Metallic solids

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:01.543
- [Instructor] Let's talk a little bit

00:00:01.543 --> 00:00:03.290
about metallic solids.

00:00:03.290 --> 00:00:05.810
And here is an example of what

00:00:05.810 --> 00:00:08.350
a metallic solid might look like.

00:00:08.350 --> 00:00:11.080
They tend to be shiny like this.

00:00:11.080 --> 00:00:12.780
Some would say lustrous.

00:00:12.780 --> 00:00:14.970
Some of you might be guessing
maybe this is some type

00:00:14.970 --> 00:00:17.990
of aluminum or silver.

00:00:17.990 --> 00:00:20.280
It actually turns out that this is sodium.

00:00:20.280 --> 00:00:23.760
Our same friend sodium that
we saw bonding with chlorine

00:00:23.760 --> 00:00:27.140
to form sodium chloride
and form ionic solids,

00:00:27.140 --> 00:00:30.360
it can actually bond with
itself with metallic bonds.

00:00:30.360 --> 00:00:31.840
This right over here,

00:00:31.840 --> 00:00:35.330
you might guess is silver or something.

00:00:35.330 --> 00:00:37.830
It actually turns out this is calcium.

00:00:37.830 --> 00:00:38.730
And I know what you're thinking.

00:00:38.730 --> 00:00:41.260
Isn't calcium kind of
this chalky white powder?

00:00:41.260 --> 00:00:43.920
Well no, those are compounds
formed with calcium,

00:00:43.920 --> 00:00:46.280
things like calcium oxide.

00:00:46.280 --> 00:00:48.240
But this right over here is pure calcium.

00:00:48.240 --> 00:00:50.200
And the reason why it has
to be in this container,

00:00:50.200 --> 00:00:52.410
it is highly reactive with oxygen.

00:00:52.410 --> 00:00:54.550
So that's not oxygen that
is in this container.

00:00:54.550 --> 00:00:56.680
It's some form of inert gas.

00:00:56.680 --> 00:00:58.730
But calcium when it just bonds to itself

00:00:58.730 --> 00:01:01.630
with metallic bonds, which we'll
talk about in a little bit,

00:01:01.630 --> 00:01:03.460
it also looks kind of similar.

00:01:03.460 --> 00:01:08.030
It's this shiny, metallic,
or lustrous look to it.

00:01:08.030 --> 00:01:10.260
And what do you think this is?

00:01:10.260 --> 00:01:12.230
Well this is something
we're used to associating

00:01:12.230 --> 00:01:14.220
with metals, this is gold.

00:01:14.220 --> 00:01:17.640
But once again, you can see
it has this lustrous property.

00:01:17.640 --> 00:01:21.200
So what is it about
metals or metallic solids

00:01:21.200 --> 00:01:23.160
that allow them to be lustrous in this way

00:01:23.160 --> 00:01:25.680
and have other properties
that we're about to see?

00:01:25.680 --> 00:01:28.210
And to understand that,
we just have to look at

00:01:28.210 --> 00:01:31.000
the periodic table of elements.

00:01:31.000 --> 00:01:33.870
And that most of the
periodic table of elements

00:01:33.870 --> 00:01:36.310
is actually some form of metal.

00:01:36.310 --> 00:01:39.360
You have in red right over
here, this group one elements,

00:01:39.360 --> 00:01:40.760
not including hydrogen.

00:01:40.760 --> 00:01:42.320
Those are your alkali metals,

00:01:42.320 --> 00:01:43.720
and you have your alkaline earth metals,

00:01:43.720 --> 00:01:44.870
your transition metals,

00:01:44.870 --> 00:01:47.610
your post-transition
metals, your metalloids.

00:01:47.610 --> 00:01:50.090
It's really only what you
see in yellow and blue here

00:01:50.090 --> 00:01:52.550
that are not your metals.

00:01:52.550 --> 00:01:55.950
So how do metals form
solids when you just have

00:01:55.950 --> 00:01:57.940
a pure sample of them?

00:01:57.940 --> 00:02:01.470
Well the general idea, you can
look at your alkali metals,

00:02:01.470 --> 00:02:03.850
they all have that one valence electron.

00:02:03.850 --> 00:02:05.450
And to get to that stable outer shell,

00:02:05.450 --> 00:02:08.800
it's much easier for them to
give away a valence electron.

00:02:08.800 --> 00:02:11.850
And that's why we often see
these folks are dissipating

00:02:11.850 --> 00:02:13.050
in ionic bonds.

00:02:13.050 --> 00:02:15.150
They can be ionized quite easily.

00:02:15.150 --> 00:02:17.240
But if you have a pure sample of them,

00:02:17.240 --> 00:02:18.870
they can contribute electrons

00:02:18.870 --> 00:02:21.510
to a sea of electron, one each.

00:02:21.510 --> 00:02:23.250
These alkaline earth metals,

00:02:23.250 --> 00:02:24.910
they have two valence electrons.

00:02:24.910 --> 00:02:27.990
They too can be ionized or
if you have a pure sample

00:02:27.990 --> 00:02:31.990
like in a calcium, they can
contribute two valence electrons

00:02:31.990 --> 00:02:33.400
to a sea of electrons.

00:02:33.400 --> 00:02:35.760
And the transition metals
here have a similar ability

00:02:35.760 --> 00:02:37.960
to contribute valence electrons.

00:02:37.960 --> 00:02:41.450
And so in general, we
can view metallic solids

00:02:41.450 --> 00:02:46.130
as having cations, these
positively charged cations

00:02:47.720 --> 00:02:50.850
in a sea of electrons.

00:02:50.850 --> 00:02:53.400
So you have all these electrons here.

00:02:53.400 --> 00:02:56.110
I'll just draw all these
minus charges that they're in.

