Triangle similarity
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Similar Triangle Basics
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Similarity Postulates
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Similar triangles 1
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Similar triangles 2
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Similar Triangle Example Problems
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Similarity Example Problems
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Solving similar triangles 1
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Similarity example where same side plays different roles
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Solving similar triangles 2
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Challenging Similarity Problem
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Finding Area Using Similarity and Congruence
Similarity Postulates Thinking about what we need to know whether two triangles are similar
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- Let's say we have triangle A, B, C, say it looks something like
- I wanna think about the minimum amount of information
- I wanna come up with a couple of postulates
- that we can use to determine whether another triangle
- is similar to triangle A-B-C
- So we already know that if, if all three angle,
- all three of the corresponding angles are congruent
- to the corresponding angles on ABC, that we know
- that we're dealing with congruent triangles
- So, for example, if this is 30 degrees, this angle is 90 degrees
- and this angle right over here is 60 degrees
- And we have another triangle that looks like this,
- that looks like this, it's clearly a smaller triangle,
- but its corresponding angles, so this is 30 degrees,
- this is 90 degrees and this is 60 degrees
- We know that X-Y-Z in this case is going to be similar
- to A-B-C
- So we would know, we would know from
- this because corresponding angles are congruent
- We would know that triangle ABC is similar to triangle XYZ
- And you gotta get the order right to make sure
- that you have the right corresponding angles
- Y corresponds to the 90 degree angle, X correspond
- to the 30 degree angle, A correspond to the 30 degree angle
- so A and X are the first two things
- B and Y, which are the 90 degrees, second two
- and then Z is the last one
- So that's what we know already, if you have three angles
- But do you need three angles?
- If we only knew two end of the angles, would
- that be enough?
- Well, sure
- Cause if you know two angles for a triangle, you know the third
- So for example, if I have another triangle, if I have a triangle
- that looks like this, I mean to look or draw it like this
- And if I told you that only two of the corresponding angles
- are congruent
- So maybe, maybe this angle right here is congruent
- to this angle and that angle right there is congruent to
- that angle
- Is that enough to say that these two triangles are similar?
- Well, sure because in a triangle, if you know two of the angles
- then you know what the last angle has to be
- I you know that this is 30 and you know that is 90
- then you know that this angle has to be 60 degrees
- Whatever you, whatever these two angles are, subtract them
- from 180 and that's going to be this angle
- So in, in general, in order to show similarity, you don't have
- to show three angles are, three corresponding angles
- are congruent
- You really just have to two, show two
- So this it'll be our, the first of our similar, similarity postulates,
- We call it Angle-Angle
- If you can show two that corresponding angles are congruent
- then we're dealing with similar triangles
- So for example, just to put some numbers here, if you showed,
- if this was 30 degrees and we know that on this triangle
- this is 90 degrees right over here
- We know that this triangle right over here is similar to
- that one there
- And you can really just go to the third angle pretty straight,
- in a pretty straight forward way
- You say this third angle is 60 degrees so all three angles
- are the same
- That's one of our constraints for similarity
- Now other thing we, we know about similarity is
- that the ratio between all of the sides are going to be the same
- So for example, if we have another triangle right over here
- Let me draw another triangle
- I'll call this triangle, I'll call this triangle X, Y and Z
- And let's say that we know that the ratio between A B
- and X Y, we know that AB over XY,
- so the ratio between this side and this side
- Notice, we're not saying that they're congruent, we're just saying
- that they're ratio, we're looking at the ratio now
- We're saying that AB over XY, let's say that
- that is equal to BC, BC over YZ
- That is equal to BC over YZ and that is equal to AC,
- that is equal to AC over XZ
- So once again, this is one of, this is one the ways we say,
- Hey! this means similarity
- So if you have all three corresponding sides,
- the ratio between all three corresponding sides are the same
- then we know that we are dealing with similar triangles
- So this is what we call Side-Side-Side Similarity
- and you don't want to get this confused
- with Side-Side-Side Congruent
- So this are all of our similarity postulates
- Similarity postulates or axioms
- or things that we're gonna assume
- and then we're gonna build off of them to solve problems
- and prove other things
- Side-Side-Side, when we're talking about congruence, means
- that the corresponding sides are congruent
- Side-Side-Side for similarity, we're saying
- that the ratio between corresponding sides are going to be the same
- So for example, if this right over here, if this right over here is,
- let's say this right over her is 10 - let me, no, think
- of a bigger number - let's say this is 60, this right over here is 30
- and this right over here is 30 square roots of 3
- And I just made those numbers right cause, you, you,
- what we will soon learn what typical ratios are of
- the sides of 30, 60, 90 triangles
- And let's say, this one over here is 6, 3 and 3 square root of 3
- Notice, AB over XY, AB over XY, 30 square root's of 3
- over 3 square root's of 3 this will be 10, this will be 10
- What is BC over XY?
- 30 divided by 3 is 10
- And what is 60 divided by 6?
