WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For this lesson, we are going to go over similar triangles.
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We already discussed what it means to be similar and the whole concept of similarity.
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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.
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The first one is angle-angle similarity (AA stands for angle-angle).
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And that just means that, if two angles are congruent to two angles of another triangle, then the two triangles are similar.
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Now, in the previous lesson, we talked about what it means to be similar.
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And remember the two things: it was that angles had to be congruent, and sides have to be proportional.
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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.
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We are talking about similarity: that two triangles are two polygons have the same shape, but different size.
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If you were to maybe draw a map of the city, then you would be using the concept of similarity,
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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,
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but then a different size--a lot smaller version of it.
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We are talking about two triangles that are similar; again, triangles only have angles and sides,
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so all angles must be congruent, and sides must be proportional.
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For this one, the AA similarity theorem, we know that A and A are both angles.
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And so, since angles have to be congruent, we are saying, "OK, well, then, the two angles that we are talking about
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have to be congruent to two angles of the other triangle."
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We are not talking about any sides--just purely, if two angles of one triangle
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are congruent to two angles of another triangle, then they are similar.
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Now, they could be congruent; but this is the bare minimum to prove that they are similar.
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Let's see: we have this angle right here at 80 degrees, and let's say that this angle right here is 55 degrees.
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Now, that doesn't really tell us much; but when we look at the other triangle, if I tell you
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that this angle right here is corresponding to this angle right here, and this is B, and this is E;
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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.
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That means that AB is the corresponding side to DE; AC is corresponding with DF; and so on.
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If A is corresponding to D, if I tell you that D is 80, and then B is 55, but then I give you
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that F is 45, now you can assume that, since this angle right here and this angle right here are congruent,
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so we have one of the A's (we have that one), and then here this is 55 and this is 45,
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but they are not corresponding, then B is corresponding with E.
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So then, I would have to subtract it from 180, and then I would get 55.
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That means I show that angle E has a measure of 55.
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Then, I know that A is congruent to D, and B is congruent to E; so automatically, I can say that,
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since I have the second angle, that these two triangles are now similar.
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So, I can draw that little symbol right there that means "similar."
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Triangle ABC is similar to triangle DEF by AA similarity.
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The next one is SSS similarity; now, don't get this confused with the SSS congruence theorem.
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SSS similarity is a little bit different; you use it the same way; it is the same concept, but for two different reasons.
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If you are trying to prove that two triangles are congruent, then you would use the SSS congruence theorem.
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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.
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Again, angles are congruent; sides are proportional; here we are talking about sides, so then it would have to be sides being proportional.
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The measures of corresponding sides (all three sides) have to be proportional to the corresponding sides of the other triangle.
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Then, the two triangles are similar by this SSS similarity.
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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.
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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;
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then I know that these two would be similar, because each corresponding pair of sides have the same scale factor; they are proportional.
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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.
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And then, the last two...the pair is 8:16; see how they are all equal to 1/2.
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That means that they all have the same scale factor, which means that they are proportional.
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And therefore, if all three sides are proportional to the three sides up here (and it has to be three;
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for the sides, it has to be all three of them), then these two are similar.
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If we had this ratio with this ratio (so then this pair of sides and this pair of sides being congruent,
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having the same ratio), but then, let's say, the third pair of sides wasn't the same;
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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.
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And the third one is the Side-Angle-Side similarity theorem.
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We have an SAS congruence theorem; you remember that one; but it is different.
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Remember again: that is the congruence theorem; that SAS congruence theorem is to prove that two triangles are congruent.
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Angles are congruent; sides are congruent; but in this one, again, sides (and I am writing this
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over and over again for each slide, so that, that way, you remember this) are proportional.
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Two things make it similar: angles are congruent, and sides proportional.
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When we talk about sides here, this S and this S, the sides, have to be proportional.
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And then, for the angle, it has to be congruent.
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So, the Side-Angle-Side similarity theorem is just saying that two sides are proportional to the corresponding sides of the other triangle.
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And then, the included angle, if you remember, is the angle between the two sides.
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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).
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It is triangle ABC with triangle DEF.
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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--
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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,
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then the included angle would be angle B--so that angle, let's say, is 120, and this is 120
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(because, remember, angles have to be congruent), then these two would be similar by SAS similarity.
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Side is proportional to side, side to side, and then the angles.
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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.
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So, 5/7 is this side, and then 5/7 again for this side.