00:02:56.110 --> 00:02:58.070
Where do those electrons come from?

00:02:58.070 --> 00:03:00.270
Well if you're looking
at the alkali metals,

00:03:00.270 --> 00:03:04.610
each of those atoms could
give one electron to that sea

00:03:04.610 --> 00:03:07.060
because it doesn't really
want that valence electron.

00:03:07.060 --> 00:03:08.750
If you're talking about
alkaline earth metals,

00:03:08.750 --> 00:03:12.050
they could each donate
two electrons to that sea.

00:03:12.050 --> 00:03:14.670
Now given that you have
this positive charge

00:03:14.670 --> 00:03:16.570
in this sea of electrons,

00:03:16.570 --> 00:03:18.400
what are you think of the properties?

00:03:18.400 --> 00:03:19.470
How good do you think this will be

00:03:19.470 --> 00:03:21.350
at conducting electricity or heat?

00:03:21.350 --> 00:03:23.550
And many of you might guessed,
if you looked at a wire,

00:03:23.550 --> 00:03:26.420
wires are made out of metals,
because they are excellent

00:03:26.420 --> 00:03:29.180
at conducting electricity,
or they tend to be excellent

00:03:29.180 --> 00:03:31.210
at conducting electricity,
because you have

00:03:31.210 --> 00:03:34.330
all of these electrons
that can move around.

00:03:34.330 --> 00:03:37.670
And so if you apply a voltage,
they will start moving

00:03:37.670 --> 00:03:39.060
and conduct electricity.

00:03:39.060 --> 00:03:41.680
And those electrons can
also be good at conducting

00:03:41.680 --> 00:03:43.950
thermal energy or heat.

00:03:43.950 --> 00:03:46.050
Now what would be, we
already talked about them

00:03:46.050 --> 00:03:48.240
having the shiny, lustrous property,

00:03:48.240 --> 00:03:51.050
but how easy would it be to bend them?

00:03:51.050 --> 00:03:52.800
Ionic solids, we talked about

00:03:52.800 --> 00:03:54.300
they can be strong but brittle.

00:03:54.300 --> 00:03:57.170
As soon as you try to shift
them around a little bit,

00:03:57.170 --> 00:03:58.280
they can break.

00:03:58.280 --> 00:04:00.060
But what do you think
is going to happen here?

00:04:00.060 --> 00:04:02.860
If let's say right over here,
I were to push really hard

00:04:02.860 --> 00:04:05.950
and on the top I would have
pushed really hard to the left.

00:04:05.950 --> 00:04:07.170
Do you think this will be brittle?

00:04:07.170 --> 00:04:09.090
Or do you think it will be malleable?

00:04:09.090 --> 00:04:10.600
It's easy to bend.

00:04:10.600 --> 00:04:12.450
Well if you have a pure metallic solid,

00:04:12.450 --> 00:04:14.210
it's actually quite malleable.

00:04:14.210 --> 00:04:18.100
If you just took this
top part and pushed it

00:04:18.100 --> 00:04:21.350
to the left like this, no big deal.

00:04:21.350 --> 00:04:23.550
You have those cations
that are still in those

00:04:23.550 --> 00:04:25.150
that sea of electrons.

00:04:25.150 --> 00:04:28.200
And that's generally
true of metallic solids.

00:04:28.200 --> 00:04:29.510
They're very malleable.

00:04:29.510 --> 00:04:30.720
They are not brittle.

00:04:30.720 --> 00:04:33.430
In fact, so much so that
often times we want them

00:04:33.430 --> 00:04:35.020
to be a little bit more rigid.

00:04:35.020 --> 00:04:37.090
We want them to be a little bit harder.

00:04:37.090 --> 00:04:40.380
And that's why we might do
things like add other elements

00:04:40.380 --> 00:04:42.270
into the metallic solid.

00:04:42.270 --> 00:04:45.080
For example, pure iron
is reasonably malleable.

00:04:45.080 --> 00:04:46.480
But if you wanna make it stronger,

00:04:46.480 --> 00:04:48.660
you could stick carbon atoms in between.

00:04:48.660 --> 00:04:51.140
For example, you could
put a carbon atom there,

00:04:51.140 --> 00:04:53.860
or carbon atom over there.

00:04:53.860 --> 00:04:56.520
And that way, it kind of disrupts

00:04:56.520 --> 00:04:59.660
this electron sea a little bit.

00:04:59.660 --> 00:05:01.660
So it's not quite as malleable.

00:05:01.660 --> 00:05:04.950
It'll be stronger and more rigid.

00:05:04.950 --> 00:05:05.920
So I'll leave you here.

00:05:05.920 --> 00:05:08.170
This is just an extension of
what we've already learned

00:05:08.170 --> 00:05:10.640
about metals and metallic bonds.

00:05:10.640 --> 00:05:13.620
To just realize why most of
the periodic table of elements

00:05:13.620 --> 00:05:16.590
that we're familiar with
has some of these properties

00:05:16.590 --> 00:05:19.483
when they are, when you
have pure solids of them.


Ionic solids

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:02.860
- [Instructor] Let's talk a
little bit about ionic solids,

00:00:02.860 --> 00:00:07.010
which you can imagine are
solids formed by ions.

00:00:07.010 --> 00:00:09.326
So, let's think a little
bit about these ions.

00:00:09.326 --> 00:00:12.610
So, for example, we could look
at group one elements here,

00:00:12.610 --> 00:00:16.430
especially things like lithium,
or sodium, or potassium.

00:00:16.430 --> 00:00:18.340
And in many other videos we have talked

00:00:18.340 --> 00:00:22.640
about these elements
wanting maybe not so much

00:00:22.640 --> 00:00:24.990
to keep their outermost electron

00:00:24.990 --> 00:00:26.490
because they only have one electron

00:00:26.490 --> 00:00:28.210
in their outermost shell.

