Welcome back.
So where we had left off, we said, OK, we have this angle
here, can we figure out if any of these angles
are equal to it?
Well we know that alternate interior angles on-- this is a
transversal line right here, and these are parallel lines.
So we know alternate interior.
So this is an interior and it's alternate interior is here.
So we know they equal each other.
I'm not going to draw it yet, because sometimes if you forget
alternate interior you could just remember, well,
corresponding angles equal each other.
So you could say that that angle is also
equal to this angle.
And then you can use opposite angles again to kind of get
back to the alternate interior.
I'll show you.
The great thing about math is it's good for people who have
trouble memorizing things, because you have to just
memorize a few things and then everything else just
kind of falls out of it.
But anyway.
So we figured out that this angle is the
same as this angle.
Right?
Because they're alternate interior angles.
And this is its corresponding side.
And then finally, what about this angle here?
I'm going to draw a triple angle.
One, two, three.
What is that one equal to on this triangle?
Well, same reason.
Alternate interior angles of two parallel lines-- and
remember, the only reason we can kind of make this claim is
because I told you at the beginning that this line right
here and this line right here are parallel.
Right?
Otherwise, you couldn't make this claim.
But since they are alternate interior we know that
this is the same angle.
All right.
So we now have shown that these are similar triangles.
And I didn't have to do all three angles.
I could have just done two, and that should have been good
enough for you to know that they're similar.
Because if two are the same then the third also
has to be the same.
And now let's see if we can use this information to
figure out our ratios.
So let's see.
Let's color the sides the same side as the angle so
we don't get confused.
So this side is the orange side.
Right?
This side is the blue.
This side is the red.
OK.
So we have everything color coded.
And it might be confusing you but it's useful, because, as
we'll see, these triangles are actually kind of flipped.
So let's see what we can do.
So we need to figure out this orange side here.
So this orange side here, let's call it x.
So x equals question mark.
This orange side here corresponds to this side here.
Right?
Because it's opposite this angle, which is
equal to this angle.
So they're opposite to the same angle.
So that's how we know they correspond to each other.
So we could say x over 6 is equal to.
And now, what other sides do we know?
Well we know this side here-- we know this 4 side.
Let me do it in that color.
We know this side is 4.
And since we've put x in the numerator on the left-hand
side, and 4 is in the same triangle as this x we're trying
to figure out, we'll put 4 in the numerator on the
right-hand side.
4 over what?
Well what side corresponds to 4?
What is opposite this angle right here?
Well it's this angle.
Right?
So the corresponding side of this side is this side-- is 5.
And now we can solve.
x is equal-- we just multiply both sides by 6.
So you get 24 over 5.
x is equal to 24 over 5.
Not too bad.
And then we could even go further.
We can now figure out what this side is right here.
This magenta side.
Let's call that, I don't know, y.
Not too creative here.
Well y corresponds to this angle.
So y corresponds to this 8 side.
Right?
So we could do y over 8 is equal to-- oh, we could
do a bunch of things.
We could say 4 over 5 or we could do-- let's do 4 over 5,
because we could do 24 over 5 over 6 and that's
kind of confusing.
So we could also do that [UNINTELLIGIBLE]
over 4 over 5.
Multiply both sides by 8.
And you get y is equal to 8 times 4, is what?
32 over 5.
And the reason why I did this example is because I want
to show you that you can't just eyeball.
Sometimes you can, if you get good at it, but it's not always
completely obvious which sides correspond to which sides.
It might have been tempting to say that, I don't know, this
side corresponds to this side or that this side
corresponds to this side.
But you really have to pay attention to which side kind of
matches up with which angles.
So any side that matches up with a certain angle, that same
angle in the other triangle, whatever side is opposite that,
that's its corresponding side.
I use a lot of words, but hopefully you have a
bit of an intuition.
Let's do another one.
First, let's take a triangle and prove to ourselves that the
two triangles are similar.
I like these parallel lines.
Let me do two parallel lines again.
And then this time around-- let's see.
I'm going to draw.
There's a line.
There we go.
First, I said these are parallel lines.
So let me mark them as such.
Parallel lines.
So what we want to do is we want to prove that this
triangle right here is similar to the bigger triangle-- is
similar to this triangle.
This is pretty interesting.
They actually overlap.
Right?
So first of all, do we know any angles of the two triangles
that equal each other?
Well, sure.
They have this angle.
They actually both share the same exact angle in common.
Right?
Because the two triangles overlap at that point.
So what else can we figure out?
So let's see.
I mean, I don't to be tacky without any
colors, but let's see.
We have this angle here.
And what other angles are equal to this angle?
Well, we can use our parallel lines and transversal of
angle rules, or theorems or whatever, and figure it out.
Well this angle corresponds to what?
Well, it corresponds to this angle.
So it's equivalent.
And you got that from your parallel lines.
Right?
So these two are the same.
And then, finally, if I have-- let me pick a good color-- if I
have this angle, draw a triple angle here.
Same thing.
This corresponding angle is going to be right here.
So there.
We know all of the three angles of this triangle are the same.
So this is a similar triangle.
Let's say we know that this side right here-- I'll give
you a little trick question.
From here to here is 5.
And from here to here is 7.
From here to here is-- I don't know; make up a
good number-- is 12.
And from here to here is, let me say, 6.
And I wanted to figure out what this is.
How do we do that?
And I've further made it more confusing by adding all
these squiggly lines.
Well, we already know that these are two
similar triangles.
So we can use that information to do our ratios.
So if we call this is equal to x.
Right?
So what do we know?
We know that this whole side corresponds to what side
on the smaller triangle?
Well, it corresponds to this side.
Right?
It corresponds to here.
So let me draw it in the correct color.
So if we do the orange, this orange corresponds to this.
Right?
Well this orange corresponds to the whole thing.
It corresponds to this whole line.
So if we take the big triangle, the big triangle
side is not just x.
Right?
Because that's not the whole side of the triangle.
It's x plus 5.
That's this whole side.
Right?
x plus 5 over the corresponding side on the smaller triangle.
Well, on the corresponding side of the smaller
triangle it's just this.
It's over 5.
Right?
Is equal to-- and then we could say, well, 12.
Is equal to 12, because this corresponds to this angle
on the big triangle.
Is equal to 12 over what?
Over 6, because this is the smaller triangle.
And then we could solve for that.
This becomes 2.
Right?
You get x plus 5 is equal to 10.
x is equal to 5.
There you go.
That's all the time I have for now.
I hope I helped you understand similar triangles
just a little bit.
I'll see you soon.