Proof: a log b = log(b^a), log a - log b = log(a/b)

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:03.500
Let's see if we can stumble
our way to another

00:00:03.500 --> 00:00:05.590
logarithm property.

00:00:05.590 --> 00:00:21.090
So let's say that the log
base x of A is equal to B.

00:00:23.690 --> 00:00:37.000
That's the same thing as saying
that x to the B is equal to A.

00:00:37.000 --> 00:00:38.900
Fair enough.

00:00:38.900 --> 00:00:40.260
So what I want to
do is experiment.

00:00:40.260 --> 00:00:43.860
What happens if I multiply this
expression by another variable?

00:00:43.860 --> 00:00:44.690
Let's call it C.

00:00:44.690 --> 00:00:49.580
So I'm going to multiply both
sides of this equation times C.

00:00:49.580 --> 00:00:51.960
And I'll just switch colors
just to keep things

00:00:51.960 --> 00:00:52.630
interesting.

00:00:52.630 --> 00:00:53.740
That's not an x that's a C.

00:00:53.740 --> 00:00:55.120
I should probably just
do a dot instead.

00:00:55.120 --> 00:00:55.860
Times C.

00:00:55.860 --> 00:00:58.610
So I'm going to multiply both
sides of this equation times C.

00:00:58.610 --> 00:01:09.940
So I get C times log base x of
A is equal to-- multiply both

00:01:09.940 --> 00:01:14.130
sides of the equation--
is equal to B times C.

00:01:14.130 --> 00:01:15.250
Fair enough.

00:01:15.250 --> 00:01:17.990
I think you realize I
have not done anything

00:01:17.990 --> 00:01:19.980
profound just yet.

00:01:19.980 --> 00:01:20.580
But let's go back.

00:01:20.580 --> 00:01:22.300
We said that this is the
same thing as this.

00:01:22.300 --> 00:01:24.210
So let's experiment
with something.

00:01:24.210 --> 00:01:27.530
Let's raise this side
to the power of C.

00:01:27.530 --> 00:01:31.120
So I'm going to raise this
side to the power of C.

00:01:31.120 --> 00:01:32.500
That's a kind of caret.

00:01:32.500 --> 00:01:33.950
And when you type
exponents that's what

00:01:33.950 --> 00:01:34.630
you would use, a caret.

00:01:34.630 --> 00:01:36.160
So I'm going to raise
it to the power of C.

00:01:36.160 --> 00:01:44.320
So then, this side is x
to the B to the C power,

00:01:44.320 --> 00:01:47.140
is equal to A to the C.

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All I did is I raised both
sides of this equation

00:01:49.680 --> 00:01:51.430
to the Cth power.

00:01:51.430 --> 00:01:54.890
And what do we know about when
you raise something to an

00:01:54.890 --> 00:01:56.610
exponent and you raise that
whole thing to another

00:01:56.610 --> 00:01:57.710
exponent, what happens
to the exponents?

00:01:57.710 --> 00:02:00.430
Well, that's just exponent
rule and you just multiply

00:02:00.430 --> 00:02:03.640
those two exponents.

00:02:03.640 --> 00:02:12.460
This just implies that x to the
BC is equal to A to the C.

00:02:12.460 --> 00:02:13.350
What can we do now?

00:02:13.350 --> 00:02:14.390
Well, I don't know.

00:02:14.390 --> 00:02:17.920
Let's take the logarithm
of both sides.

00:02:17.920 --> 00:02:20.010
Or let's just write this--
let's not take the

00:02:20.010 --> 00:02:20.990
logarithm of both sides.

00:02:20.990 --> 00:02:23.170
Let's write this as a
logarithm expression.

00:02:23.170 --> 00:02:27.280
We know that x to the BC
is equal to A to the C.

00:02:27.280 --> 00:02:31.240
Well, that's the exact same
thing as saying that the

00:02:31.240 --> 00:02:43.400
logarithm base x of A to
the C is equal to BC.

00:02:43.400 --> 00:02:44.220
Correct?

00:02:44.220 --> 00:02:48.310
Because all I did is I rewrote
this as a logarithm expression.

00:02:48.310 --> 00:02:50.460
And I think now you
realized that something

00:02:50.460 --> 00:02:52.170
interesting has happened.

00:02:52.170 --> 00:02:58.120
That BC, well, of course, it's
the same thing as this BC.

00:02:58.120 --> 00:03:04.350
So this expression must be
equal to this expression.

00:03:04.350 --> 00:03:08.200
And I think we have another
logarithm property.

00:03:08.200 --> 00:03:11.750
That if I have some kind of a
coefficient in front of the

00:03:11.750 --> 00:03:15.740
logarithm where I'm multiplying
the logarithm, so if I have C

00:03:15.740 --> 00:03:20.820
log-- Clog base x of A,
but that's C times the

00:03:20.820 --> 00:03:22.660
logarithm base x of A.

00:03:22.660 --> 00:03:29.730
That equals the log
base x of A to the C.

00:03:29.730 --> 00:03:32.410
So you could take this
coefficient and instead make

00:03:32.410 --> 00:03:36.490
it an exponent on the term
inside the logarithm.

00:03:36.490 --> 00:03:38.850
That is another
logarithm property.

00:03:38.850 --> 00:03:41.220
So let's review what we know
so far about logarithms.

00:03:45.420 --> 00:03:54.320
We know that if I write-- let
me say-- well, let me just with

00:03:54.320 --> 00:03:55.120
the letters I've been using.

00:03:55.120 --> 00:04:04.430
C times logarithm base x of
A is equal to logarithm

00:04:04.430 --> 00:04:07.480
base x of A to the C.

00:04:07.480 --> 00:04:09.710
We know that.

00:04:09.710 --> 00:04:16.880
And we know-- we just learned
that logarithm base x of A plus

00:04:16.880 --> 00:04:23.360
logarithm base x of B is equal
to the logarithm base

00:04:23.360 --> 00:04:27.130
x of A times B.

00:04:27.130 --> 00:04:28.340
Now let me ask you a question.

00:04:28.340 --> 00:04:33.850
What happens if instead of
a positive sign here we

00:04:33.850 --> 00:04:35.530
put a negative sign?

00:04:35.530 --> 00:04:38.180
Well, you could probably figure
it out yourself but we could do

00:04:38.180 --> 00:04:40.320
that same exact proof that
we did in the beginning.

00:04:40.320 --> 00:04:45.000
But in this time we will
set it up with a negative.

00:04:45.000 --> 00:04:53.235
Let's just say that log
base x of A is equal to l.

00:04:53.235 --> 00:04:58.845
Let's say that log base
x of B is equal to m.

00:05:01.690 --> 00:05:14.330
Let's say that log base x of A
divided by B is equal to n.

00:05:14.330 --> 00:05:16.550
How can we write all of these
expressions as exponents?

00:05:16.550 --> 00:05:23.340
Well, this just says that
x to the l is equal to A.

00:05:23.340 --> 00:05:24.190
Let me switch colors.

00:05:24.190 --> 00:05:26.230
That keeps it interesting.

00:05:26.230 --> 00:05:30.840
This is just saying that x
to the m is equal to B.

00:05:30.840 --> 00:05:39.600
And this is just saying that
x to the n is equal to A/B.

00:05:45.000 --> 00:05:46.410
So what can we do here?

00:05:46.410 --> 00:05:49.660
Well what's another
way of writing A/B?

00:05:49.660 --> 00:05:53.580
Well, that's just the same
thing as writing x to the l

00:05:53.580 --> 00:05:59.250
because that's A,
over x to the m.

00:05:59.250 --> 00:05:59.770
That's B.

00:06:03.290 --> 00:06:07.980
And this we know from our
exponent rules-- this could

00:06:07.980 --> 00:06:14.800
also be written as x to the
l, x to the negative m.

00:06:14.800 --> 00:06:20.780
Or that also equals
x to the l minus m.

00:06:20.780 --> 00:06:21.490
So what do we know?

00:06:21.490 --> 00:06:25.020
We know that x to the n is
equal to x to the l minus m.

00:06:31.140 --> 00:06:32.070
Those equal each other.

00:06:32.070 --> 00:06:33.900
I just made a big
equal chain here.

00:06:33.900 --> 00:06:40.226
So we know that n is
equal to l minus m.

00:06:40.226 --> 00:06:42.340
Well, what does that do for us?

00:06:42.340 --> 00:06:44.340
Well, what's another
way of writing n?

00:06:44.340 --> 00:06:47.370
I'm going to do it up here
because I think we have

00:06:47.370 --> 00:06:50.920
stumbled upon another
logarithm rule.

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What's another way
of writing n?

00:06:52.660 --> 00:06:55.160
Well, I did it right here.

00:06:55.160 --> 00:06:57.690
This is another
way of writing n.

00:06:57.690 --> 00:07:07.130
So logarithm base x of
A/B-- this is an x over

00:07:07.130 --> 00:07:09.610
here-- is equal to l.

00:07:09.610 --> 00:07:12.550
l is this right here.

00:07:12.550 --> 00:07:14.640
Log base x of A is equal to l.

00:07:14.640 --> 00:07:22.250
The log base x of A minus m.

00:07:22.250 --> 00:07:24.300
I wrote m right here.

00:07:24.300 --> 00:07:26.390
That's log base x of B.

00:07:30.430 --> 00:07:31.910
There you go.

00:07:31.910 --> 00:07:33.060
I probably didn't
have to prove it.

00:07:33.060 --> 00:07:37.085
You could've probably tried
it out with dividing

00:07:37.085 --> 00:07:38.310
it, but whatever.

00:07:38.310 --> 00:07:46.060
But you know are hopefully
satisfied that we have this new

00:07:46.060 --> 00:07:49.080
logarithm property right there.

00:07:49.080 --> 00:07:52.240
Now I have one more logarithm
property to show you, but I

00:07:52.240 --> 00:07:54.100
don't think I have time to
show it in this video.

00:07:54.100 --> 00:07:56.190
So I will do it in
the next video.

00:07:56.190 --> 00:07:57.811
I'll see you soon.


Proof: log a + log b = log ab

WEBVTT
Kind: captions
Language: en

00:00:00.950 --> 00:00:02.160
Hello.

00:00:02.160 --> 00:00:05.230
Let's do some work on
logarithm properties.

00:00:05.230 --> 00:00:07.700
So, let's just review real
quick what a logarithm even is.

00:00:07.700 --> 00:00:19.230
So if I write, let's say I
write log base x of a is

00:00:19.230 --> 00:00:22.020
equal to, I don't know,
make up a letter, n.

00:00:22.020 --> 00:00:23.550
What does this mean?

00:00:23.550 --> 00:00:35.800
Well, this just means that
x to the n equals a.

00:00:35.800 --> 00:00:37.880
I think we already know that.

00:00:37.880 --> 00:00:40.150
We've learned that in
the logarithm video.

00:00:40.150 --> 00:00:42.860
And so it is very important to
realize that when you evaluate

00:00:42.860 --> 00:00:49.170
a logarithm expression, like
log base x of a, the answer

00:00:49.170 --> 00:00:52.350
when you evaluate, what
you get, is an exponent.

00:00:52.350 --> 00:00:54.231
This n is really
just an exponent.

00:00:54.231 --> 00:00:56.820
This is equal to this thing.

00:00:56.820 --> 00:00:58.910
You could've written
it just like this.

00:00:58.910 --> 00:01:02.190
You could have, because this n
is equal to this, you could

00:01:02.190 --> 00:01:10.140
just write x, it's going to get
a little messy, to the log

00:01:10.140 --> 00:01:13.930
base x of a, is equal to a.

00:01:13.930 --> 00:01:17.000
All I did is I, took this n and
I replaced it with this term.

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And I wanted to write it this
way because I want you to

00:01:19.530 --> 00:01:22.580
really get an intuitive
understanding of the notion

00:01:22.580 --> 00:01:24.390
that a logarithm, when
you evaluate it, it

00:01:24.390 --> 00:01:25.745
really an exponent.

00:01:25.745 --> 00:01:27.420
And we're going to
take that notion.

00:01:27.420 --> 00:01:29.910
And that's where, really,
all of the logarithm

00:01:29.910 --> 00:01:32.380
properties come from.

00:01:32.380 --> 00:01:35.130
So let me just do -- what I
actually want to do is, I

00:01:35.130 --> 00:01:37.760
want to to stumble upon the
logarithm properties

00:01:37.760 --> 00:01:38.540
by playing around.

00:01:38.540 --> 00:01:40.405
And then, later on, I'll
summarize it and then

00:01:40.405 --> 00:01:41.120
clean it all up.

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But I want to show maybe
how people originally

00:01:45.100 --> 00:01:47.040
discovered this stuff.

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So, let's say that x,
let me switch colors.

00:01:52.960 --> 00:01:55.600
I think that that keeps
things interesting.

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So let's say that x to
the l is equal to a.

00:02:05.190 --> 00:02:07.680
Well, if we write that as
a logarithm, that same

00:02:07.680 --> 00:02:14.900
relationship as a logarithm, we
could write that log base x of

00:02:14.900 --> 00:02:19.410
a is equal to l, right?

00:02:19.410 --> 00:02:22.530
I just rewrote what I
wrote on the top line.

00:02:22.530 --> 00:02:25.010
Now, let me switch colors.

00:02:25.010 --> 00:02:33.100
And if I were to say that x to
the m is equal to b, it's the

00:02:33.100 --> 00:02:34.620
same thing, I just
switched letters.

00:02:34.620 --> 00:02:41.980
But that just means that
log base x of b is

00:02:41.980 --> 00:02:43.730
equal to m, right?