00:00:28.210 --> 00:00:31.310
And it'd be pretty easy for
them to loose that electron

00:00:31.310 --> 00:00:33.300
to get to a noble gas configuration

00:00:33.300 --> 00:00:35.180
to have a full outer shell.

00:00:35.180 --> 00:00:37.600
So, these characters
like to lose one electron

00:00:37.600 --> 00:00:39.991
the group two elements
like to lose two electrons.

00:00:39.991 --> 00:00:43.130
While if you go on the other
side of the periodic table,

00:00:43.130 --> 00:00:45.990
if you look at the
halogens right over here,

00:00:45.990 --> 00:00:47.770
they're one electron away

00:00:47.770 --> 00:00:50.560
from having a noble gas
electron configuration,

00:00:50.560 --> 00:00:52.740
from having a full outer shell.

00:00:52.740 --> 00:00:55.860
So, they really like to grab electrons.

00:00:55.860 --> 00:00:58.913
And, if you look at elements
like oxygen and sulfur,

00:00:58.913 --> 00:01:03.180
they really like to grab
two electrons, if they can.

00:01:03.180 --> 00:01:06.750
So, what do you think happens
if you have some metals

00:01:06.750 --> 00:01:09.890
on the left end here
mixed with some nonmetals

00:01:09.890 --> 00:01:11.310
on the right end here?

00:01:11.310 --> 00:01:14.670
Well, you might imagine
there would be a reaction.

00:01:14.670 --> 00:01:18.039
So, for example, if you
mixed sodium with chlorine,

00:01:18.039 --> 00:01:22.690
the sodiums might lose an
electron to the chlorines,

00:01:22.690 --> 00:01:26.120
in which case you're going
to have sodium cations,

00:01:26.120 --> 00:01:28.080
positively charged ions.

00:01:28.080 --> 00:01:31.660
And if the chlorines are
now taking those electrons,

00:01:31.660 --> 00:01:34.437
they then become chloride anions.

00:01:35.320 --> 00:01:37.470
And now if you have a
bunch of positive ions

00:01:37.470 --> 00:01:40.750
hanging around a bunch of negative ions,

00:01:40.750 --> 00:01:42.440
what do you think is going to happen?

00:01:42.440 --> 00:01:44.370
They're going to get
attracted to each other.

00:01:44.370 --> 00:01:46.040
And they're going to get
attracted to each other

00:01:46.040 --> 00:01:49.000
and form a lattice structure, like this.

00:01:49.000 --> 00:01:51.330
I like to use sodium
chloride as an example

00:01:51.330 --> 00:01:52.380
because this is probably the one

00:01:52.380 --> 00:01:55.800
that we see most in our
life, this is table salt.

00:01:55.800 --> 00:01:58.290
If you were to lick it, it'd taste salty.

00:01:58.290 --> 00:02:00.304
But, there's many other ionic salt solids,

00:02:00.304 --> 00:02:03.030
many of them would actually be categorized

00:02:03.030 --> 00:02:04.600
as salt, generally.

00:02:04.600 --> 00:02:07.130
You could have a potassium chloride.

00:02:07.130 --> 00:02:10.360
You could have a sodium chloride.

00:02:10.360 --> 00:02:14.330
You could have, for
example, a magnesium oxide.

00:02:14.330 --> 00:02:15.710
What's going on there?

00:02:15.710 --> 00:02:19.750
Well, in that situation, the
magnesium, each magnesium

00:02:19.750 --> 00:02:22.130
might lose two electrons,

00:02:22.130 --> 00:02:26.320
so they become a ion with
a positive two charge,

00:02:26.320 --> 00:02:30.760
and each of the oxygens
would gain two electrons.

00:02:30.760 --> 00:02:35.690
So then they are anions
with a negative two charge.

00:02:35.690 --> 00:02:38.640
And these characters once
again are going to be attracted

00:02:38.640 --> 00:02:42.380
to each other and form an
ionic solid in a regular

00:02:42.380 --> 00:02:44.640
lattice structure like this.

00:02:44.640 --> 00:02:47.670
So let's think a little
bit about their properties.

00:02:47.670 --> 00:02:51.070
So first of all, let's think
about the melting points.

00:02:51.070 --> 00:02:55.808
So, these solids, the
electrostatic attraction between

00:02:55.808 --> 00:02:58.230
these ions is strong.

00:02:58.230 --> 00:03:01.920
And so they tend to have
high melting points.

00:03:01.920 --> 00:03:03.680
Now what if we were to
compare melting points

00:03:03.680 --> 00:03:05.630
between ionic solids?

00:03:05.630 --> 00:03:08.130
So for example, if you wanted
to compare the melting point

00:03:08.130 --> 00:03:11.980
of sodium chloride to the melting point

00:03:11.980 --> 00:03:15.670
of magnesium oxide which one do you think

00:03:15.670 --> 00:03:16.760
has a higher melting point?

00:03:16.760 --> 00:03:19.000
Pause this video and think about it.

00:03:19.000 --> 00:03:21.640
Well, as you can imagine the
electrostatic attraction,

00:03:21.640 --> 00:03:23.840
it's going to be dependent on two things.

00:03:23.840 --> 00:03:27.590
The magnitude of the charge and the radius

00:03:27.590 --> 00:03:30.630
of the atoms that ae forming
this lattice structure.

00:03:30.630 --> 00:03:33.230
And the magnitude of the
charge here is clear.