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There are three of them: angle-angle similarity, SSS (side-side-side) similarity, and side-angle-side similarity.
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With those three, let's use them to solve our examples.
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Determine whether each pair of triangles is similar.
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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.
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And so, here are my triangles that I can base corresponding parts to.
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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.
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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.
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6/9 is 2/3, and 7/10.5...if you just want to check those, what you can do...
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OK, let's do this a different way, because you have that decimal, so it is not like you can easily simplify that.
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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.
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Remember: a proportion is when you have two equal ratios.
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I am just going to make them into a proportion, just to see if they are equal ratios.
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I am going to solve them out and see if I get the correct answer.
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Remember: with cross-products, I have to multiply my extremes with my means, so 6 times 10.5.
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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.
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That means that this is a correct proportion, meaning that this ratio equals this ratio; they are equal to each other.
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Then, those two work (so far, so good); and then, we have to try our last pair of corresponding sides.
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That is AC, which is 10, to DF; that is 15.
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Here, this is going to be 2/3; remember how this one was 2/3; so then, if this comes out to 2/3,
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then this also has to come out to 2/3, because they are equal; and this comes out to 2/3.
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So then, for this one, this one is "yes"; they work.
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And normally you can just simplify; but the reason why we had to multiply this out is that, if you have decimals,
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or you have fractions that make it hard for you to just look at it and simplify, then you can just solve it
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as a proportion to see if those two are the correct ratio.
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The next one: now, here I see that I have 95 and 95; so automatically, I know that these are congruent angles.
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Now, I look at the next one; they are not the same--do you automatically assume that they are not similar now?
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No, because we have to check to see if they are even corresponding angles.
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They kind of look like they would be, but you would have to check these triangles.
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Angle A is corresponding to angle D; see, it is not corresponding--this angle is corresponding to this angle.
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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."
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You have to check to see if they are supposed to be the same, first of all.
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I am going to find the missing measure here: 95 + 53 is 148.
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And then, I take 180, and I subtract that, and I get 32 degrees.
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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."
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And for this one, what was the rule there? It was SSS similarity.
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The next one: Determine which triangles are similar.
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Here, we have a couple different shapes; we have three triangles--I have triangle ABF;
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I have the bigger one, triangle ACD; and then, I have this triangle right here, FED.
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And then, I have a parallelogram, parallelogram BEDC.
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Now, of course, I am not going to use a parallelogram to prove anything; I am not going to use that to show similarity.
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But I do need it...I am probably going to need it to determine which triangles are similar.
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Let's see...let's look at our parallelogram: now, we have parallel lines here...
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Or, no; back to the parallelogram: you know that opposite angles are congruent.
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So then, I don't want to say that this whole thing is equal to this whole thing.
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I could, but then here, see how I can say that this angle E is congruent to angle C, because opposite angles are congruent.
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Now, I wouldn't want to say that of angles B and D, because this is cutting into two different triangles.
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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.
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I have an angle; since I have an A, what two things could that be?
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That can be...remember: I can use the three different similarity theorems.
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The two that use an angle are angle-angle similarity and side-angle-side similarity.
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So, we know that we are going to use one of these two.
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And then, the other one: let's see: we can say that these two angles are congruent, because they are vertical angles.
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But then, remember: I need two pairs, so since my first pair, this one right here that I just marked,
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has to do with this big triangle right here (that is an angle from the big triangle and an angle from this triangle),
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let's see if we can find something from those two triangles--another angle from this big one, and another one from this one.
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This one...even though I can say that these are vertical angles, I have two angles here, but then I only have one here.
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So, I can't really use this triangle if only one of the angles is congruent to this triangle.
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That is why I want to try to see about this one, because I need two angles.
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If you look at this very closely, if I extend this out, we have a line...I have two parallel lines.
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And then, try to ignore this line right there, BE, so that all you see is CA and DE.
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And then, right here, you see a transversal.
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Let me just draw it out for you on the side: here is CA; here is DE; and here is a transversal.
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Now, these lines are parallel; that means that we can say that this angle right here is congruent to this angle,
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because alternate interior angles are congruent when the lines are parallel.
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Those are alternate interior angles; so what angles are those now, in here?
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That would be this angle and this angle; see, now we have two angles from this big triangle,
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and they are congruent to the corresponding angles of this triangle right here.
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So, you can say that triangle ACD is similar (be careful that you don't put "congruent") to triangle...
