Geometry 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 about0124

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 triangle0134

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 you0166

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 you0210

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 this0505

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 1200610

(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 it0852

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