00:03:33.230 --> 00:03:36.020
Here you have a plus two
charge being attracted

00:03:36.020 --> 00:03:39.210
to a negative two charge
so this has a stronger

00:03:39.210 --> 00:03:41.580
electrostatic attraction
and so you're going to have

00:03:41.580 --> 00:03:44.710
a higher melting point right over here.

00:03:44.710 --> 00:03:47.320
The melting point of magnesium oxide?

00:03:47.320 --> 00:03:52.320
2,825 degrees Celsius, while
the melting point of table salt

00:03:54.050 --> 00:03:58.670
or sodium chloride is 801 degrees Celsius.

00:03:58.670 --> 00:04:01.760
You could also try to
compare sodium chloride

00:04:01.760 --> 00:04:06.346
to something like sodium fluoride.

00:04:06.346 --> 00:04:07.730
Which one do you think is going to

00:04:07.730 --> 00:04:09.650
have a higher melting point?

00:04:09.650 --> 00:04:13.031
Sodium chloride or sodium fluoride?

00:04:13.031 --> 00:04:16.300
Well fluorines are smaller
than chlorines and each of them

00:04:16.300 --> 00:04:19.490
gain an electron, then the fluoride anion

00:04:19.490 --> 00:04:21.160
is still going to be a
reasonable bit smaller

00:04:21.160 --> 00:04:22.870
than the chloride anion.

00:04:22.870 --> 00:04:25.630
Or when you have smaller constituent ions,

00:04:25.630 --> 00:04:28.268
the electrostatic attraction
is actually stronger.

00:04:28.268 --> 00:04:31.110
Remember, we've seen in Coulomb's law,

00:04:31.110 --> 00:04:33.870
that the closer two
charges are to each other,

00:04:33.870 --> 00:04:36.710
the stronger the attractive
or the repulsive force,

00:04:36.710 --> 00:04:37.870
and if they're opposite charges,

00:04:37.870 --> 00:04:39.570
it's going to be an attractive force.

00:04:39.570 --> 00:04:43.090
So, sodium fluoride is
actually gonna have a higher

00:04:43.090 --> 00:04:46.410
melting point than sodium
chloride, by a little bit.

00:04:46.410 --> 00:04:48.210
It actually turns out
that the melting point

00:04:48.210 --> 00:04:52.500
of sodium fluoride is 996 degrees Celsius.

00:04:52.500 --> 00:04:54.280
But if you're comparing these three,

00:04:54.280 --> 00:04:57.100
the highest melting
point is magnesium oxide,

00:04:57.100 --> 00:05:01.110
followed by sodium fluoride,
followed by sodium chloride.

00:05:01.110 --> 00:05:04.770
So charge is what's really
dominating over here.

00:05:04.770 --> 00:05:06.590
Now the next question
you might be wondering

00:05:06.590 --> 00:05:10.040
is all right I can imagine
these solids are really hard,

00:05:10.040 --> 00:05:12.290
but what would happen if
I were to try to break it?

00:05:12.290 --> 00:05:14.890
Would it bend like a lot
of the metals we know

00:05:14.890 --> 00:05:16.540
and we'll study that in other videos,

00:05:16.540 --> 00:05:18.200
or would something else happen?

00:05:18.200 --> 00:05:20.980
And to understand that, let
me draw a two dimensional

00:05:20.980 --> 00:05:22.140
representation of this.

00:05:22.140 --> 00:05:24.360
So let me draw the chlorine,

00:05:24.360 --> 00:05:26.577
or I should say the chloride anions.

00:05:27.652 --> 00:05:31.810
And this is just a two dimensional
version of that lattice.

00:05:31.810 --> 00:05:34.150
Obviously not drawing it to scale.

00:05:34.150 --> 00:05:36.870
And then let me draw the sodiums.

00:05:37.874 --> 00:05:39.413
Sodium cations.

00:05:40.665 --> 00:05:42.960
As you can see, the positives
are attracted to the negative,

00:05:42.960 --> 00:05:44.220
that's why they're next to each other,

00:05:44.220 --> 00:05:46.060
the negatives aren't next each other

00:05:46.060 --> 00:05:47.280
because they repel each other.

00:05:47.280 --> 00:05:49.310
The positives aren't next to each other,

00:05:49.310 --> 00:05:53.140
but what would happen if I were to try to,

00:05:53.140 --> 00:05:55.760
or I were to press down
really hard on this side

00:05:55.760 --> 00:06:00.690
and if I were to press
really hard up on this side?

00:06:00.690 --> 00:06:02.451
So what would happen if I press had enough

00:06:02.451 --> 00:06:06.010
that this side begins to budge?

00:06:06.010 --> 00:06:07.540
So it begins to budge.

00:06:07.540 --> 00:06:10.780
Would it just bend, or what
do you think's gonna happen

00:06:10.780 --> 00:06:13.340
when I get right about there?

00:06:13.340 --> 00:06:15.630
Well, when I get right about there,

00:06:15.630 --> 00:06:19.170
all of a sudden I've, not only
have I broken the lattice,

00:06:19.170 --> 00:06:20.800
but the negatives are
next to the negatives

00:06:20.800 --> 00:06:22.362
and the positives are
next to the positives

00:06:22.362 --> 00:06:24.870
and so it's not just going to bend,

00:06:24.870 --> 00:06:27.700
and be malleable like a lot
of the metals we've seen,

00:06:27.700 --> 00:06:29.680
it's just going to break.

00:06:29.680 --> 00:06:31.910
So this is going to be, even
though it's going to be hard,

00:06:31.910 --> 00:06:33.800
it is going to be brittle.

00:06:33.800 --> 00:06:36.610
Now the last question
we'll address in this video

00:06:36.610 --> 00:06:40.600
is how good do you think ionic
solids conduct electricity?

00:06:40.600 --> 00:06:43.130
Pause this video and think about that.