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what is corresponding with A?...D...there is our D...where is C?...E, and F.
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Now, another pair of angles...back to this angle: now, because I solved this angle first (that angle was this angle right here),
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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.
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Now, since we have another pair of congruent angles, I can also say that this triangle right here,
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because it has the two angles congruent to two angles of this triangle (this angle is congruent to this angle,
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and then these are congruent to each other)--that means that these two triangles are also similar.
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I can also say that triangle AFB is similar to this triangle: it is DFE.
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So, because of this theorem, where both of these pairs are, you can say that those are similar--
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this big one with this one, and then the second would be this one with this one.
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The third example: Determine if the statement is true or false; if false, show a counterexample.
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Remember: a counterexample was an example of the opposite; you are showing an example of the statement that is false.
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And here is the statement: If the measures of the sides of a triangle are x, y, and z,
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and the measures of the sides of a second triangle are (x + 1), (y + 1), and (z + 1), the two triangles are similar.
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Here is triangle 1, and say this is triangle 2.
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That means that, if this is x, y, and z, this is (x + 1), (y + 1), and (z + 1); are they similar?
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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.
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Let's say that we are going to use the numbers 4, 5, and 6.
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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.
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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.
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I don't know if they are equal yet...6:7.
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Now, I don't think that they are the same; if you want to double-check, well, let's work with this first.
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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.
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A correct proportion would be that this ratio is equal to this ratio.
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We are going to see if that works; so then, the cross-product: 4 times 6 is 24...equal to 25.
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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.
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Here is my counter-example; it is an example of the statement that shows that it is false.
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And this shows that it is false; so that is my answer, and that is my counterexample; it is false.
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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.
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If I multiplied each one of these by 2, then it would have the same scale factor.
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But again, if you add a number, then it is not going to be the same.
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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.
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This is a proportion, and we have to prove that this proportion is correct;
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so BE to ED, the ratio of this to this, is going to be equal to the ratio of this to this.
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The only thing that I am given is that these two lines are parallel.
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From there, if I have parallel lines, I can say a lot.
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Here, I have parallel lines, and then my transversal; so then, I can say that this angle is congruent to this angle.
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I can also say that this angle is congruent to this angle.
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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.
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Now, from here, I can also say that these two are vertical angles, and they are congruent.
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But I don't need to, because all I need to prove that these two triangles are similar is angle-angle, our two angles.
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So, I don't have to say that; that would be an extra step that is not necessary.
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And why do I want to say that these two triangles are similar?
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It is because, see, look at these parts: this BE is a side from this triangle; that is a part of this triangle;
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ED is a part of this triangle; CE is from this triangle, and EA from this triangle.
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These are the parts of these two triangles, and then these are scale factors; they are ratios.
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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.
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Step 1: Remember: for a two-column proof, you write your statements on one side, and then your reasons.
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What are your reasons for that statement?
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The first statement...
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And if you haven't really understood proofs, just remember that your given statement is your starting step.
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You are starting here, and then this statement right here, the "prove" statement--that is your ending; that is your last step.
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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.
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Then, our first statement is going to be BC parallel to AD; what is the reason for that?--the reason is always "given."
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And then, from there, what did I say about these angles--what angles are congruent?
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So then, here you can't just say angle B, because angle B has all of these different angles here.
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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.
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Angle CBE is congruent to angle ADE; what is the reason for that?--"alternate interior angles theorem."
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You might have to write it out; you might have to say, "If lines are parallel, then alternate interior angles are congruent."
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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.
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If lines are parallel, then alternate interior angles are congruent.
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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.
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So then, just keep it under step 2: Angle BCA is congruent to angle DAE--and again, it is the same reason.
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So then, see how I already have the angle here and the angle here.
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Automatically, my third step is going to be that the triangles are similar.
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So, triangle BCE is similar to triangle DAE; now, the reason for that would be angle-angle similarity.
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And then, my last step (because it doesn't end there; this has to be my last step):
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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,
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is going to have the same ratio as CE, that side, to the side of the other triangle.
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So then, the reason for that: well, corresponding sides of similar triangles (that is a triangle; it looks like an A)
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are...not congruent...remember, what do we know about the sides of similar triangles?...they are proportional.
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So then, that is it; here is our given statement: we start there, and then, given that,
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we are going to take all of the steps that we need to end up here, and that should be our last step.
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That is it for this lesson; thank you for watching Educator.com.