00:02:43.730 --> 00:02:46.280
I just did the same thing
that I did in this line,

00:02:46.280 --> 00:02:47.452
I just switched letters.

00:02:47.452 --> 00:02:49.620
So let's just keep going
and see what happens.

00:02:49.620 --> 00:02:52.770
So let's say, let me
get another color.

00:02:56.380 --> 00:03:03.010
So let's say I have x to the n,
and you're saying, Sal, where

00:03:03.010 --> 00:03:03.710
are you going with this.

00:03:03.710 --> 00:03:04.710
But you'll see.

00:03:04.710 --> 00:03:12.360
It's pretty neat. x to the
n is equal to a times b.

00:03:12.360 --> 00:03:15.260
x to the n is equal
to a times b.

00:03:15.260 --> 00:03:22.730
And that's just like
saying that log base x

00:03:22.730 --> 00:03:26.420
is equal to a times b.

00:03:26.420 --> 00:03:28.460
So what can we do
with all of this?

00:03:28.460 --> 00:03:31.010
Well, let's start with
with this right here.

00:03:31.010 --> 00:03:33.420
x to the n is equal
to a times b.

00:03:33.420 --> 00:03:35.670
So, how could we rewrite this?

00:03:35.670 --> 00:03:38.910
Well, a is this.

00:03:38.910 --> 00:03:41.670
And b is this, right?

00:03:41.670 --> 00:03:43.010
So let's rewrite that.

00:03:43.010 --> 00:03:49.770
So we know that x to
the n is equal to a.

00:03:49.770 --> 00:03:51.480
a is this.

00:03:51.480 --> 00:03:55.120
x to the l.

00:03:55.120 --> 00:03:57.370
x to the l.

00:03:57.370 --> 00:03:59.500
And what's b?

00:03:59.500 --> 00:04:01.190
Times b.

00:04:01.190 --> 00:04:04.740
Well, b is x to the m, right?

00:04:04.740 --> 00:04:07.380
Not doing anything
fancy right now.

00:04:07.380 --> 00:04:09.320
But what's x to the
l times x to the m?

00:04:09.320 --> 00:04:13.730
Well, we know from the
exponents, when you multiply

00:04:13.730 --> 00:04:17.390
two expressions that have the
same base and different

00:04:17.390 --> 00:04:19.025
exponents, you just
add the exponents.

00:04:19.025 --> 00:04:22.830
So this is equal to, let
me take a neutral color.

00:04:22.830 --> 00:04:24.660
I don't know if I said that
verbally correct, but

00:04:24.660 --> 00:04:25.300
you get the point.

00:04:25.300 --> 00:04:27.560
When you have the same base and
you're multiplying, you can

00:04:27.560 --> 00:04:28.930
just add the exponents.

00:04:28.930 --> 00:04:32.390
That equals x to the, I want to
keep switching colors, because

00:04:32.390 --> 00:04:33.870
I think that's useful.

00:04:33.870 --> 00:04:39.590
l, l plus m.

00:04:39.590 --> 00:04:42.520
That's kind of onerous to
keep switching colors, but.

00:04:42.520 --> 00:04:43.820
You get what I'm saying.

00:04:43.820 --> 00:04:47.590
So, x to the n is equal
to x to the l plus m.

00:04:47.590 --> 00:04:49.790
Let me put the x here.

00:04:49.790 --> 00:04:51.350
Oh, I wanted that to be green.

00:04:51.350 --> 00:04:53.530
x to the l plus n.

00:04:53.530 --> 00:04:54.050
So what do we know?

00:04:54.050 --> 00:04:58.980
We know x to the n is equal
to x to the l plus m.

00:04:58.980 --> 00:05:00.220
Right?

00:05:00.220 --> 00:05:02.510
Well, we have the same base.

00:05:02.510 --> 00:05:06.370
These exponents must
equal each other.

00:05:06.370 --> 00:05:18.863
So we know that n is
equal to l l plus m.

00:05:18.863 --> 00:05:21.270
What does that do for us?

00:05:21.270 --> 00:05:23.590
I've kind of just been playing
around with logarithms.

00:05:23.590 --> 00:05:25.840
Am I getting anywhere?

00:05:25.840 --> 00:05:27.590
I think you'll see that I am.

00:05:27.590 --> 00:05:31.140
Well, what's another
way of writing n?

00:05:31.140 --> 00:05:34.510
So we said, x to the n is
equal to a times b -- oh, I

00:05:34.510 --> 00:05:37.350
actually skipped a step here.

00:05:37.350 --> 00:05:40.080
So that means -- so, going
back here, x to the n

00:05:40.080 --> 00:05:40.710
is equal to a times b.

00:05:40.710 --> 00:05:44.640
That means that log base x
of a times b is equal to n.

00:05:44.640 --> 00:05:45.170
You knew that.

00:05:45.170 --> 00:05:45.890
I didn't.

00:05:45.890 --> 00:05:47.880
I hope you don't realize I'm
not backtracking or anything.

00:05:47.880 --> 00:05:52.360
I just forgot to write that
down when I first did it.

00:05:52.360 --> 00:05:53.250
But, anyway.

00:05:53.250 --> 00:05:54.070
So, what's n?

00:05:54.070 --> 00:05:55.520
What's another way
of writing n?

00:05:55.520 --> 00:05:58.400
Well, another way of
writing n is right here.

00:05:58.400 --> 00:06:01.640
Log base x of a times b.

00:06:01.640 --> 00:06:04.840
So, now we know that if we just
substitute n for that, we

00:06:04.840 --> 00:06:11.690
get log base x of a times b.

00:06:11.690 --> 00:06:13.080
And what does that equal?

00:06:13.080 --> 00:06:14.500
Well, that equals l.

00:06:14.500 --> 00:06:18.230
Another way to write
l is right up here.

00:06:18.230 --> 00:06:25.570
It equals log base
x of a, plus m.

00:06:25.570 --> 00:06:27.710
And what's m?

00:06:27.710 --> 00:06:30.792
m is right here.

00:06:30.792 --> 00:06:35.970
So log base x of b.

00:06:35.970 --> 00:06:38.990
And there we have our
first logarithm property.

00:06:38.990 --> 00:06:44.620
The log base x of a times b --
well that just equals the log

00:06:44.620 --> 00:06:48.130
base x of a plus the
log base x of b.

00:06:48.130 --> 00:06:50.880
And this, hopefully,
proves that to you.

00:06:50.880 --> 00:06:55.460
And if you want the intuition
of why this works out it falls

00:06:55.460 --> 00:07:00.400
from the fact that logarithms
are nothing but exponents.

00:07:00.400 --> 00:07:02.250
So, with that, I'll leave
you with this video.

00:07:02.250 --> 00:07:04.470
And in the next video,
I will prove another

00:07:04.470 --> 00:07:05.900
logarithm property.

00:07:05.900 --> 00:07:07.670
I'll see you soon.


Similar triangles

WEBVTT
Kind: captions
Language: en

00:00:01.040 --> 00:00:01.760
Hello.

00:00:01.760 --> 00:00:06.230
I will now introduce you to the
concept of similar triangles.

00:00:06.230 --> 00:00:07.210
Let me write that down.

00:00:14.150 --> 00:00:16.350
So in everyday life what
does similar mean?

00:00:26.890 --> 00:00:29.470
Well, if two things are similar
they're kind of the same but

00:00:29.470 --> 00:00:32.620
they're not the same thing or
they're not identical, right?

00:00:32.620 --> 00:00:34.650
That's the same thing
for triangles.

00:00:34.650 --> 00:00:40.960
So similar triangles are
two triangles that have

00:00:40.960 --> 00:00:42.270
all the same angles.

00:00:50.460 --> 00:00:57.350
For example, let me draw
two similar triangles.

00:00:57.350 --> 00:00:59.543
I'll try to make them look kind
of the same because they're

00:00:59.543 --> 00:01:02.350
supposed to look kind of the
same, but just maybe

00:01:02.350 --> 00:01:04.980
be different sizes.

00:01:04.980 --> 00:01:12.350
So that's one, and I'll draw
another one that's right here.

00:01:12.350 --> 00:01:13.900
I'm going to draw it a little
smaller to show you that

00:01:13.900 --> 00:01:17.120
they're not necessarily the
same size, they just are

00:01:17.120 --> 00:01:19.980
same shape essentially.

00:01:19.980 --> 00:01:22.020
One way I like to think about
similar triangles are they're

00:01:22.020 --> 00:01:25.080
just triangles that could be
kind of scaled up or down in

00:01:25.080 --> 00:01:28.260
size or flipped around or
rotated, but they all have

00:01:28.260 --> 00:01:30.500
the same angles so they're
essentially the same shape.

00:01:30.500 --> 00:01:33.470
For example, these two
triangles, if I were tell you

00:01:33.470 --> 00:01:36.240
that this angle -- and this is
how they do it in class.

00:01:39.990 --> 00:01:44.270
If I were to tell you this
angle is equal to this angle

00:01:44.270 --> 00:01:49.640
and I told you that this angle
here is equal to this angle.

00:01:52.520 --> 00:01:54.010
Well, a couple of things.

00:01:54.010 --> 00:01:56.020
You already know that this
angle's going to be equal to

00:01:56.020 --> 00:01:58.430
this angle, and why is that?

00:01:58.430 --> 00:02:02.170
Well because if two angles
are the same, then the third

00:02:02.170 --> 00:02:03.400
has to be the same, right?

00:02:03.400 --> 00:02:06.540
Because all three
angles add up to 180.

00:02:06.540 --> 00:02:11.870
For example, if this is x,
this is y, this one has to be

00:02:11.870 --> 00:02:16.060
180 minus x minus y, right?

00:02:16.060 --> 00:02:17.550
That's probably too
small for you to see.

00:02:17.550 --> 00:02:19.300
But that's the same thing here.

00:02:19.300 --> 00:02:23.420
If this is x and this is
y, then this angle right

00:02:23.420 --> 00:02:28.200
here is going to be 180
minus x minus y, right?

00:02:28.200 --> 00:02:30.880
So if we know that two angles
are the same in two triangles,

00:02:30.880 --> 00:02:33.712
so we know that the third one's
also going to be to same.

00:02:33.712 --> 00:02:38.270
So we could also say this angle
is identical to this angle.

00:02:38.270 --> 00:02:42.160
And if all the angles are the
same, then we know that we are

00:02:42.160 --> 00:02:45.970
dealing with similar triangles.

00:02:45.970 --> 00:02:49.590
What useful thing can we
now do once we know that

00:02:49.590 --> 00:02:51.320
a triangle is similar?

00:02:51.320 --> 00:02:54.150
Well, we can use that
information to kind of figure

00:02:54.150 --> 00:02:55.690
out some of the sides.

00:02:55.690 --> 00:03:00.210
So, even though they don't have
the same sides, the ratio

00:03:00.210 --> 00:03:03.550
of corresponding side
lengths is the same.

00:03:03.550 --> 00:03:04.750
I know I've just confused you.

00:03:04.750 --> 00:03:07.340
Let me give you an example.

00:03:07.340 --> 00:03:15.970
For example, let's say that
this side is -- this side is 5.

00:03:15.970 --> 00:03:19.167
Let's say that this side is,
I don't know, I'm just going

00:03:19.167 --> 00:03:21.370
to make up some number, 6.

00:03:21.370 --> 00:03:26.630
And let's say that this
side is 7, right?

00:03:26.630 --> 00:03:30.840
And let's say we know that, I
don't know, let's say we know

00:03:30.840 --> 00:03:34.970
that this side here is 2.

00:03:37.990 --> 00:03:40.180
So we know the ratio
of corresponding

00:03:40.180 --> 00:03:40.950
sides is the same.

00:03:40.950 --> 00:03:43.990
So, if we look at these two
triangles, they have completely

00:03:43.990 --> 00:03:47.400
different sizes but they
have corresponding sides.

00:03:47.400 --> 00:03:53.010
For example, this side
corresponds to this side.

00:03:53.010 --> 00:03:54.130
How do we know that?

00:03:54.130 --> 00:03:55.560
Well, in this case, they
just happen to have

00:03:55.560 --> 00:03:56.340
the same orientation.

00:03:56.340 --> 00:03:59.330
But we know that because
these sides are opposite

00:03:59.330 --> 00:04:00.940
the same angle, right?

00:04:00.940 --> 00:04:03.940
This is opposite angle y,
and then this side is

00:04:03.940 --> 00:04:05.350
opposite angle y again.

00:04:05.350 --> 00:04:07.850
This whole triangle might be
too small for you to see, but

00:04:07.850 --> 00:04:09.650
hopefully you're getting
what I'm saying.

00:04:09.650 --> 00:04:12.180
So these are
corresponding sides.

00:04:12.180 --> 00:04:20.490
Similarly, this side, this
blue side, and this blue side

00:04:20.490 --> 00:04:21.730
are corresponding sides.

00:04:21.730 --> 00:04:22.160
Why?

00:04:22.160 --> 00:04:25.180
Not because they're kind of on
the top left because we could

00:04:25.180 --> 00:04:27.940
have rotated this and flipped
it and whatever else.

00:04:27.940 --> 00:04:29.980
It's because it's
opposite the same angle.

00:04:32.810 --> 00:04:33.895
That's the way I always
think about triangles.

00:04:33.895 --> 00:04:35.160
It's a good way to think about
it, especially when you

00:04:35.160 --> 00:04:37.100
start doing trigonometry.

00:04:37.100 --> 00:04:39.310
So what does that us?

00:04:39.310 --> 00:04:42.220
Well, the ratio between
corresponding sides

00:04:42.220 --> 00:04:43.810
is always the same.

00:04:43.810 --> 00:04:48.270
So let's say we want to figure
out how long this side of

00:04:48.270 --> 00:04:50.110
the small triangle is.