00:06:43.130 --> 00:06:44.720
Well, in order to conduct electricity,

00:06:44.720 --> 00:06:47.900
either electrons or charge
generally has to be able

00:06:47.900 --> 00:06:48.950
to move about.

00:06:48.950 --> 00:06:52.010
And when it's just in
its solid form like this,

00:06:52.010 --> 00:06:54.037
the, even though you do have these ions

00:06:54.037 --> 00:06:56.100
they're not going to move about.

00:06:56.100 --> 00:06:58.406
So ionic solids in their solid form,

00:06:58.406 --> 00:07:01.580
they aren't good at
conducting electricity.

00:07:01.580 --> 00:07:04.150
They can be good at conducting electricity

00:07:04.150 --> 00:07:06.410
if you were to dissolve it in a solution.

00:07:06.410 --> 00:07:08.410
For example, if you were
to dissolve this salt

00:07:08.410 --> 00:07:10.708
in water, now the ions can move around

00:07:10.708 --> 00:07:13.580
and then they're good at
conducting electricity.

00:07:13.580 --> 00:07:16.210
Or, if you were to heat
this sodium chloride

00:07:16.210 --> 00:07:20.270
up beyond 801 degrees Celsius
and it turns into a liquid,

00:07:20.270 --> 00:07:23.290
then once again the ions can move around

00:07:23.290 --> 00:07:26.120
and you can actually conduct electricity.

00:07:26.120 --> 00:07:29.020
Take everything I say
with a grain of salt.

00:07:29.020 --> 00:07:31.210
Sorry, I know, I couldn't help it.

00:07:31.210 --> 00:07:32.380
But hopefully you know a little bit more

00:07:32.380 --> 00:07:33.913
about ionic solids now.


Information for congruency

WEBVTT
Kind: captions
Language: en

00:00:00.140 --> 00:00:02.260
- [Instructor] So I have
two triangles depicted here

00:00:02.260 --> 00:00:05.340
and we have some information
about each of those triangles.

00:00:05.340 --> 00:00:08.390
We know that this side
of this left triangle

00:00:08.390 --> 00:00:09.960
has length eight.

00:00:09.960 --> 00:00:12.440
We know that this side has length seven

00:00:12.440 --> 00:00:15.330
and then we know that
this angle is 50 degrees.

00:00:15.330 --> 00:00:18.600
And on this triangle we
see some things that look

00:00:18.600 --> 00:00:20.130
a little bit familiar.

00:00:20.130 --> 00:00:23.740
This triangle, this side has length eight.

00:00:23.740 --> 00:00:26.320
This side has length seven.

00:00:26.320 --> 00:00:30.730
And this angle right over here
has a measure of 50 degrees.

00:00:30.730 --> 00:00:32.570
So my question to you is,

00:00:32.570 --> 00:00:35.440
can you definitively say, not assuming

00:00:35.440 --> 00:00:36.660
that these are drawn to scale,

00:00:36.660 --> 00:00:39.580
because they actually aren't,
can you definitively say

00:00:39.580 --> 00:00:41.760
that these triangles are congruent?

00:00:41.760 --> 00:00:44.470
Or could you definitively say
that they aren't congruent?

00:00:44.470 --> 00:00:46.460
Or can you not say either?

00:00:46.460 --> 00:00:49.010
Would you have to say that
there's not enough information?

00:00:49.010 --> 00:00:51.210
Pause this video and think about that.

00:00:51.210 --> 00:00:54.440
So essentially what we have
here are two pairs of sides

00:00:54.440 --> 00:00:56.580
that have the same length and and angle,

00:00:56.580 --> 00:00:59.830
but that angle is not
between those two sides.

00:00:59.830 --> 00:01:02.400
If the angle were here and here,

00:01:02.400 --> 00:01:07.130
then we could use side,
angle, side, side, angle, side

00:01:07.130 --> 00:01:09.970
to deduce that, hey, these are congruent,

00:01:09.970 --> 00:01:11.350
but that's not what we're dealing with.

00:01:11.350 --> 00:01:14.240
We are dealing with side, side, angle.

00:01:14.240 --> 00:01:16.110
Side, side, angle.

00:01:16.110 --> 00:01:18.230
I'm saying the side and
the side before the angle,

00:01:18.230 --> 00:01:21.440
because if I don't do that it
becomes a little bit crass.

00:01:21.440 --> 00:01:22.273
So what we're really saying

00:01:22.273 --> 00:01:27.273
is side, side, angle
sufficient to prove congruency?

00:01:27.330 --> 00:01:28.590
And the reason why it's not,

00:01:28.590 --> 00:01:31.680
is that you can actually
construct different triangles

00:01:31.680 --> 00:01:33.800
with the same constraints.

00:01:33.800 --> 00:01:36.060
For example, on this right most triangle

00:01:36.060 --> 00:01:37.490
it could look like this,

00:01:37.490 --> 00:01:39.290
or it could look like this.

00:01:39.290 --> 00:01:43.480
The seven side could go down like this

00:01:43.480 --> 00:01:46.400
and intersect just like that.

00:01:46.400 --> 00:01:48.380
Now you might be saying, hey,
that's not what it looks like.

00:01:48.380 --> 00:01:50.390
It looks very similar, but remember,

00:01:50.390 --> 00:01:52.120
we're not going on looks.

00:01:52.120 --> 00:01:54.940
We have to go based on the
information they've given us

00:01:54.940 --> 00:01:57.760
and so you could just as easily,
based on the information,

00:01:57.760 --> 00:01:59.440
the constraints they've given us,

00:01:59.440 --> 00:02:01.350
have a triangle like this.

