young lee on December 21, 2012

what difference with that three similarity, i dont understand.