00:04:50.110 --> 00:04:52.040
Well there's a bunch of
ways we could do it.

00:04:52.040 --> 00:05:00.450
We could say that the ratio of
this side to this side, so x to

00:05:00.450 --> 00:05:07.505
7 is going to be equal to the
ratio of this side to this side

00:05:07.505 --> 00:05:11.680
-- is equal to the
ratio of 2 to 5.

00:05:11.680 --> 00:05:12.440
And then we could solve it.

00:05:12.440 --> 00:05:14.150
And the only reason why we can
do this -- you can't do this

00:05:14.150 --> 00:05:16.150
with just random triangles, you
can only do this with

00:05:16.150 --> 00:05:18.100
similar triangles.

00:05:18.100 --> 00:05:21.090
So we could then solve for x,
multiply both sides but 7 and

00:05:21.090 --> 00:05:26.200
you get x is equal
to 14 over 5.

00:05:26.200 --> 00:05:27.910
So it's a little
bit less than 3.

00:05:27.910 --> 00:05:32.180
So 14 over 5, so 2.8 or
something like that,

00:05:32.180 --> 00:05:33.550
that equals x.

00:05:33.550 --> 00:05:36.640
And we could do the same thing
to figure out this yellow side.

00:05:36.640 --> 00:05:39.200
So if you know two triangles
are similar, you know all the

00:05:39.200 --> 00:05:41.775
sides of one of the triangles,
you know one of the sides of

00:05:41.775 --> 00:05:44.760
the other triangle, you can
figure out all the sides.

00:05:44.760 --> 00:05:47.720
I think I just confused
you with that comment.

00:05:47.720 --> 00:05:50.730
So, this side, so
let's call this y.

00:05:50.730 --> 00:05:52.000
So we could do the same way.

00:05:52.000 --> 00:06:00.230
We could say y over 6 is equal
to 2 over 5, Right Make sure if

00:06:00.230 --> 00:06:02.710
you're doing one triangle's
going to be the denominator

00:06:02.710 --> 00:06:05.260
here, then that same
triangle has to be the

00:06:05.260 --> 00:06:06.520
denominator on the--.

00:06:06.520 --> 00:06:10.400
If one triangle is the
numerator on the left hand side

00:06:10.400 --> 00:06:12.590
of the equal sign, right, so
the smaller one's

00:06:12.590 --> 00:06:13.570
the numerator.

00:06:13.570 --> 00:06:15.900
Then it's also going to be the
numerator on the right hand

00:06:15.900 --> 00:06:18.030
side of the equal sign.

00:06:18.030 --> 00:06:19.620
I just want to make sure
you're consistent that way.

00:06:19.620 --> 00:06:21.870
If you flip it then you're
going to mess everything up.

00:06:21.870 --> 00:06:25.180
And then we can just solve for,
so y is equal to 12 over 5.

00:06:30.736 --> 00:06:33.920
So, let's use this information
about similar triangles

00:06:33.920 --> 00:06:35.300
just to do some problems.

00:06:44.750 --> 00:06:47.680
So let's use some of the
geometry we've already learned.

00:06:47.680 --> 00:06:58.340
I have two parallel lines, then
I have a line like that, then

00:06:58.340 --> 00:07:00.650
I have a line like this.

00:07:00.650 --> 00:07:04.390
What did I say, I said that the
lines are parallel, so this

00:07:04.390 --> 00:07:09.010
line is parallel to this line.

00:07:09.010 --> 00:07:24.990
And I want to know if this side
is length 5, what is -- well,

00:07:24.990 --> 00:07:28.180
let's say this length is length
5, let's say that this length

00:07:28.180 --> 00:07:32.030
is -- let me draw
another color.

00:07:32.030 --> 00:07:37.790
This length is, I
don't know, 8.

00:07:45.370 --> 00:07:48.330
I want to know what
this side is.

00:07:48.330 --> 00:07:52.030
Actually no, let me give you
one more side just to make sure

00:07:52.030 --> 00:07:53.320
you know all of one triangle.

00:07:53.320 --> 00:07:58.090
Let's say that this side is 6,
and what I want to do is I want

00:07:58.090 --> 00:08:05.570
to figure out what this side is
right here, this purple side.

00:08:05.570 --> 00:08:07.540
So how do we do this?

00:08:07.540 --> 00:08:10.390
So before we start using any of
that ratio stuff, we have to

00:08:10.390 --> 00:08:15.610
prove to ourselves and prove in
general, that these are

00:08:15.610 --> 00:08:16.580
similar triangles.

00:08:16.580 --> 00:08:18.280
So how can we do that?

00:08:18.280 --> 00:08:20.510
Let's see if we can figure
out which angles are

00:08:20.510 --> 00:08:23.090
equal to other angles.

00:08:23.090 --> 00:08:26.020
So we have this angle here.

00:08:26.020 --> 00:08:29.330
Is this angle equal to any
of these three angles

00:08:29.330 --> 00:08:30.820
in this triangle?

00:08:30.820 --> 00:08:31.455
Well, yeah sure.

00:08:31.455 --> 00:08:33.990
It's opposite this angle right
here, so this is going to be

00:08:33.990 --> 00:08:37.570
equal to this angle
right here, right?

00:08:37.570 --> 00:08:39.900
So we know that its opposite
side is it's corresponding

00:08:39.900 --> 00:08:43.380
side, so we know that it
corresponds to -- we don't know

00:08:43.380 --> 00:08:46.040
its length, but we know it
corresponds to this

00:08:46.040 --> 00:08:48.170
8 length, right?

00:08:48.170 --> 00:08:50.200
I forgot to give you
some information.

00:08:50.200 --> 00:08:52.860
I forgot to tell you that
this side is -- let me

00:08:52.860 --> 00:08:54.150
give it a neutral color.

00:08:54.150 --> 00:08:56.340
Let's say that this side is 4.

00:08:56.340 --> 00:08:57.470
Let's go back to the problem.

00:08:57.470 --> 00:09:00.340
So we just figured out these
two angles are the same, and

00:09:00.340 --> 00:09:02.570
that this is that angle's
corresponding side.

00:09:02.570 --> 00:09:05.590
Can we figure out any other
angles are the same?

00:09:05.590 --> 00:09:09.430
Let's say we know
what this angle is.

00:09:12.200 --> 00:09:15.100
I'm going to do kind of a
double angle measure here.

00:09:15.100 --> 00:09:18.480
So what angle in this triangle
-- does any angle here

00:09:18.480 --> 00:09:19.990
equal that angle?

00:09:19.990 --> 00:09:20.410
Sure.

00:09:20.410 --> 00:09:23.850
We know that these are parallel
lines, so we can use alternate

00:09:23.850 --> 00:09:26.180
interior angles to figure out
which of these angles

00:09:26.180 --> 00:09:27.830
equals that one.

00:09:27.830 --> 00:09:29.430
But I just saw the time
and I realize I'm

00:09:29.430 --> 00:09:30.390
running out of time.

00:09:30.390 --> 00:09:33.140
So I will continue this
in the next video.


Similar triangles (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:01.810
Welcome back.

00:00:01.810 --> 00:00:04.820
So where we had left off, we
said, OK, we have this angle

00:00:04.820 --> 00:00:07.720
here, can we figure out if
any of these angles

00:00:07.720 --> 00:00:08.930
are equal to it?

00:00:08.930 --> 00:00:14.750
Well we know that alternate
interior angles on-- this is a

00:00:14.750 --> 00:00:17.780
transversal line right here,
and these are parallel lines.

00:00:17.780 --> 00:00:18.940
So we know alternate interior.

00:00:18.940 --> 00:00:21.320
So this is an interior and it's
alternate interior is here.

00:00:21.320 --> 00:00:23.340
So we know they
equal each other.

00:00:23.340 --> 00:00:25.520
I'm not going to draw it yet,
because sometimes if you forget

00:00:25.520 --> 00:00:27.680
alternate interior you could
just remember, well,

00:00:27.680 --> 00:00:29.460
corresponding angles
equal each other.

00:00:29.460 --> 00:00:31.410
So you could say that
that angle is also

00:00:31.410 --> 00:00:32.880
equal to this angle.

00:00:32.880 --> 00:00:35.460
And then you can use opposite
angles again to kind of get

00:00:35.460 --> 00:00:37.660
back to the alternate interior.

00:00:37.660 --> 00:00:38.490
I'll show you.

00:00:38.490 --> 00:00:40.900
The great thing about math is
it's good for people who have

00:00:40.900 --> 00:00:42.520
trouble memorizing things,
because you have to just

00:00:42.520 --> 00:00:45.530
memorize a few things and
then everything else just

00:00:45.530 --> 00:00:46.520
kind of falls out of it.

00:00:46.520 --> 00:00:47.220
But anyway.

00:00:47.220 --> 00:00:51.020
So we figured out that
this angle is the

00:00:51.020 --> 00:00:52.440
same as this angle.

00:00:52.440 --> 00:00:52.650
Right?

00:00:52.650 --> 00:00:55.550
Because they're alternate
interior angles.

00:00:55.550 --> 00:01:00.320
And this is its
corresponding side.

00:01:00.320 --> 00:01:03.030
And then finally, what
about this angle here?

00:01:03.030 --> 00:01:05.270
I'm going to draw
a triple angle.

00:01:05.270 --> 00:01:08.640
One, two, three.

00:01:08.640 --> 00:01:11.400
What is that one equal
to on this triangle?

00:01:11.400 --> 00:01:13.230
Well, same reason.

00:01:13.230 --> 00:01:15.990
Alternate interior angles of
two parallel lines-- and

00:01:15.990 --> 00:01:18.420
remember, the only reason we
can kind of make this claim is

00:01:18.420 --> 00:01:21.810
because I told you at the
beginning that this line right

00:01:21.810 --> 00:01:25.030
here and this line right
here are parallel.

00:01:25.030 --> 00:01:25.310
Right?

00:01:25.310 --> 00:01:27.330
Otherwise, you couldn't
make this claim.

00:01:27.330 --> 00:01:29.010
But since they are alternate
interior we know that

00:01:29.010 --> 00:01:34.570
this is the same angle.

00:01:34.570 --> 00:01:35.560
All right.

00:01:35.560 --> 00:01:39.080
So we now have shown that
these are similar triangles.

00:01:39.080 --> 00:01:40.630
And I didn't have to
do all three angles.

00:01:40.630 --> 00:01:42.960
I could have just done two, and
that should have been good

00:01:42.960 --> 00:01:44.380
enough for you to know
that they're similar.

00:01:44.380 --> 00:01:46.180
Because if two are the
same then the third also

00:01:46.180 --> 00:01:47.370
has to be the same.

00:01:47.370 --> 00:01:49.740
And now let's see if we can
use this information to

00:01:49.740 --> 00:01:51.980
figure out our ratios.

00:01:51.980 --> 00:01:53.560
So let's see.

00:01:53.560 --> 00:01:58.040
Let's color the sides the
same side as the angle so

00:01:58.040 --> 00:01:58.980
we don't get confused.

00:01:58.980 --> 00:02:02.970
So this side is
the orange side.

00:02:02.970 --> 00:02:04.800
Right?

00:02:04.800 --> 00:02:05.810
This side is the blue.

00:02:05.810 --> 00:02:06.390
This side is the red.

00:02:06.390 --> 00:02:06.650
OK.

00:02:06.650 --> 00:02:08.810
So we have everything
color coded.

00:02:08.810 --> 00:02:13.320
And it might be confusing you
but it's useful, because, as

00:02:13.320 --> 00:02:16.220
we'll see, these triangles are
actually kind of flipped.

00:02:16.220 --> 00:02:17.290
So let's see what we can do.

00:02:17.290 --> 00:02:21.470
So we need to figure out
this orange side here.

00:02:21.470 --> 00:02:24.980
So this orange side
here, let's call it x.

00:02:24.980 --> 00:02:28.850
So x equals question mark.

00:02:28.850 --> 00:02:31.820
This orange side here
corresponds to this side here.

00:02:31.820 --> 00:02:32.000
Right?

00:02:32.000 --> 00:02:34.730
Because it's opposite
this angle, which is

00:02:34.730 --> 00:02:36.090
equal to this angle.

00:02:36.090 --> 00:02:38.760
So they're opposite
to the same angle.

00:02:38.760 --> 00:02:40.940
So that's how we know they
correspond to each other.

00:02:40.940 --> 00:02:47.960
So we could say x
over 6 is equal to.

00:02:47.960 --> 00:02:50.260
And now, what other
sides do we know?

00:02:50.260 --> 00:02:53.410
Well we know this side
here-- we know this 4 side.

00:02:53.410 --> 00:02:55.240
Let me do it in that color.

00:02:55.240 --> 00:02:57.310
We know this side is 4.

00:02:57.310 --> 00:02:59.570
And since we've put x in the
numerator on the left-hand

00:02:59.570 --> 00:03:03.070
side, and 4 is in the same
triangle as this x we're trying

00:03:03.070 --> 00:03:04.900
to figure out, we'll put 4 in
the numerator on the

00:03:04.900 --> 00:03:06.590
right-hand side.

00:03:06.590 --> 00:03:09.250
4 over what?

00:03:09.250 --> 00:03:10.880
Well what side
corresponds to 4?

00:03:10.880 --> 00:03:14.290
What is opposite this
angle right here?

00:03:14.290 --> 00:03:15.000
Well it's this angle.

00:03:17.720 --> 00:03:19.050
Right?

00:03:19.050 --> 00:03:24.690
So the corresponding side of
this side is this side-- is 5.

00:03:24.690 --> 00:03:26.310
And now we can solve.