00:02:01.350 --> 00:02:03.320
And so the very fact that you can create

00:02:03.320 --> 00:02:06.240
two different triangles that
are clearly not congruent,

00:02:06.240 --> 00:02:08.440
based on the exact same information,

00:02:08.440 --> 00:02:10.580
the exact same constraints, tells you that

00:02:10.580 --> 00:02:12.070
that information, those constraints,

00:02:12.070 --> 00:02:13.990
are not enough to tell you

00:02:13.990 --> 00:02:15.763
that these are congruent triangles.


Non-congruent shapes & transformations

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:02.510
- [Instructor] We are told,
Brenda was able to map

00:00:02.510 --> 00:00:06.050
circle M onto circle N

00:00:06.050 --> 00:00:09.690
using a translation and a dilation.

00:00:09.690 --> 00:00:11.550
This is circle M right over here.

00:00:11.550 --> 00:00:13.100
Here's the center of it.

00:00:13.100 --> 00:00:17.310
This is circle M, this
circle right over here.

00:00:17.310 --> 00:00:19.850
It looks like at first, she translates it.

00:00:19.850 --> 00:00:23.390
The center goes from this
point to this point here.

00:00:23.390 --> 00:00:27.230
After the translation, we have the circle

00:00:27.230 --> 00:00:28.800
right over here.

00:00:28.800 --> 00:00:30.820
Then she dilates it.

00:00:30.820 --> 00:00:34.170
The center of dilation
looks like it is point N.

00:00:34.170 --> 00:00:36.990
She dilates it with some
type of a scale factor

00:00:36.990 --> 00:00:40.600
in order to map it exactly onto N.

00:00:40.600 --> 00:00:42.990
That all seems right.

00:00:42.990 --> 00:00:47.177
Brenda concluded, "I
was able to map circle M

00:00:47.177 --> 00:00:50.307
"onto circle N using a sequence

00:00:50.307 --> 00:00:52.357
"of rigid transformations,

00:00:52.357 --> 00:00:56.020
"so the figures are congruent."

00:00:56.020 --> 00:00:57.723
Is she correct?

00:00:59.218 --> 00:01:01.813
Pause this video and think about that.

00:01:03.210 --> 00:01:05.920
Let's work on this together.

00:01:05.920 --> 00:01:10.560
She was able to map circle M onto circle N

00:01:10.560 --> 00:01:13.700
using a sequence of transformations.

00:01:13.700 --> 00:01:15.660
She did a translation and then a dilation.

00:01:15.660 --> 00:01:17.330
Those are all transformations,

00:01:17.330 --> 00:01:19.910
but they are not all
rigid transformations.

00:01:19.910 --> 00:01:21.950
I'll put a question mark right over there.

00:01:21.950 --> 00:01:25.000
A translation is a rigid transformation.

00:01:25.000 --> 00:01:26.670
Remember, rigid transformations are ones

00:01:26.670 --> 00:01:28.100
that preserve distances,

00:01:28.100 --> 00:01:30.850
preserve angle measures, preserve lengths,

00:01:30.850 --> 00:01:33.700
while a dilation is not
a rigid transformation.

00:01:33.700 --> 00:01:35.240
As you can see very clearly,

00:01:35.240 --> 00:01:36.920
it is not preserving lengths.

00:01:36.920 --> 00:01:39.050
It is not, for example,
preserving the radius

00:01:39.050 --> 00:01:40.270
of the circle.

00:01:40.270 --> 00:01:43.230
In order for two figures to be congruent,

00:01:43.230 --> 00:01:47.630
the mapping has to be only
with rigid transformations.

00:01:47.630 --> 00:01:49.760
Because she used a dilation,

00:01:49.760 --> 00:01:51.730
in fact, you have to use a dilation

00:01:51.730 --> 00:01:53.890
if you wanna be able to map M onto N

00:01:53.890 --> 00:01:55.633
because they have different radii,

00:01:56.510 --> 00:01:58.320
then she's not correct.

00:01:58.320 --> 00:01:59.740
These are not congruent figures.

00:01:59.740 --> 00:02:01.843
She cannot make this conclusion.


Congruent shapes and transformations

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:02.497
- [Instructor] We're
told, "Kason was curious

00:00:02.497 --> 00:00:06.837
"if triangle ABC and
triangle GFE were congruent,

00:00:06.837 --> 00:00:10.107
"so he tried to map one
figure onto the other

00:00:10.107 --> 00:00:12.140
"using a rotation."

00:00:12.140 --> 00:00:15.610
So let's see, this is triangle
ABC, and it looks like,

00:00:15.610 --> 00:00:20.610
at first, he rotates
triangle ABC about point C,

00:00:21.150 --> 00:00:24.350
to get it right over here, so
that's what they're depicting

00:00:24.350 --> 00:00:25.610
in this diagram.

00:00:25.610 --> 00:00:28.717
And then they say, "Kason concluded:

00:00:28.717 --> 00:00:31.697
"It is not possible to map triangle ABC

00:00:31.697 --> 00:00:34.647
"onto triangle GFE using a sequence

00:00:34.647 --> 00:00:36.407
"of rigid transformations,

00:00:36.407 --> 00:00:39.560
"so the triangles are not congruent."

00:00:39.560 --> 00:00:41.580
So what I want you to
do is pause this video

00:00:41.580 --> 00:00:43.870
and think about, is Kason correct

00:00:43.870 --> 00:00:46.048
that they are not congruent,
because you can not map

00:00:46.048 --> 00:00:49.180
ABC, triangle ABC onto triangle GFE

00:00:49.180 --> 00:00:50.703
with rigid transformations?