- What, or, AC over XZ, AC over XZ,
- Well that's going to be 10
- So in general to go from the corresponding side here
- to the corresponding side there, we always multiply
- by 10 on every side
- So, we're not saying that they're congruent or,
- or we're not saying are the side
- for this Side- Side-Side for similarity
- We're saying that we're really just scaling them up by
- the same amount
- Or another way to think about it,
- the ratio between corresponding sides are the same
- Now what about, what about if we had,
- if we had, let's start another triangle right over here
- Let me draw it like this
- I shall not leave this here so we can have our list
- So let me draw another triangle ABC
- Let's draw another triangle ABC, So this is A,B and C
- And let's say that we know, let's say that we know
- that this side to go, when we go to another triangle,
- when we go to another triangle that,
- we know that XY, that we know
- that XY is AB multiplied by some constant
- So, A, so I can write it over here, XY is equal
- to some constant times AB
- Actually let me make XY bigger so it actually, it doesn't have to be,
- that constant can be less than 1, in which case
- it would be a smaller value, but let me just do it that way
- So let me just make XY look a little bit bigger
- So let's say that this is X and that is Y
- So let's say that we know, that XY, XY over AB, over AB is equal
- to some constant
- Or if you multiply both sides by AB,
- you would get XY subscaled up version of AB
- So, you know, maybe this is, maybe AB is 5, XY is 10
- then our constant would be 2
- We scaled it up by a factor of two
- And let's say we also know, we also know
- that angle ABC is congruent to XYZ
- and I'll had another point over here
- So let me draw another side, right over here, so this is Z
- So let's say we also know that angle ABC is congruent to XYZ
- Now let's say we know that the ratio
- between BC and YZ is also this constant
- The ratio between BC and YZ is also equal to the same constant
- So an example where this is 5 and 10, maybe this is 3 and 6
- The constant we're kind of doubling the length of the side
- So is this triangle, is triangle XYZ going to be similar
- Well if you think about it there's only one, if you say
- that this is some multiple, if X, if XY is, is
- the same multiple of AB as YZ is the multiple of BC
- and this, the angle in between is congruent,
- there's only one triangle we can set-up over here
- We're, we're only constraint to one triangle right over here,
- and that, and so we're completely constraining
- the length of this side
- And the length of this side is going to have to be
- that same scale as that over there
- And so we call that Side-Angle-Side Similarity, Side-Angle-Side
- So, once again, we saw SSS and SAS in our congruent's postulates
- but we're saying something different here
- We're saying that in, in SAS,
- if the ratio between one corresponding side and
- the other corresponding, one corres
- if the ratio between corresponding sides of
- the two triangle are the same
- So AB and XY of one corresponding side,
- And then another corresponding side, so
- that's that second side, so that's between BC and YZ
- and the angle between them are congruent
- then we're saying it's similar
- For SAS, for congruency we said that the sides actually had
- to be congruent
- Here, we're saying that the ratio between
- the corresponding sides just has to be the same
- So for example, SAS just to apply it,
- if I have, let me just draw, show some examples here
- So let's say I have an exam a triangle here that is 3,2,4
- And let's say we have another triangle here,
- we have another triangle here that has length, that has length 9,6
- We also know that the angle in between i are congruent
- So that, that angle is equal to that angle
- What SAS ins ins in the similarity world tells you is
- that these triangles are definitely going to be similar triangles
- That we're actually constraining because
- there's actually only one triangle we can draw right over here
- It's the triangle where all of the sides are going to have
- to be scaled up by the same amount
- So there's only one long side right here that we can actually draw
- And that's going to have to be scaled up by three as well
- There's only have,thi thi this is the only one possible triangle,
- if you constrain this side, if you're saying look, this is 3 times
- that side, this is 3 times that side and the angle between them
- is congruent, there's only one triangle we can dra we can make
- And we know there, there is a similar triangle
- there where everything is scaled up by factor of 3
- So that one triangle we can draw, has to be that one similar triangle
- So this is what we're talking about, SAS
- We're not saying that this side is congruent to that side or
- that side is congruent to that side
- We're saying that they're scaled up by the same factor
- If we had another triangle, if we had another triangle
- that looked like this, So maybe this is 9, this is 4
- and the angle between them were congruent
- You couldn't say that they're similar because this side is,
- is scaled up by a factor of 3
- This side is only scaled up by a factor of two
- So this, when we write over there, you could not say
- that it is neccess necessarily similar
- And like wise, if you had a triangle that had length 9 here
- and length 6 there but you did not know, you did not know
- that these two angles are the same
- Once again, you're not constraining this enough
- and you would not know that those two triangles
- are necessarily similar
- Cause you don't know those, that middle angle is the same
- Now you might be saying there's a few other postulates
- that we had, we had, we had A AS when we dealt with congruency
- but if you think about it,
- we've already shown that two angles by themselves
- are enough to show similarity
- So why worry about an angle, angle and a side or
- the ratio between the sides
- Why even worry about that
- And we also had Angle-Side-Angle in congruence
- but once again, we already know that two angles are enough
- so we don't need to throw in this extra side
- So we don't even need this right over here
- So these are going to be our similarity postulates
- And I want to remind you, Side-Side-Side, this is different
- than the Side-Side-Side for congruence
- We're talking about the ratio between corresponding sides
- We're not saying that they're actually congruent
- And here, Side-Angle-Side, it's, it's different
- than the Side-Angle-Side for congruence
- It's kind of related but we're here, we're talking about
- the ratio between the sides, not the actual measures
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