00:03:26.310 --> 00:03:29.010
x is equal-- we just
multiply both sides by 6.

00:03:29.010 --> 00:03:31.310
So you get 24 over 5.

00:03:31.310 --> 00:03:35.745
x is equal to 24 over 5.

00:03:38.760 --> 00:03:40.040
Not too bad.

00:03:40.040 --> 00:03:41.650
And then we could
even go further.

00:03:41.650 --> 00:03:44.170
We can now figure out what
this side is right here.

00:03:44.170 --> 00:03:45.770
This magenta side.

00:03:45.770 --> 00:03:48.340
Let's call that,
I don't know, y.

00:03:48.340 --> 00:03:50.000
Not too creative here.

00:03:50.000 --> 00:03:53.250
Well y corresponds
to this angle.

00:03:53.250 --> 00:03:55.550
So y corresponds
to this 8 side.

00:03:55.550 --> 00:03:57.060
Right?

00:03:57.060 --> 00:04:03.120
So we could do y over 8 is
equal to-- oh, we could

00:04:03.120 --> 00:04:03.680
do a bunch of things.

00:04:03.680 --> 00:04:07.090
We could say 4 over 5 or we
could do-- let's do 4 over 5,

00:04:07.090 --> 00:04:09.870
because we could do 24 over
5 over 6 and that's

00:04:09.870 --> 00:04:10.520
kind of confusing.

00:04:10.520 --> 00:04:11.980
So we could also do
that [UNINTELLIGIBLE]

00:04:11.980 --> 00:04:15.380
over 4 over 5.

00:04:15.380 --> 00:04:17.000
Multiply both sides by 8.

00:04:17.000 --> 00:04:24.770
And you get y is equal
to 8 times 4, is what?

00:04:24.770 --> 00:04:27.160
32 over 5.

00:04:31.980 --> 00:04:33.825
And the reason why I did this
example is because I want

00:04:33.825 --> 00:04:37.170
to show you that you
can't just eyeball.

00:04:37.170 --> 00:04:39.860
Sometimes you can, if you get
good at it, but it's not always

00:04:39.860 --> 00:04:42.710
completely obvious which sides
correspond to which sides.

00:04:42.710 --> 00:04:45.612
It might have been tempting to
say that, I don't know, this

00:04:45.612 --> 00:04:48.270
side corresponds to this
side or that this side

00:04:48.270 --> 00:04:49.500
corresponds to this side.

00:04:49.500 --> 00:04:53.150
But you really have to pay
attention to which side kind of

00:04:53.150 --> 00:04:55.000
matches up with which angles.

00:04:55.000 --> 00:04:58.180
So any side that matches up
with a certain angle, that same

00:04:58.180 --> 00:05:02.610
angle in the other triangle,
whatever side is opposite that,

00:05:02.610 --> 00:05:04.300
that's its corresponding side.

00:05:04.300 --> 00:05:07.800
I use a lot of words, but
hopefully you have a

00:05:07.800 --> 00:05:09.670
bit of an intuition.

00:05:09.670 --> 00:05:12.230
Let's do another one.

00:05:12.230 --> 00:05:16.970
First, let's take a triangle
and prove to ourselves that the

00:05:16.970 --> 00:05:18.160
two triangles are similar.

00:05:20.710 --> 00:05:21.800
I like these parallel lines.

00:05:21.800 --> 00:05:25.830
Let me do two parallel
lines again.

00:05:25.830 --> 00:05:28.520
And then this time
around-- let's see.

00:05:28.520 --> 00:05:31.480
I'm going to draw.

00:05:31.480 --> 00:05:34.450
There's a line.

00:05:34.450 --> 00:05:35.300
There we go.

00:05:39.140 --> 00:05:41.240
First, I said these
are parallel lines.

00:05:41.240 --> 00:05:45.110
So let me mark them as such.

00:05:45.110 --> 00:05:46.220
Parallel lines.

00:05:46.220 --> 00:05:49.990
So what we want to do is we
want to prove that this

00:05:49.990 --> 00:05:58.300
triangle right here is similar
to the bigger triangle-- is

00:05:58.300 --> 00:06:00.310
similar to this triangle.

00:06:00.310 --> 00:06:01.190
This is pretty interesting.

00:06:01.190 --> 00:06:02.490
They actually overlap.

00:06:02.490 --> 00:06:02.830
Right?

00:06:08.070 --> 00:06:10.970
So first of all, do we know any
angles of the two triangles

00:06:10.970 --> 00:06:12.420
that equal each other?

00:06:12.420 --> 00:06:13.010
Well, sure.

00:06:13.010 --> 00:06:13.880
They have this angle.

00:06:13.880 --> 00:06:16.730
They actually both share the
same exact angle in common.

00:06:16.730 --> 00:06:17.230
Right?

00:06:17.230 --> 00:06:20.250
Because the two triangles
overlap at that point.

00:06:20.250 --> 00:06:22.000
So what else can we figure out?

00:06:22.000 --> 00:06:23.950
So let's see.

00:06:23.950 --> 00:06:25.530
I mean, I don't to be
tacky without any

00:06:25.530 --> 00:06:26.930
colors, but let's see.

00:06:26.930 --> 00:06:31.550
We have this angle here.

00:06:31.550 --> 00:06:33.470
And what other angles are
equal to this angle?

00:06:33.470 --> 00:06:37.320
Well, we can use our parallel
lines and transversal of

00:06:37.320 --> 00:06:42.350
angle rules, or theorems or
whatever, and figure it out.

00:06:42.350 --> 00:06:44.860
Well this angle
corresponds to what?

00:06:44.860 --> 00:06:46.620
Well, it corresponds
to this angle.

00:06:46.620 --> 00:06:48.320
So it's equivalent.

00:06:48.320 --> 00:06:49.750
And you got that from
your parallel lines.

00:06:49.750 --> 00:06:50.090
Right?

00:06:50.090 --> 00:06:52.000
So these two are the same.

00:06:52.000 --> 00:06:57.150
And then, finally, if I have--
let me pick a good color-- if I

00:06:57.150 --> 00:06:59.550
have this angle, draw
a triple angle here.

00:06:59.550 --> 00:07:00.110
Same thing.

00:07:00.110 --> 00:07:02.610
This corresponding angle is
going to be right here.

00:07:05.250 --> 00:07:05.830
So there.

00:07:05.830 --> 00:07:10.450
We know all of the three angles
of this triangle are the same.

00:07:10.450 --> 00:07:11.760
So this is a similar triangle.

00:07:16.540 --> 00:07:18.780
Let's say we know that this
side right here-- I'll give

00:07:18.780 --> 00:07:19.920
you a little trick question.

00:07:19.920 --> 00:07:24.430
From here to here is 5.

00:07:24.430 --> 00:07:29.530
And from here to here is 7.

00:07:41.250 --> 00:07:46.825
From here to here is-- I
don't know; make up a

00:07:46.825 --> 00:07:49.820
good number-- is 12.

00:07:49.820 --> 00:08:01.430
And from here to here
is, let me say, 6.

00:08:01.430 --> 00:08:04.920
And I wanted to figure
out what this is.

00:08:04.920 --> 00:08:06.080
How do we do that?

00:08:06.080 --> 00:08:08.720
And I've further made it more
confusing by adding all

00:08:08.720 --> 00:08:10.050
these squiggly lines.

00:08:10.050 --> 00:08:11.460
Well, we already know
that these are two

00:08:11.460 --> 00:08:12.460
similar triangles.

00:08:12.460 --> 00:08:14.910
So we can use that information
to do our ratios.

00:08:14.910 --> 00:08:20.110
So if we call this
is equal to x.

00:08:20.110 --> 00:08:21.700
Right?

00:08:21.700 --> 00:08:23.320
So what do we know?

00:08:23.320 --> 00:08:31.350
We know that this whole side
corresponds to what side

00:08:31.350 --> 00:08:33.250
on the smaller triangle?

00:08:33.250 --> 00:08:34.580
Well, it corresponds
to this side.

00:08:34.580 --> 00:08:34.820
Right?

00:08:34.820 --> 00:08:37.085
It corresponds to here.

00:08:37.085 --> 00:08:39.220
So let me draw it in
the correct color.

00:08:39.220 --> 00:08:42.780
So if we do the orange, this
orange corresponds to this.

00:08:42.780 --> 00:08:44.030
Right?

00:08:44.030 --> 00:08:47.190
Well this orange corresponds
to the whole thing.

00:08:47.190 --> 00:08:49.900
It corresponds to
this whole line.

00:08:49.900 --> 00:08:52.770
So if we take the big
triangle, the big triangle

00:08:52.770 --> 00:08:54.210
side is not just x.

00:08:54.210 --> 00:08:54.490
Right?

00:08:54.490 --> 00:08:55.875
Because that's not the whole
side of the triangle.

00:08:55.875 --> 00:08:56.933
It's x plus 5.

00:09:00.850 --> 00:09:02.060
That's this whole side.

00:09:02.060 --> 00:09:02.450
Right?

00:09:06.116 --> 00:09:11.340
x plus 5 over the corresponding
side on the smaller triangle.

00:09:11.340 --> 00:09:12.660
Well, on the corresponding
side of the smaller

00:09:12.660 --> 00:09:14.630
triangle it's just this.

00:09:14.630 --> 00:09:16.610
It's over 5.

00:09:16.610 --> 00:09:17.870
Right?

00:09:17.870 --> 00:09:22.180
Is equal to-- and then
we could say, well, 12.

00:09:22.180 --> 00:09:25.740
Is equal to 12, because this
corresponds to this angle

00:09:25.740 --> 00:09:27.332
on the big triangle.

00:09:27.332 --> 00:09:30.540
Is equal to 12 over what?

00:09:30.540 --> 00:09:33.980
Over 6, because this is
the smaller triangle.

00:09:33.980 --> 00:09:34.930
And then we could
solve for that.

00:09:34.930 --> 00:09:35.900
This becomes 2.

00:09:35.900 --> 00:09:36.860
Right?

00:09:36.860 --> 00:09:40.936
You get x plus 5
is equal to 10.

00:09:40.936 --> 00:09:43.530
x is equal to 5.

00:09:43.530 --> 00:09:46.300
There you go.

00:09:46.300 --> 00:09:48.560
That's all the time
I have for now.

00:09:48.560 --> 00:09:51.540
I hope I helped you
understand similar triangles

00:09:51.540 --> 00:09:52.580
just a little bit.

00:09:52.580 --> 00:09:54.720
I'll see you soon.


Angle game (part 2)

WEBVTT
Kind: captions
Language: en

00:00:01.010 --> 00:00:02.000
Welcome back.

00:00:02.000 --> 00:00:04.830
Let's do a couple more angle
game problems, and hopefully

00:00:04.830 --> 00:00:07.746
this will make you an
angle game expert.

00:00:07.746 --> 00:00:11.710
So let's start, I have the star
drawn again, and let's say we

00:00:11.710 --> 00:00:13.120
know the following angles.

00:00:13.120 --> 00:00:18.316
We know this angle right
here is 41 degrees.

00:00:21.140 --> 00:00:25.710
We know this angle
here is 113 degrees.

00:00:29.960 --> 00:00:35.920
We know this angle
here is 101 degrees.

00:00:35.920 --> 00:00:40.070
And what we have to figure out
-- this is the goal of this

00:00:40.070 --> 00:00:44.360
angle game -- we want to figure
out what this angle is.

00:00:44.360 --> 00:00:48.300
And like always, I encourage
you to try it on your own.

00:00:48.300 --> 00:00:51.500
Pause the video and then just
try to work it through.

00:00:51.500 --> 00:00:54.520
If you get stuck, then play
the video again and hopefully

00:00:54.520 --> 00:00:56.450
I'll have a solution for you.

00:00:56.450 --> 00:00:59.300
So pause right now, but
otherwise let me explain

00:00:59.300 --> 00:01:00.300
how to do this.

00:01:00.300 --> 00:01:02.895
So let's see, we know this,
this and this, and we're going

00:01:02.895 --> 00:01:05.740
to figure out this angle.

00:01:05.740 --> 00:01:07.060
So how can we figure
out this angle?

00:01:07.060 --> 00:01:08.490
What are the possible
strategies?

00:01:08.490 --> 00:01:10.850
Well, if we knew this
angle here, we could say

00:01:10.850 --> 00:01:12.140
they're supplementary.

00:01:12.140 --> 00:01:14.060
But that angle seems like a
hard angle to figure out

00:01:14.060 --> 00:01:16.610
too, because it's not a
part of any triangles.

00:01:16.610 --> 00:01:19.260
But this angle is a
part of this triangle

00:01:19.260 --> 00:01:20.550
right here, right?

00:01:20.550 --> 00:01:23.940
So if we were able to figure
out this angle and this angle,

00:01:23.940 --> 00:01:26.540
these green angles, if we're
able to figure out these green

00:01:26.540 --> 00:01:32.840
angles, then we could figure
out this brown angle, which is

00:01:32.840 --> 00:01:35.290
the goal of this angle game.

00:01:35.290 --> 00:01:37.920
So, this could also be a good
time to pause because I

00:01:37.920 --> 00:01:40.390
just gave you a hint.

00:01:40.390 --> 00:01:43.930
This green angle, well it's
supplementary to this angle

00:01:43.930 --> 00:01:47.300
right here, so that means it
adds up to 180 degrees, and

00:01:47.300 --> 00:01:49.732
that's clear because it's
on kind of the same line.

00:01:49.732 --> 00:01:52.180
So this is 101 degrees
and this is going to

00:01:52.180 --> 00:01:53.900
be 79 degrees, right?

00:01:53.900 --> 00:01:56.510
So it adds up to 180 degrees.