00:00:51.890 --> 00:00:54.410
All right, so the way I think about it,

00:00:54.410 --> 00:00:55.792
he was able to do the rotation

00:00:55.792 --> 00:00:58.740
that got us right over here,

00:00:58.740 --> 00:01:01.040
so it is rotation about point C,

00:01:01.040 --> 00:01:02.590
and so this point right over here,

00:01:02.590 --> 00:01:03.650
let me make sure I get this right,

00:01:03.650 --> 00:01:05.830
this would've become B prime,

00:01:05.830 --> 00:01:07.650
and then this is A prime,

00:01:07.650 --> 00:01:12.110
and then C is mapped to itself,
so C is equal to C prime.

00:01:12.110 --> 00:01:15.040
But he's not done, there's
another rigid transformation

00:01:15.040 --> 00:01:17.470
he could do, and that
would be a reflection

00:01:17.470 --> 00:01:20.090
about the line FG.

00:01:20.090 --> 00:01:22.610
So if he reflects about the line FG,

00:01:22.610 --> 00:01:25.620
then this point is going
to be mapped to point E,

00:01:25.620 --> 00:01:26.530
just like that.

00:01:26.530 --> 00:01:28.720
And then if you did that,
you would see that there is

00:01:28.720 --> 00:01:30.340
a series of rigid transformations

00:01:30.340 --> 00:01:34.340
that maps triangle ABC onto triangle GFE.

00:01:34.340 --> 00:01:38.980
So Kason is not correct, he
missed one more transformation

00:01:38.980 --> 00:01:41.703
he could've done, which is a reflection.


Dilating triangles: find the error

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:01.810
- [Instructor] We are told triangle

00:00:01.810 --> 00:00:04.380
A-prime, B-prime, C-prime is the image

00:00:04.380 --> 00:00:07.290
of triangle ABC under a dilation

00:00:07.290 --> 00:00:12.270
whose center is P and scale factor is 3/4.

00:00:12.270 --> 00:00:14.880
Which figure correctly shows triangle

00:00:14.880 --> 00:00:18.990
A-prime, B-prime, C-prime
using the solid line.

00:00:18.990 --> 00:00:20.370
So pause this video and see

00:00:20.370 --> 00:00:22.320
if you can figure this out on your own.

00:00:23.280 --> 00:00:25.760
All right, now before I
even look at the choices,

00:00:25.760 --> 00:00:26.710
I like to just think about,

00:00:26.710 --> 00:00:29.240
what would that dilation
actually look like?

00:00:29.240 --> 00:00:31.680
So our center of dilation is P.

00:00:31.680 --> 00:00:34.560
And it's a scale factor of 3/4.

00:00:34.560 --> 00:00:36.140
So one way to think about it is,

00:00:36.140 --> 00:00:38.730
however far any point was from P before,

00:00:38.730 --> 00:00:43.270
is now going to be 3/4 as
far, but along the same line.

00:00:43.270 --> 00:00:44.900
So I'm just going to estimate it.

00:00:44.900 --> 00:00:48.400
So if C was there, 3/2 would be this far.

00:00:48.400 --> 00:00:50.190
So 3/4 would be right about there.

00:00:50.190 --> 00:00:52.580
So C-prime should be about there.

00:00:52.580 --> 00:00:57.580
If we have this line
connecting B and P like this,

00:00:58.200 --> 00:01:00.070
let's see, half of that is there.

00:01:00.070 --> 00:01:02.170
3/4 is going to be there.

00:01:02.170 --> 00:01:04.610
So B-prime should be there.

00:01:04.610 --> 00:01:09.020
And then on this line,
halfway is roughly there.

00:01:09.020 --> 00:01:09.980
I'm just eyeballing it.

00:01:09.980 --> 00:01:11.690
So 3/4 is there.

00:01:11.690 --> 00:01:15.240
So A-prime, A-prime, should be there.

00:01:15.240 --> 00:01:18.280
And so A-prime, B-prime, C-prime

00:01:18.280 --> 00:01:22.930
should look something like this.

00:01:22.930 --> 00:01:27.930
Which we can see is exactly
what we see for choice C.

00:01:28.700 --> 00:01:30.470
So choice C, it looks correct.

00:01:30.470 --> 00:01:32.400
So I'm gonna just circle that,

00:01:32.400 --> 00:01:34.390
or select it just like that.

00:01:34.390 --> 00:01:35.800
But let's just make sure we understand why

00:01:35.800 --> 00:01:38.820
these other two choices were not correct.

00:01:38.820 --> 00:01:42.450
So choice A, it looks
like it is a dilation

00:01:42.450 --> 00:01:44.600
with a 3/4 scale factor.

00:01:44.600 --> 00:01:46.540
Each of the dimensions, each of the sides

00:01:46.540 --> 00:01:48.360
of these triangles, of this triangle,

00:01:48.360 --> 00:01:51.310
looks like it's about 3/4
of what it originally was.

00:01:51.310 --> 00:01:54.620
But it doesn't look like
the center of dilation is P.

00:01:54.620 --> 00:01:56.690
Here the center of dilation looks

00:01:56.690 --> 00:02:01.690
like it is probably the
midpoint of segment AC.

00:02:02.570 --> 00:02:05.410
Because now it looks
like everything is 3/4

00:02:05.410 --> 00:02:08.140
of the distance it was to that point.

00:02:08.140 --> 00:02:12.290
So they have this other center
of dilation in choice A.

00:02:12.290 --> 00:02:14.040
The center of dilation is not P,

00:02:14.040 --> 00:02:16.020
and that's why we can rule that one out.

00:02:16.020 --> 00:02:19.240
And then for choice B right over here,

00:02:19.240 --> 00:02:21.290
it looks like they just
got the scale factor wrong.

00:02:21.290 --> 00:02:22.590
Actually they got the center of dilation

00:02:22.590 --> 00:02:23.670
and the scale factor wrong.