00:01:56.510 --> 00:01:58.890
That's 79.

00:01:58.890 --> 00:02:03.480
Now how can we figure
out this angle?

00:02:03.480 --> 00:02:07.730
Well, it's kind of left by
itself out in the corner of

00:02:07.730 --> 00:02:10.215
some place, so we could see if
it's part of any triangles.

00:02:10.215 --> 00:02:13.030
But we already said it's
part of this triangle.

00:02:13.030 --> 00:02:14.370
But that doesn't help us
because we don't know

00:02:14.370 --> 00:02:16.330
this angle and that's
actually our goal.

00:02:16.330 --> 00:02:18.560
What other triangles
is it a part of?

00:02:18.560 --> 00:02:21.540
Well, it's a part of this
triangle right here.

00:02:21.540 --> 00:02:24.120
That's why I like the star
problem because it has all

00:02:24.120 --> 00:02:26.190
these triangles in it that
might not be obvious to you the

00:02:26.190 --> 00:02:27.880
first time you look at it.

00:02:27.880 --> 00:02:29.300
But the more you look at you
see all these triangles.

00:02:29.300 --> 00:02:31.760
So it's part of this
triangle, and it's also

00:02:31.760 --> 00:02:33.890
part of this triangle.

00:02:33.890 --> 00:02:38.060
I'm going to draw this triangle
another color because I think

00:02:38.060 --> 00:02:40.260
it'll be clear to you that this
is a useful triangle to see

00:02:40.260 --> 00:02:41.020
that's it's a part of.

00:02:55.670 --> 00:02:57.180
So we have that triangle.

00:02:57.180 --> 00:02:59.790
So do we know two of the
angles of that triangle?

00:02:59.790 --> 00:03:00.230
Well sure.

00:03:00.230 --> 00:03:02.790
We know this angle and
we know this angle.

00:03:02.790 --> 00:03:07.740
So we know that this angle plus
113 plus 41 is going to equal

00:03:07.740 --> 00:03:11.050
180 degrees because of the
three angles of a triangle.

00:03:11.050 --> 00:03:15.870
So let me call this, I
don't know, g for green.

00:03:18.610 --> 00:03:20.850
Let's call this g for green.

00:03:20.850 --> 00:03:28.970
So we know g plus 113 degrees,
that's this one right here,

00:03:28.970 --> 00:03:32.800
plus 41 -- remember, we're
looking at this triangle;

00:03:32.800 --> 00:03:34.810
that's the hardest part just
keeping track of which triangle

00:03:34.810 --> 00:03:39.970
we're looking at -- is
going to equal 180 degrees.

00:03:39.970 --> 00:03:46.300
g plus, what is this, 154?

00:03:46.300 --> 00:03:46.580
Right?

00:03:46.580 --> 00:03:51.140
40, 50, 154 equals 180 degrees.

00:03:51.140 --> 00:03:53.170
That's always where I
mess up on the addition.

00:03:53.170 --> 00:03:57.200
And so g is equal to, what is
this, 26 degrees, right,

00:03:57.200 --> 00:03:59.510
because I just subtract
154 from both sides.

00:04:02.790 --> 00:04:03.890
So we're almost there.

00:04:03.890 --> 00:04:05.940
So we figured out g, we
know this green angle.

00:04:05.940 --> 00:04:07.430
We just have to figure out
this, and they're all part

00:04:07.430 --> 00:04:10.490
of this triangle, this
small one right here.

00:04:10.490 --> 00:04:13.570
This small triangle.

00:04:13.570 --> 00:04:17.460
So our goal, which is
let's call this x.

00:04:17.460 --> 00:04:21.910
x plus g, which is 26 degrees
-- we just figured that out.

00:04:21.910 --> 00:04:26.820
26 plus this angle, 79 -- and
we figured that out because it

00:04:26.820 --> 00:04:29.010
was supplementary to this
angle -- is going to

00:04:29.010 --> 00:04:31.320
equal 180 degrees.

00:04:31.320 --> 00:04:38.536
So x plus, what is this,
105 equal to 180.

00:04:38.536 --> 00:04:43.000
So x is equal to 75 degrees,
if I did my addition and

00:04:43.000 --> 00:04:44.530
subtraction correctly.

00:04:44.530 --> 00:04:49.660
So x is equal to 75 degrees.

00:04:49.660 --> 00:04:51.520
And then we are done.

00:04:51.520 --> 00:04:53.940
Let's do another one
of these problems.

00:04:53.940 --> 00:04:55.300
These problems are all
generated on the [? Card ?]

00:04:55.300 --> 00:04:58.220
Academy website dynamically
by the computer.

00:04:58.220 --> 00:05:01.400
Whoever wrote this software
must be a genius.

00:05:01.400 --> 00:05:02.530
But anyway, back
to the problem.

00:05:07.110 --> 00:05:09.550
Let me draw some more.

00:05:09.550 --> 00:05:13.790
So this is going to be a pretty
straightforward drawing.

00:05:13.790 --> 00:05:18.270
It's pretty much just two
triangles next to each other.

00:05:18.270 --> 00:05:20.860
Like that and then let me draw
another line that goes like

00:05:20.860 --> 00:05:24.160
that, and then we draw a line
that goes like that, and I

00:05:24.160 --> 00:05:25.940
think I have done my drawing.

00:05:25.940 --> 00:05:27.060
There you go.

00:05:27.060 --> 00:05:28.590
I'm have done my drawing.

00:05:28.590 --> 00:05:30.000
So let's see.

00:05:30.000 --> 00:05:32.610
What do we know about this
triangle and what do

00:05:32.610 --> 00:05:34.960
we need to figure out?

00:05:34.960 --> 00:05:39.960
I'm going to tell you that this
angle here, this big angle

00:05:39.960 --> 00:05:42.495
here, is 86 degrees.

00:05:46.360 --> 00:05:50.355
We also know that this
angle here is 28 degrees.

00:05:54.150 --> 00:06:00.590
And we also know that this
angle here is 122 degrees.

00:06:00.590 --> 00:06:04.810
And our goal, our mission
in this round is to figure

00:06:04.810 --> 00:06:06.390
out what this angle is.

00:06:06.390 --> 00:06:08.670
And maybe we can do it, we
can do it in a good color.

00:06:08.670 --> 00:06:12.560
Maybe we can do it in a
couple of different ways.

00:06:12.560 --> 00:06:17.720
So one thing we could do is we
could figure out what this

00:06:17.720 --> 00:06:20.470
angle is, so we could just
subtract this green angle from

00:06:20.470 --> 00:06:23.270
86 and we would get our answer.

00:06:23.270 --> 00:06:25.940
Well, this angle's easy, right,
because we know two angles

00:06:25.940 --> 00:06:28.390
of this triangle, so we
could figure that out.

00:06:28.390 --> 00:06:30.730
Let's just call this, I don't
know, let's call this y.

00:06:30.730 --> 00:06:41.900
So y plus 122 plus 28 degrees
is going to equal 180.

00:06:41.900 --> 00:06:48.420
So y plus 150 is equal to 180.

00:06:48.420 --> 00:06:51.170
So y is equal to 30
degrees, right?

00:06:51.170 --> 00:06:55.780
So this is equal to 30 degrees.

00:06:55.780 --> 00:06:59.980
So this is 30 degrees, and
this big angle here is 86.

00:06:59.980 --> 00:07:06.130
So our goal, let's call that x,
so x is going to just be equal

00:07:06.130 --> 00:07:11.230
to the big angle, 86 minus this
angle we just figured

00:07:11.230 --> 00:07:14.200
out, minus 30.

00:07:14.200 --> 00:07:17.430
So x is going to be
equal to 50 degrees.

00:07:17.430 --> 00:07:17.830
Done.

00:07:17.830 --> 00:07:19.860
That was a pretty
straightforward problem.

00:07:19.860 --> 00:07:22.260
Let's see if we could figure
that out any other way.

00:07:22.260 --> 00:07:25.963
Well, we could say instead
of doing it that way --

00:07:25.963 --> 00:07:28.490
let's forget we just
solved it that way.

00:07:28.490 --> 00:07:33.660
We could say this angle here
is supplementary to this

00:07:33.660 --> 00:07:37.230
122 degree angle, right, so
it has to add up to 180.

00:07:37.230 --> 00:07:41.180
So this plus 122 is 180, so
what does that make this?

00:07:41.180 --> 00:07:46.390
It makes this 58
degrees, right?

00:07:46.390 --> 00:07:47.830
This plus this is
going to be 180.

00:07:47.830 --> 00:07:49.040
So we figured out this.

00:07:49.040 --> 00:07:50.740
If we could figure out
this, then we could

00:07:50.740 --> 00:07:52.190
use this triangle.

00:07:52.190 --> 00:07:54.800
How do we figure
out this angle?

00:07:54.800 --> 00:08:02.340
Well, we could look at this big
triangle here, and we know

00:08:02.340 --> 00:08:06.540
this side, right, and we
could figure out this.

00:08:06.540 --> 00:08:08.410
Let's call this z.

00:08:08.410 --> 00:08:14.130
So we know that z plus this
angle, plus 28, plus this big

00:08:14.130 --> 00:08:18.620
angle, plus 86 is equal to 180.

00:08:18.620 --> 00:08:26.070
So z plus, what is this,
106, 114 is equal to 180.

00:08:26.070 --> 00:08:30.450
So z is equal to, what
is this, 66 degrees.

00:08:30.450 --> 00:08:32.250
I don't know if I'm doing
any of my math correctly,

00:08:32.250 --> 00:08:33.770
but let's just hope.

00:08:33.770 --> 00:08:35.580
z equals 66.

00:08:35.580 --> 00:08:41.000
So z is 66, this angle is 58,
and now we can use this

00:08:41.000 --> 00:08:47.960
triangle here to figure out
what this angle is, our x.

00:08:47.960 --> 00:09:00.842
So x plus 66 plus 58
is equal to 180.

00:09:00.842 --> 00:09:03.440
I already think I might
have made a mistake some

00:09:03.440 --> 00:09:04.150
place in the addition.

00:09:04.150 --> 00:09:08.080
So this time around I get x
is equal to -- let's see,

00:09:08.080 --> 00:09:13.600
66 plus 58 is 110 plus 14.

00:09:13.600 --> 00:09:17.110
So 180 minus 124.

00:09:17.110 --> 00:09:21.600
So now I got it, x is
equal to 56 degrees.

00:09:21.600 --> 00:09:23.350
Oh great, I actually
got the right answer.

00:09:23.350 --> 00:09:24.500
I was looking at this, I
thought it was 50, but this

00:09:24.500 --> 00:09:26.790
was 56, right -- 86 minus 30.

00:09:26.790 --> 00:09:29.150
So x is equal to
56 degrees again.

00:09:29.150 --> 00:09:30.420
So we did it two
different ways.

00:09:30.420 --> 00:09:31.110
That's what I wanted
to show you.

00:09:31.110 --> 00:09:32.790
There's actually not a right
answer, as long as you kind

00:09:32.790 --> 00:09:34.970
of get there eventually.

00:09:34.970 --> 00:09:36.900
We solved it two different ways
and I did all my addition and

00:09:36.900 --> 00:09:40.940
subtraction correctly, and you
get the exact same answer.

00:09:40.940 --> 00:09:45.730
So hopefully you find the angle
game fun and you'll be playing

00:09:45.730 --> 00:09:47.000
this with your friends.

00:09:47.000 --> 00:09:48.640
I'll see you later.


The angle game

WEBVTT
Kind: captions
Language: en

00:00:00.930 --> 00:00:03.630
Let's play the angle game.

00:00:03.630 --> 00:00:07.170
So I've drawn this crazy figure
here and I'm going to give you

00:00:07.170 --> 00:00:09.590
a couple of angles and then I
want you to figure

00:00:09.590 --> 00:00:10.950
out another angle.

00:00:10.950 --> 00:00:13.350
So let me give you some angles.

00:00:13.350 --> 00:00:24.970
So let's say that this angle
up here is 56 degrees.

00:00:28.800 --> 00:00:37.580
Then I also tell you this
angle here is 115 degrees.

00:00:41.970 --> 00:00:45.980
What I would like you to figure
out -- this is the object of

00:00:45.980 --> 00:00:50.040
the angle game -- I want you to
figure out what this

00:00:50.040 --> 00:00:51.960
angle is right here.

00:00:55.990 --> 00:00:59.050
If you are brave, you can
pause the video and try to

00:00:59.050 --> 00:01:00.520
figure it out yourself.

00:01:00.520 --> 00:01:02.815
If you would like me to walk
you through it -- and maybe I

00:01:02.815 --> 00:01:04.520
can give you a couple of steps
and then you pause it and you

00:01:04.520 --> 00:01:05.970
get the rest of it by yourself.

00:01:05.970 --> 00:01:08.410
But I will now show you
how I would have solved

00:01:08.410 --> 00:01:10.810
this in the angle game.

00:01:10.810 --> 00:01:14.370
You have all the tools
necessary to already solve it.

00:01:14.370 --> 00:01:16.420
I want you to be able to get
good at this, because this is

00:01:16.420 --> 00:01:19.800
kind of like the key
skill on the SAT.

00:01:19.800 --> 00:01:22.080
Oh, I didn't give you a key
piece of -- you're probably

00:01:22.080 --> 00:01:23.560
saying I can't solve this.

00:01:23.560 --> 00:01:25.250
You probably can't because
I haven't given you a key

00:01:25.250 --> 00:01:26.760
piece of information.

00:01:26.760 --> 00:01:29.510
This line here and this line
here, so this line and this

00:01:29.510 --> 00:01:32.130
line, they're parallel.

00:01:32.130 --> 00:01:34.330
I was telling you to solve
it before giving you a

00:01:34.330 --> 00:01:35.290
key piece of information.

00:01:35.290 --> 00:01:37.120
That means that
they are parallel.

00:01:37.120 --> 00:01:38.740
So what can we do this figure?

00:01:38.740 --> 00:01:41.500
So whenever I see these type of
problems, either while playing

00:01:41.500 --> 00:01:46.890
the angle game or on, say, an
SAT, I just literally kind of

00:01:46.890 --> 00:01:50.320
figure out every angle that I
can figure out and slowly try

00:01:50.320 --> 00:01:53.570
to make my way to
the goal angle.

00:01:53.570 --> 00:01:54.950
Let's see what we can
figure out here.

00:01:54.950 --> 00:01:58.080
So I'm going to do it in this
blue-green color anything

00:01:58.080 --> 00:02:00.070
that I can figure out.

00:02:00.070 --> 00:02:04.260
So this angle is 56
degrees, right?

00:02:04.260 --> 00:02:06.700
These lines are parallel.

00:02:06.700 --> 00:02:08.535
This line here looks
like a transversal.

00:02:12.300 --> 00:02:14.670
Well, let's see, what's a
corresponding angle to

00:02:14.670 --> 00:02:16.420
this angle right here?

00:02:16.420 --> 00:02:18.830
Well, it's the angle, right?

00:02:18.830 --> 00:02:21.230
What do we know about
corresponding angles for

00:02:21.230 --> 00:02:23.000
parallel lines when you
have a transversal.

00:02:23.000 --> 00:02:30.110
Well, that's 56 degrees because
corresponding angles are equal.

00:02:30.110 --> 00:02:31.320
We could have done a
lot of other stuff.

00:02:31.320 --> 00:02:33.730
We could have figured out that
this angle is 56 degrees, but

00:02:33.730 --> 00:02:37.430
that probably wouldn't have
gotten us closer to our goal.

00:02:37.430 --> 00:02:40.150
That angle's 56 degrees and
its corresponding angle

00:02:40.150 --> 00:02:41.580
is also 56 degrees.

00:02:41.580 --> 00:02:43.460
That wouldn't have gotten
us any closer to our goal.

00:02:43.460 --> 00:02:46.910
We could have figured out that
this is 180 minus 56, right,

00:02:46.910 --> 00:02:49.710
which is, what, 124 degrees.

00:02:49.710 --> 00:02:51.270
That really wouldn't
have helped us much.

00:02:51.270 --> 00:02:53.540
I'm showing you, these are all
things that you can do while

00:02:53.540 --> 00:02:54.870
playing the angle game.

00:02:54.870 --> 00:02:57.380
But anyway, the first step
-- I said well, these are

00:02:57.380 --> 00:03:00.810
corresponding angles,
so that's 56 degrees.

00:03:00.810 --> 00:03:04.160
So let's see, I need to figure
out this angle right here.

00:03:04.160 --> 00:03:07.720
I know this one, and they're
in a triangle, right?

00:03:07.720 --> 00:03:09.120
You see this triangle.

00:03:09.120 --> 00:03:10.885
If only I knew this angle.

00:03:14.570 --> 00:03:16.910
Can you figure out this angle?

00:03:16.910 --> 00:03:22.410
Well, it is supplementary to
this 115 degrees, right?

00:03:22.410 --> 00:03:26.000
So this green angle plus this
purple angle is equal to 180.

00:03:26.000 --> 00:03:29.240
So this is 180 minus 115.

00:03:29.240 --> 00:03:29.870
So what's that?

00:03:29.870 --> 00:03:32.355
180 minus -- so this
is 65 degrees.

00:03:35.110 --> 00:03:35.940
So what have we done so far?

00:03:35.940 --> 00:03:38.060
We just said well these
are parallel lines, so

00:03:38.060 --> 00:03:40.010
corresponding angles are equal.

00:03:40.010 --> 00:03:43.900
So this 56 degrees is
equal to this 56 degrees.

00:03:43.900 --> 00:03:47.240
Then we said, well, this green
angle and this purple angle are

00:03:47.240 --> 00:03:49.110
supplementary, so they
have to add up to 180.

00:03:49.110 --> 00:03:51.010
So this is 115.

00:03:51.010 --> 00:03:55.870
But this is 65, which
is just 180 minus 115.

00:03:55.870 --> 00:03:57.690
I think you might see
where I'm going now.

00:03:57.690 --> 00:04:02.640
Now we know two angles
of a triangle.

00:04:02.640 --> 00:04:05.440
If we know two angles of a
triangle, what can figure

00:04:05.440 --> 00:04:05.960
out about the third?

00:04:05.960 --> 00:04:09.870
Well, we know the angles of a
triangle add up to 180, right?

00:04:09.870 --> 00:04:13.210
So let's called this x.

00:04:13.210 --> 00:04:23.350
We know that x plus 56
plus 65 equals 180.

00:04:23.350 --> 00:04:25.210
What's 56 plus 65?

00:04:25.210 --> 00:04:27.635
This is where I always
mess up, on the addition

00:04:27.635 --> 00:04:28.820
and the subtraction.

00:04:28.820 --> 00:04:32.340
So, 5 plus 6 is 110.

00:04:32.340 --> 00:04:33.990
This is 121 I believe.

00:04:37.400 --> 00:04:40.900
121 equals 180.

00:04:40.900 --> 00:04:43.160
Then x is equal to --
let's see, 180 minus

00:04:43.160 --> 00:04:46.040
20 is 60, so it's 59.

00:04:46.040 --> 00:04:48.570
x is equal to 59 degrees.

00:04:52.710 --> 00:04:53.590
There we go.

00:04:53.590 --> 00:04:59.430
We have accomplished our first
goal in the angle game.

00:04:59.430 --> 00:05:00.090
There you saw it.

00:05:00.090 --> 00:05:03.005
So let's do a tougher
angle problem.

00:05:06.530 --> 00:05:09.580
This one maybe won't
involve parallel lines.

00:05:09.580 --> 00:05:11.400
But I just want to show you,
everything really just boils

00:05:11.400 --> 00:05:15.280
down to everything we learned
about parallel lines and

00:05:15.280 --> 00:05:18.340
triangles and angles
adding up to each other.

00:05:18.340 --> 00:05:20.180
So this one involves a star.

00:05:23.780 --> 00:05:28.600
So, a line from there to there,
draw a line from there to

00:05:28.600 --> 00:05:33.650
there, draw a line from there
to there, draw a line

00:05:33.650 --> 00:05:36.240
from there to there.

00:05:36.240 --> 00:05:39.840
Draw a line from
there to there.

00:05:39.840 --> 00:05:41.510
What do we know about this?

00:05:41.510 --> 00:05:44.400
We know that this angle
is 75 -- oh boy, I'm

00:05:44.400 --> 00:05:45.410
using the wrong tool.

00:05:45.410 --> 00:05:48.750
This angle is 75 degrees.

00:05:48.750 --> 00:05:54.280
We also know that this
angle is 75 degrees.

00:05:54.280 --> 00:06:00.300
We know this angle
here is 101 degrees.

00:06:00.300 --> 00:06:06.640
Your mission in this angle
game is to figure out

00:06:06.640 --> 00:06:08.920
this angle right here.

00:06:08.920 --> 00:06:10.910
What is this angle?

00:06:10.910 --> 00:06:13.030
This is a good time to
pause because I will now

00:06:13.030 --> 00:06:14.550
show you the solution.

00:06:14.550 --> 00:06:15.560
So what can we do here?

00:06:15.560 --> 00:06:19.230
So this angle, well jeez, I
just like to just mess around

00:06:19.230 --> 00:06:20.690
and see what I can figure out.

00:06:20.690 --> 00:06:23.800
So, if this angle here is
101 degrees, what other

00:06:23.800 --> 00:06:24.880
angles can we figure out?

00:06:24.880 --> 00:06:27.550
We could figure out -- well, we
could figure out this angle.

00:06:27.550 --> 00:06:28.540
We could figure out
a bunch of angles.

00:06:28.540 --> 00:06:31.640
We could figure out that -- let
me switch the color, these

00:06:31.640 --> 00:06:33.720
are my figure out angles.

00:06:33.720 --> 00:06:37.010
So that's 101, then this
is supplementary, that's

00:06:37.010 --> 00:06:39.110
79 degrees, right?

00:06:39.110 --> 00:06:44.190
That's also 79 degrees because
this is also supplementary.

00:06:44.190 --> 00:06:46.100
This angle right here is
opposite to it, so this

00:06:46.100 --> 00:06:48.010
angle right here is
going to be 101 degrees.

00:06:52.190 --> 00:06:54.340
What else can figure out?

00:06:54.340 --> 00:06:56.510
We could figure out this angle
because it's supplementary, we

00:06:56.510 --> 00:06:57.910
could figure out this angle.

00:06:57.910 --> 00:06:59.700
We could also figure out this
angle because we see this

00:06:59.700 --> 00:07:01.380
triangle right here.

00:07:01.380 --> 00:07:07.860
This angle plus 75 plus 75 is
going to equal 180, right?

00:07:07.860 --> 00:07:12.180
So let's call this
angle b, b for blue.

00:07:12.180 --> 00:07:19.910
So b plus 75 plus 75 is
going to equal 180.

00:07:19.910 --> 00:07:25.050
And I'm just using this
triangle right here.

00:07:25.050 --> 00:07:33.200
So b plus 150 is equal to 180,
or b is equal to 30 degrees.

00:07:33.200 --> 00:07:35.960
So we're able to
figure this out.

00:07:35.960 --> 00:07:39.120
Now, what will you do if I told
you that we are now ready to

00:07:39.120 --> 00:07:41.280
figure out this yellow angle?

00:07:41.280 --> 00:07:42.260
It might not be obvious to you.

00:07:42.260 --> 00:07:43.965
You kind of have to look at the
triangle in the right way, and

00:07:43.965 --> 00:07:46.850
the SAT will do this to you
all the time, all the time.

00:07:46.850 --> 00:07:49.170
That's why I'm testing
you this way.

00:07:49.170 --> 00:07:51.740
Well, let me give
you a little hint.

00:07:51.740 --> 00:07:52.555
Look at this triangle.

00:07:56.290 --> 00:08:00.360
Non-ideal color, let me do it
in red so it really stands out.

00:08:00.360 --> 00:08:02.590
Look at this triangle.

00:08:02.590 --> 00:08:04.290
I'll tell you, the hardest
thing about these problems is

00:08:04.290 --> 00:08:07.140
just looking at the right
triangle and kind of seeing

00:08:07.140 --> 00:08:11.260
that oh wow, I actually
can figure out something.

00:08:11.260 --> 00:08:14.800
Look at this triangle
right here.

00:08:14.800 --> 00:08:17.380
We know this angle
of it, 101 degrees.

00:08:17.380 --> 00:08:19.040
We know this angle, we
just figured it out,

00:08:19.040 --> 00:08:20.970
it was 30 degrees.

00:08:20.970 --> 00:08:23.210
So all we have left is to
figure out this yellow

00:08:23.210 --> 00:08:24.405
angle, call it x.

00:08:28.230 --> 00:08:35.740
So x plus 101 plus 30 is
equal to 180 degrees because

00:08:35.740 --> 00:08:38.430
the angles in a triangle
add up to 180 degrees.

00:08:38.430 --> 00:08:43.222
So x plus 131 is equal to 180.

00:08:43.222 --> 00:08:45.300
x is equal to what?

00:08:45.300 --> 00:08:47.580
49 degrees.

00:08:47.580 --> 00:08:49.190
There you go.

00:08:49.190 --> 00:08:53.520
We've done the second
problem in the angle game.

00:08:53.520 --> 00:08:56.230
I think that's all of the time
I have now in this video.

00:08:56.230 --> 00:08:58.100
In the next video maybe I'll
do a couple more of these

00:08:58.100 --> 00:08:59.600
angle game problems.

00:08:59.600 --> 00:09:01.030
See you soon.


Angles (part 3)

WEBVTT
Kind: captions
Language: en

00:00:01.020 --> 00:00:01.990
Welcome back.

00:00:01.990 --> 00:00:06.130
We're almost done learning all
the rules or laws of angles

00:00:06.130 --> 00:00:09.420
that we need to start
playing the angle game.

00:00:09.420 --> 00:00:11.550
So let's just teach
you a couple of more.

00:00:11.550 --> 00:00:15.200
So let's say I have two
parallel lines, and you may not

00:00:15.200 --> 00:00:17.700
know what a parallel line is
and I will explain

00:00:17.700 --> 00:00:18.850
it to you now.

00:00:18.850 --> 00:00:23.570
So I have one line like this --
you probably have an intuition

00:00:23.570 --> 00:00:26.330
what a parallel line means.

00:00:26.330 --> 00:00:29.140
That's one of my parallel
lines, and let me make the

00:00:29.140 --> 00:00:32.540
green one the other
parallel line.

00:00:32.540 --> 00:00:34.910
So parallel lines, and I'm
just drawing part of them.

00:00:34.910 --> 00:00:37.320
We assume that they keep on
going forever because these are

00:00:37.320 --> 00:00:42.080
abstract notions -- this light
blue line keeps going and going

00:00:42.080 --> 00:00:44.880
on and on and on off the screen
and same for this green line.

00:00:44.880 --> 00:00:47.930
And parallel lines are two
lines in the same plane.

00:00:47.930 --> 00:00:50.310
And a plane is just kind of
you can kind of use like a

00:00:50.310 --> 00:00:53.270
flat surface is a plane.

00:00:53.270 --> 00:00:56.630
We won't go into
three-dimensional space

00:00:56.630 --> 00:00:58.450
in geometry class.

00:00:58.450 --> 00:01:00.990
But they're on the same plane
and you can view this plane as

00:01:00.990 --> 00:01:03.130
the screen of your computer
right now or the piece of paper

00:01:03.130 --> 00:01:05.610
you're working on that never
intersect each other and

00:01:05.610 --> 00:01:06.960
they're two separate lines.

00:01:06.960 --> 00:01:09.620
Obviously if they were drawn
on top of each other then

00:01:09.620 --> 00:01:11.410
they intersect each
other everywhere.

00:01:11.410 --> 00:01:13.500
So it's really just two
lines on a plane that never

00:01:13.500 --> 00:01:14.640
intersect each other.

00:01:14.640 --> 00:01:15.840
That's a parallel line.

00:01:15.840 --> 00:01:18.210
If you've already learned your
algebra and you're familiar

00:01:18.210 --> 00:01:21.190
with slope, parallel lines are
two lines that have the

00:01:21.190 --> 00:01:22.430
same slope, right?

00:01:22.430 --> 00:01:26.160
They kind of increase or
decrease at the same rate.

00:01:26.160 --> 00:01:27.540
But they have different
y intercepts.

00:01:27.540 --> 00:01:28.800
If you don't know what
I'm talking about,

00:01:28.800 --> 00:01:29.510
don't worry about it.

00:01:29.510 --> 00:01:31.670
I think you know what a
parallel line means.

00:01:31.670 --> 00:01:33.840
You've seen this -- parallel
parking, what's parallel

00:01:33.840 --> 00:01:37.080
parking is when you park a car
right next to another car

00:01:37.080 --> 00:01:39.970
without having the two cars
intersect, because if the cars

00:01:39.970 --> 00:01:42.690
did intersect you would have to
call your insurance company.

00:01:42.690 --> 00:01:44.710
But anyway, so those
are parallel lines.

00:01:44.710 --> 00:01:48.440
The blue and the green
lines are parallel.

00:01:48.440 --> 00:01:51.210
And I will introduce you to
a new complicated geometry

00:01:51.210 --> 00:01:54.050
term called a transversal.

00:01:54.050 --> 00:01:58.800
All a transversal is is
another line that actually

00:01:58.800 --> 00:02:01.940
intersects those two lines.

00:02:01.940 --> 00:02:03.320
That's a transversal.

00:02:03.320 --> 00:02:07.310
Fancy word for something
very simple, transversal.

00:02:07.310 --> 00:02:10.370
Let me write it down just
to write something down.

00:02:10.370 --> 00:02:10.745
Transversal.

00:02:18.690 --> 00:02:23.510
It crosses the other two lines.

00:02:23.510 --> 00:02:25.640
I was thinking of mnemonics
for transversals, but I

00:02:25.640 --> 00:02:27.390
probably was thinking of
things inappropriate.

00:02:27.390 --> 00:02:31.710
But anyway

00:02:31.710 --> 00:02:33.810
going on with the geometry.

00:02:33.810 --> 00:02:36.710
So we have a transversal
that intersects the

00:02:36.710 --> 00:02:38.660
two parallel lines.

00:02:38.660 --> 00:02:40.910
What we're going to do is think
of a bunch of -- and actually

00:02:40.910 --> 00:02:42.060
if it intersects one
of them it's going to

00:02:42.060 --> 00:02:43.320
intersect the other.

00:02:43.320 --> 00:02:44.380
I'll let you think about that.

00:02:44.380 --> 00:02:46.940
There's no way that I can draw
something that intersects one

00:02:46.940 --> 00:02:49.750
parallel line that doesn't
intersect the other, as long as

00:02:49.750 --> 00:02:51.800
this line keeps going forever.

00:02:51.800 --> 00:02:53.790
I think that that might be
pretty obvious to you.

00:02:53.790 --> 00:02:56.690
But what I want to do
is explore the angles

00:02:56.690 --> 00:02:58.640
of a transversal.

00:02:58.640 --> 00:03:03.180
So the first thing I'm
going to do is explore

00:03:03.180 --> 00:03:05.490
the corresponding angles.

00:03:05.490 --> 00:03:08.500
So let's say corresponding
angles are kind of the

00:03:08.500 --> 00:03:10.890
same angle at each of
the parallel lines.

00:03:10.890 --> 00:03:13.000
That's how I think of them.

00:03:13.000 --> 00:03:17.240
So this angle, this angle,
and this angle are

00:03:17.240 --> 00:03:20.260
corresponding angles.

00:03:20.260 --> 00:03:22.890
They kind of play the same
role where the transversal

00:03:22.890 --> 00:03:24.830
intersects each of the lines.

00:03:24.830 --> 00:03:28.820
As you can imagine, and as it
looks from my amazingly neat

00:03:28.820 --> 00:03:31.390
drawing -- I'm normally not
this good -- that these are

00:03:31.390 --> 00:03:32.780
going to be equal
to each other.

00:03:32.780 --> 00:03:38.500
So if this is x, this
is also going to be x.

00:03:38.500 --> 00:03:42.500
If we know that then we could
use, actually the rules that we

00:03:42.500 --> 00:03:44.510
just learned to figure out
everything else about

00:03:44.510 --> 00:03:46.390
all of these lines.

00:03:46.390 --> 00:03:51.740
Because if this is x then what
is this going to be right here?

00:03:51.740 --> 00:03:55.260
What is this angle going
to be in magenta?

00:03:58.970 --> 00:04:00.990
Well, these are opposite
angles, right?

00:04:00.990 --> 00:04:02.785
They're on opposite
side of crossing lines

00:04:02.785 --> 00:04:03.810
so this is also x.

00:04:06.940 --> 00:04:08.410
And similar we can do
the same thing here.

00:04:08.410 --> 00:04:12.030
This is the opposite angle of
this angle, so this is also x.

00:04:18.580 --> 00:04:21.010
Let me pick a good color.

00:04:21.010 --> 00:04:23.520
What is yellow?

00:04:23.520 --> 00:04:26.180
What is this angle going to be?

00:04:26.180 --> 00:04:27.310
Well, just like we
were doing before.

00:04:27.310 --> 00:04:30.090
Look, we have this huge
angle here, right?

00:04:30.090 --> 00:04:33.910
This angle, this whole
angle is 180 degrees.

00:04:33.910 --> 00:04:38.860
So x and this yellow angle are
supplementary, so we could call

00:04:38.860 --> 00:04:47.000
this angle y, and this is
equal to 180 minus x, right?

00:04:47.000 --> 00:04:49.300
And we're just using
supplementary angles here.

00:04:49.300 --> 00:04:53.260
Well, if this angle is y, then
this angle is opposite to y.

00:04:53.260 --> 00:04:57.100
So this angle is also y.

00:04:57.100 --> 00:04:58.560
Fascinating.

00:04:58.560 --> 00:05:03.220
And similarly, if we have x up
here and x is supplementary to

00:05:03.220 --> 00:05:05.920
this angle as well, right?

00:05:05.920 --> 00:05:10.600
So this is equal to 180 minus
x where it also equals y.

00:05:10.600 --> 00:05:15.330
And then opposite angles,
this is also equal to y.

00:05:15.330 --> 00:05:19.170
So there's all sorts of
geometry words and rules that

00:05:19.170 --> 00:05:21.170
fall out of this, and I'll
review them real fast but

00:05:21.170 --> 00:05:22.090
it's really nothing fancy.

00:05:22.090 --> 00:05:23.850
All I did is I started
off with the notion of

00:05:23.850 --> 00:05:24.850
corresponding angles.

00:05:24.850 --> 00:05:28.320
I said well, this x
is equal to this x.

00:05:28.320 --> 00:05:32.350
I said, oh well, if those are
equal to each other, well not

00:05:32.350 --> 00:05:34.810
even if -- I mean if this is x
and this is also x because

00:05:34.810 --> 00:05:37.590
they're opposite, and the
same thing for this.

00:05:37.590 --> 00:05:40.260
Then, well, if this is x and
this is x and those equal

00:05:40.260 --> 00:05:42.750
each other, as they should
because those are also

00:05:42.750 --> 00:05:44.750
corresponding angles.

00:05:44.750 --> 00:05:48.310
These two magenta angles
are playing the same role.

00:05:48.310 --> 00:05:50.270
They're both kind of
the bottom left angle.

00:05:50.270 --> 00:05:51.970
That's how I think about it.

00:05:51.970 --> 00:05:54.420
We went around, we used
supplementary angles to kind

00:05:54.420 --> 00:05:56.820
of derive well, these y
angles are also the same.

00:05:56.820 --> 00:05:59.000
This y angle is equal to
this y angle because

00:05:59.000 --> 00:06:00.290
they're corresponding.

00:06:00.290 --> 00:06:02.270
This y angle is equal to
this y angle because

00:06:02.270 --> 00:06:03.660
it's corresponding.

00:06:03.660 --> 00:06:06.800
So corresponding angles
are equal to each other.

00:06:06.800 --> 00:06:09.820
It makes sense, they're kind
of playing the same role.

00:06:09.820 --> 00:06:12.270
The bottom right, if you look
at the bottom right angle.

00:06:12.270 --> 00:06:14.020
So corresponding
angles are equal.

00:06:22.870 --> 00:06:25.130
That's my shorthand notation.

00:06:25.130 --> 00:06:27.360
And we've really just
derived everything already.

00:06:27.360 --> 00:06:28.650
That's all you really
have to know.

00:06:28.650 --> 00:06:31.040
But if you wanted to kind of
skip a step, you also know

00:06:31.040 --> 00:06:46.530
the alternate interior
angles are equal.

00:06:46.530 --> 00:06:50.320
So what do I mean by
alternate interior angles?

00:06:50.320 --> 00:06:53.980
Well, the interior angles are
kind of the angles that are

00:06:53.980 --> 00:06:57.560
closer to each other in the two
parallel lines, but they're on

00:06:57.560 --> 00:06:59.410
opposite side of
the transversal.

00:06:59.410 --> 00:07:01.850
That's a very complicated way
of saying this orange angle and

00:07:01.850 --> 00:07:03.300
this magenta angle right here.

00:07:03.300 --> 00:07:05.760
These are alternate interior
angles, and we've already

00:07:05.760 --> 00:07:08.630
proved if this is
x then that is x.

00:07:08.630 --> 00:07:11.420
So these are alternate
interior angles.

00:07:11.420 --> 00:07:17.570
This x and then that x
are alternate interior.

00:07:17.570 --> 00:07:22.220
And actually this y and this y
are also alternate interior,

00:07:22.220 --> 00:07:24.120
and we already proved that
they equal each other.

00:07:24.120 --> 00:07:29.520
Then the last term that you'll
see in geometry is alternate --

00:07:29.520 --> 00:07:31.360
I'm not going to write the
whole thing -- alternate

00:07:31.360 --> 00:07:33.800
exterior angle.

00:07:33.800 --> 00:07:37.760
Alternate exterior
angles are also equal.

00:07:37.760 --> 00:07:40.970
That's the angles on the kind
of further away from each other

00:07:40.970 --> 00:07:43.270
on the parallel lines, but
they're still alternate.

00:07:43.270 --> 00:07:48.790
So an example of that is this x
up here and this x down here,

00:07:48.790 --> 00:07:53.540
right, because they're on the
outsides of the two parallel

00:07:53.540 --> 00:07:56.000
lines, right -- one's on
top, one's on bottom.

00:07:56.000 --> 00:07:58.470
Then they're on opposite
sides, on alternate sides,

00:07:58.470 --> 00:07:59.680
of the transversal.

00:07:59.680 --> 00:08:01.720
These are just fancy words,
but I think hopefully

00:08:01.720 --> 00:08:03.770
you have the intuition.

00:08:03.770 --> 00:08:06.410
Corresponding a angles make
the most sense to me.

00:08:06.410 --> 00:08:09.180
Then everything else proves out
just through opposite angles

00:08:09.180 --> 00:08:10.450
and supplementary angles.

00:08:10.450 --> 00:08:18.150
But alternate exterior is
that angle and that angle.

00:08:18.150 --> 00:08:22.880
Then the other alternate
exterior is this y and this y.

00:08:22.880 --> 00:08:23.870
Those are also equal.

00:08:23.870 --> 00:08:27.150
So if you know these, you know
pretty much everything you need

00:08:27.150 --> 00:08:29.190
to know about parallel lines.

00:08:29.190 --> 00:08:32.300
The last thing I'm going to
teach you in order to play the

00:08:32.300 --> 00:08:35.780
geometry game with full force
is just that the angles in a

00:08:35.780 --> 00:08:38.140
triangle add up to 180 degrees.

00:08:41.770 --> 00:08:45.580
So let me just draw a
triangle, a kind of

00:08:45.580 --> 00:08:48.580
random looking triangle.

00:08:48.580 --> 00:08:51.300
That's my random
looking triangle.

00:08:51.300 --> 00:08:57.690
And if this is x, this
is y, and this is z.

00:08:57.690 --> 00:09:01.380
We know that the angles of a
triangle -- x degrees plus y

00:09:01.380 --> 00:09:06.910
degrees plus z degrees are
equal to 180 degrees.

00:09:06.910 --> 00:09:09.580
So if I said that this is
equal to, I don't know, 30

00:09:09.580 --> 00:09:15.240
degrees, this is equal to,
I don't know, 70 degrees.

00:09:15.240 --> 00:09:16.170
Then what does z equal?

00:09:16.170 --> 00:09:23.650
Well, we would say 30 plus 70
plus z is equal to 180, or

00:09:23.650 --> 00:09:27.740
100 plus z is equal to 180.

00:09:27.740 --> 00:09:29.150
Subtract 100 from both sides.

00:09:29.150 --> 00:09:33.480
z would be equal to 80 degrees.

00:09:33.480 --> 00:09:36.150
We'll see variations of this
where you get two of the angles

00:09:36.150 --> 00:09:39.250
and you can use this property
to figure out the third.

00:09:39.250 --> 00:09:41.450
With everything we've now
learned, I think we're

00:09:41.450 --> 00:09:45.290
ready to kind of ease
into the angle game.

00:09:45.290 --> 00:09:47.510
I'll see you in the next video.


Angles (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.950 --> 00:00:03.460
So let's review everything that
we know so far, because it's

00:00:03.460 --> 00:00:05.470
good to keep reviewing.

00:00:05.470 --> 00:00:07.360
Because these are things
you should never forget

00:00:07.360 --> 00:00:08.870
the rest of your life.

00:00:08.870 --> 00:00:14.330
So if I have a line and if I
draw an angle that goes--

00:00:14.330 --> 00:00:16.530
let's say this is the
pivot point, right?

00:00:16.530 --> 00:00:19.790
If I go all the way around
the line, or in a circle,

00:00:19.790 --> 00:00:20.975
that's 360 degrees.

00:00:23.970 --> 00:00:27.430
We learned that there are
360 degrees in a circle.

00:00:27.430 --> 00:00:28.750
Right?

00:00:28.750 --> 00:00:32.440
We also learned that if
I have lines like this.

00:00:32.440 --> 00:00:41.140
If I have two angles-- let
me draw it like that.

00:00:43.710 --> 00:00:49.490
And this is angle x.

00:00:49.490 --> 00:00:50.625
This is angle y.

00:00:54.122 --> 00:00:57.825
x and y are supplementary.

00:01:04.240 --> 00:01:07.630
And that just means that
they add up to 180 degrees.

00:01:07.630 --> 00:01:10.810
x plus y is equal
to 180 degrees.

00:01:10.810 --> 00:01:11.920
And why does that make sense?

00:01:11.920 --> 00:01:16.200
Because look, if we add
up x plus y we have gone

00:01:16.200 --> 00:01:19.450
halfway around the circle.

00:01:19.450 --> 00:01:22.040
So that's 180 degrees, right?

00:01:22.040 --> 00:01:25.000
So this is part of the way, and
this is the rest of the way.

00:01:25.000 --> 00:01:28.620
So x plus y are going
to equal 180 degrees.

00:01:28.620 --> 00:01:31.380
So hopefully we
have learned that.

00:01:31.380 --> 00:01:35.316
And then let me switch colors
for the sake of variety.

00:01:35.316 --> 00:01:39.080
Let me use my line tool.

00:01:39.080 --> 00:01:44.140
If I have-- let's see,
I'm going to draw

00:01:44.140 --> 00:01:46.050
perpendicular lines.

00:01:46.050 --> 00:01:50.220
If I have that line, and
then I have that line.

00:01:50.220 --> 00:01:52.080
And they are perpendicular.

00:01:52.080 --> 00:01:55.270
And then I have another line.

00:01:55.270 --> 00:01:56.870
Let's say it goes like that.

00:01:59.830 --> 00:02:03.730
And then I say that
this is angle x.

00:02:03.730 --> 00:02:05.030
Woops.

00:02:05.030 --> 00:02:07.650
This is angle x.

00:02:07.650 --> 00:02:09.240
And this is angle y.

00:02:12.290 --> 00:02:16.040
Well, I said this line and this
line are perpendicular, right?

00:02:16.040 --> 00:02:18.390
So that means that they
intersect at a 90 degree angle.

00:02:18.390 --> 00:02:20.710
So we know that this whole
thing is 90 degrees.

00:02:24.070 --> 00:02:26.010
And so what do we
know about x plus y?

00:02:26.010 --> 00:02:29.435
Well, x plus y is going
to equal 90 degrees.

00:02:34.130 --> 00:02:41.330
Or we could say that x
and y are complementary.

00:02:41.330 --> 00:02:43.920
And I always get confused
between supplementary

00:02:43.920 --> 00:02:44.670
and complementary.

00:02:44.670 --> 00:02:45.910
You just got to memorize it.

00:02:45.910 --> 00:02:47.470
I don't know if there's
any-- let's see, is

00:02:47.470 --> 00:02:48.650
there any easy way?

00:02:48.650 --> 00:02:51.950
180, supplementary.

00:02:51.950 --> 00:02:56.890
You could say that 180-- 100
starts with an O, which

00:02:56.890 --> 00:02:58.950
supplementary does
not start with.

00:02:58.950 --> 00:03:00.080
So there.

00:03:00.080 --> 00:03:01.500
There's your mnemonic.

00:03:01.500 --> 00:03:02.910
Complementary.

00:03:02.910 --> 00:03:05.100
And 90 starts with an N,
and complementary does

00:03:05.100 --> 00:03:05.920
not start with an N.

00:03:05.920 --> 00:03:07.450
That's your other mnemonic.

00:03:07.450 --> 00:03:07.930
Complementary.

00:03:15.420 --> 00:03:16.590
I don't know if I'm
spelling it right.

00:03:16.590 --> 00:03:17.590
Who cares?

00:03:17.590 --> 00:03:18.750
Let's move on.

00:03:18.750 --> 00:03:20.190
So let's learn some more
stuff about angles.

00:03:20.190 --> 00:03:22.490
And what I'm going to do is I'm
going to give you an arsenal,

00:03:22.490 --> 00:03:25.640
and then once you have that
arsenal you can just tackle

00:03:25.640 --> 00:03:28.280
these beastly problems that
I'm going to throw at you.

00:03:28.280 --> 00:03:31.960
So just take these for granted
right now, and then in a few

00:03:31.960 --> 00:03:34.630
videos, probably, we're
going to tackle some

00:03:34.630 --> 00:03:35.505
beastly problems.

00:03:38.220 --> 00:03:40.480
And you know, I'm
using variables here.

00:03:40.480 --> 00:03:41.486
And if you're not familiar
with variables you

00:03:41.486 --> 00:03:42.135
can put numbers here.

00:03:42.135 --> 00:03:46.050
If x was 30 degrees, then y
is going to be 60 degrees.

00:03:46.050 --> 00:03:46.670
Right?

00:03:46.670 --> 00:03:51.050
Or in this case, if x is, I
don't know, 45 degrees, then y

00:03:51.050 --> 00:03:53.960
is going to be 135 degrees.

00:03:53.960 --> 00:03:54.560
That other way.

00:03:54.560 --> 00:03:59.290
Let me draw another property of
angles of intersecting lines.

00:03:59.290 --> 00:04:05.890
So if I have two angles, two
lines that intersect like this.

00:04:08.560 --> 00:04:10.760
So a couple of
interesting things.

00:04:10.760 --> 00:04:14.695
So first, I'm going to teach
you about opposite angles.

00:04:17.490 --> 00:04:19.510
Let me switch colors.

00:04:19.510 --> 00:04:22.860
Let me switch to yellow.

00:04:22.860 --> 00:04:30.710
So if this is x degrees, then
it turns out that the angle

00:04:30.710 --> 00:04:34.110
opposite to it is also
equal to x degrees.

00:04:40.430 --> 00:04:42.180
And you don't believe me?

00:04:42.180 --> 00:04:44.550
Well let me prove it to you.

00:04:44.550 --> 00:04:49.900
Let's say we call this,
I don't know, let's

00:04:49.900 --> 00:04:52.800
call this y degrees.

00:04:52.800 --> 00:04:53.560
Right?

00:04:53.560 --> 00:04:55.520
And I'm going to prove
to you that the x and

00:04:55.520 --> 00:04:57.020
the y are the same.

00:04:57.020 --> 00:04:58.680
Well what do we know already?

00:04:58.680 --> 00:05:01.640
Let's call this other angle--
and I'm doing this to

00:05:01.640 --> 00:05:10.530
confuse you-- angle z.

00:05:10.530 --> 00:05:14.940
Well what do we know about
angle x and angle z?

00:05:14.940 --> 00:05:17.160
It may not be obvious to you
because I've drawn it slightly

00:05:17.160 --> 00:05:21.350
different, but I'll give you
a small hint with

00:05:21.350 --> 00:05:26.310
an appropriately
interesting color.

00:05:26.310 --> 00:05:31.870
So what angle is this
whole thing right here?

00:05:31.870 --> 00:05:33.550
Well I'm just going
along a line, right?

00:05:33.550 --> 00:05:35.600
That's halfway around a circle.

00:05:35.600 --> 00:05:37.000
So what angle is that?

00:05:37.000 --> 00:05:39.190
Well that's 180 degrees.

00:05:39.190 --> 00:05:41.250
So what does x plus z equal?

00:05:41.250 --> 00:05:44.820
Well, x plus z is going to
equal that larger angle.

00:05:44.820 --> 00:05:54.040
x plus purple z is going to
equal-- I think I'll switch to

00:05:54.040 --> 00:05:57.030
the blue; maybe it's taking too
much time for me to switch--

00:05:57.030 --> 00:05:58.890
is equal to 180 degrees.

00:05:58.890 --> 00:06:03.580
Or x and z are supplementary.

00:06:09.240 --> 00:06:10.630
I've run out of space.

00:06:10.630 --> 00:06:12.900
So what do we know about z?

00:06:12.900 --> 00:06:20.400
Well z is equal to 180 minus x.

00:06:20.400 --> 00:06:20.750
Right?

00:06:20.750 --> 00:06:23.070
Because x plus z is 180.

00:06:23.070 --> 00:06:24.540
Fine.

00:06:24.540 --> 00:06:27.850
Now, what's the relationship
between z and y?

00:06:27.850 --> 00:06:31.770
Well, z and y are
also supplementary.

00:06:31.770 --> 00:06:37.210
Because look, if I
drew this angle here.

00:06:37.210 --> 00:06:38.792
Look at this big angle.

00:06:42.070 --> 00:06:42.690
What angle is that?

00:06:42.690 --> 00:06:45.260
Well once again I'm still going
halfway around the circle.

00:06:45.260 --> 00:06:45.990
Right?

00:06:45.990 --> 00:06:48.390
But now I'm using this
line right here.

00:06:48.390 --> 00:06:51.230
So that's 180 degrees.

00:06:51.230 --> 00:06:56.440
So we know that angle z
plus angle y is also

00:06:56.440 --> 00:06:57.685
equal to 180 degrees.

00:07:05.540 --> 00:07:06.820
Right?

00:07:06.820 --> 00:07:08.930
Or, I don't want to keep
writing it, but z and y

00:07:08.930 --> 00:07:11.670
are also supplementary.

00:07:11.670 --> 00:07:14.500
But we just figured out
that z is 180 minus x.

00:07:14.500 --> 00:07:15.140
Right?

00:07:15.140 --> 00:07:18.960
So let's just substitute
that back in here.

00:07:18.960 --> 00:07:29.420
So we get 180 minus x plus
y is equal to 180 degrees.

00:07:29.420 --> 00:07:31.690
Why don't we subtract 180
degrees from both sides

00:07:31.690 --> 00:07:32.960
of this equation.

00:07:32.960 --> 00:07:39.540
That cancels out, and we get
minus x plus y is equal to 0.

00:07:39.540 --> 00:07:41.730
And then add x to both sides
of this equation, and

00:07:41.730 --> 00:07:46.030
we get y is equal to x.

00:07:46.030 --> 00:07:49.000
That was a very long way of
showing you something that is

00:07:49.000 --> 00:07:52.080
fairly simple-- that opposite
angles are equal to each other.

00:07:52.080 --> 00:07:54.950
So x is equal to y.

00:07:54.950 --> 00:07:56.730
And if you've played around
with this, if you just drew a

00:07:56.730 --> 00:07:59.160
bunch of straight lines and
they intersected at different

00:07:59.160 --> 00:08:02.720
angles, I think when you
eyeball it it would make sense.

00:08:02.720 --> 00:08:06.590
And then similarly, if that's z
then the other opposite angle

00:08:06.590 --> 00:08:15.510
here is also z degrees.

00:08:15.510 --> 00:08:16.790
So what do we know now?

00:08:16.790 --> 00:08:20.870
The total angles in a
circle, 360 degrees.

00:08:20.870 --> 00:08:24.380
When two angles kind of
combine, go halfway around the

00:08:24.380 --> 00:08:28.520
circle-- or they combine,
kind of form a line.

00:08:28.520 --> 00:08:30.530
There's different ways
you can think about it.

00:08:30.530 --> 00:08:31.370
We know they're supplementary.

00:08:31.370 --> 00:08:33.350
They add up to 180 degrees.

00:08:33.350 --> 00:08:34.920
x plus y is 180 degrees.

00:08:34.920 --> 00:08:38.360
If they add up to 90
it's complementary.

00:08:38.360 --> 00:08:39.570
x plus y is 90.

00:08:39.570 --> 00:08:42.020
And then opposite angles
are equal to each other.

00:08:42.020 --> 00:08:42.350
Right?

00:08:42.350 --> 00:08:46.130
This angle is equal
to this angle.

00:08:46.130 --> 00:08:49.310
And then this angle is going to
be equal to this angle for the

00:08:49.310 --> 00:08:50.990
same reason-- because
it's opposite.

00:08:50.990 --> 00:08:54.355
In the next video I'm going
to show you about parallel

00:08:54.355 --> 00:08:55.940
lines and transversals.

00:08:55.940 --> 00:08:59.170
More fancy words for what
I think are fairly

00:08:59.170 --> 00:09:00.520
straightforward concepts.

00:09:00.520 --> 00:09:01.350
I'll see in the next video.