00:02:23.670 --> 00:02:26.320
It still looks like they are using this

00:02:26.320 --> 00:02:27.830
as a center of dilation.

00:02:27.830 --> 00:02:30.680
But this scale factor
looks like it's much closer

00:02:30.680 --> 00:02:34.290
to 1/4 or 1/3, not 3/4.

00:02:34.290 --> 00:02:36.940
So that's why we can rule
that one out as well.

00:02:36.940 --> 00:02:38.483
We like our choice, C.


Proving the ASA and AAS triangle congruence criteria using transformations

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.550
- [Instructor] What we're going
to do in this video is show

00:00:02.550 --> 00:00:05.450
that if we have two different triangles

00:00:05.450 --> 00:00:09.980
that have one pair of sides
that have the same length,

00:00:09.980 --> 00:00:11.530
so these blue sides in each

00:00:11.530 --> 00:00:13.740
of these triangles have the same length,

00:00:13.740 --> 00:00:18.420
and they have two pairs of
angles where, for each pair,

00:00:18.420 --> 00:00:21.090
the corresponding angles
have the same measure,

00:00:21.090 --> 00:00:23.370
so this gray angle here
has the same measure

00:00:23.370 --> 00:00:27.350
as this angle here, and then
these double orange arcs show

00:00:27.350 --> 00:00:31.040
that this angle ACB has the same measure

00:00:31.040 --> 00:00:32.123
as angle DFE.

00:00:33.520 --> 00:00:36.950
And so we're gonna show that
if you have two of your angles

00:00:36.950 --> 00:00:40.570
and a side that had the
same measure or length,

00:00:40.570 --> 00:00:44.820
that we can always create a
series of rigid transformations

00:00:44.820 --> 00:00:46.930
that maps one triangle onto the other.

00:00:46.930 --> 00:00:49.400
Or another way to say it,
they must be congruent

00:00:49.400 --> 00:00:53.700
by the rigid transformation
definition of congruency.

00:00:53.700 --> 00:00:56.510
And the reason why I wrote
angle side angle here

00:00:56.510 --> 00:00:59.240
and angle angle side is to realize

00:00:59.240 --> 00:01:01.060
that these are equivalent.

00:01:01.060 --> 00:01:03.500
Because if you have two angles,

00:01:03.500 --> 00:01:06.300
then you know what the
third angle is going to be.

00:01:06.300 --> 00:01:08.660
So for example, in this
case right over here,

00:01:08.660 --> 00:01:10.170
if we know that we have two pairs

00:01:10.170 --> 00:01:12.230
of angles that have the same measure,

00:01:12.230 --> 00:01:14.060
then that means that the third pair

00:01:14.060 --> 00:01:17.170
must have the same measure as well.

00:01:17.170 --> 00:01:18.670
So we'll know this as well.

00:01:18.670 --> 00:01:20.090
So if you really think about it,

00:01:20.090 --> 00:01:22.860
if you have the side
between the two angles,

00:01:22.860 --> 00:01:26.310
that's equivalent to having an
angle, an angle, and a side.

00:01:26.310 --> 00:01:28.170
Because as long as you have two angles,

00:01:28.170 --> 00:01:30.600
the third angle is also going
to have the same measure

00:01:30.600 --> 00:01:34.040
as the corresponding third
angle on the other triangle.

00:01:34.040 --> 00:01:36.820
So let's just show a series
of rigid transformations

00:01:36.820 --> 00:01:39.583
that can get us from ABC to DEF.

00:01:41.030 --> 00:01:44.180
So the first step, you might
imagine, we've already shown

00:01:44.180 --> 00:01:46.660
that if you have two
segments of equal length

00:01:46.660 --> 00:01:48.010
that they are congruent.

00:01:48.010 --> 00:01:50.300
You can have a series
of rigid transformations

00:01:50.300 --> 00:01:52.000
that maps one onto the other.

00:01:52.000 --> 00:01:54.440
So what I want to do is

00:01:54.440 --> 00:01:56.440
map segment AC

00:01:58.710 --> 00:02:00.383
onto DF.

00:02:01.285 --> 00:02:03.780
And the way that I could do that

00:02:03.780 --> 00:02:07.330
is I could translate point A

00:02:07.330 --> 00:02:09.630
to be on top of point D,

00:02:09.630 --> 00:02:11.980
so then I'll call this A prime.

00:02:11.980 --> 00:02:13.630
And then when I do that,

00:02:13.630 --> 00:02:16.250
this segment AC is going to
look something like this.

00:02:16.250 --> 00:02:17.410
I'm just sketching it right now.

00:02:17.410 --> 00:02:19.020
It's going to be in that direction.

00:02:19.020 --> 00:02:20.000
But then, and the whole,

00:02:20.000 --> 00:02:21.700
the rest of the triangle
is going to come with it.

00:02:21.700 --> 00:02:25.060
So let's see, the rest of
that orange side, side AB,

00:02:25.060 --> 00:02:27.920
is going to look something like that.

00:02:27.920 --> 00:02:29.810
But then we could do another
rigid transformation,

00:02:29.810 --> 00:02:33.080
which is rotate about
point D or point A prime,

00:02:33.080 --> 00:02:34.560
they're the same point now,

00:02:34.560 --> 00:02:39.060
so that point C coincides with point F.

00:02:39.060 --> 00:02:40.490
And so just like that,

00:02:40.490 --> 00:02:43.120
you would have two rigid
transformations that get us,

00:02:43.120 --> 00:02:46.800
that map AC onto DF.

00:02:46.800 --> 00:02:50.220
And so A prime, where A is
mapped, is now equal to D,

00:02:50.220 --> 00:02:52.940
and F is now equal to C prime.

00:02:52.940 --> 00:02:57.690
But the question is where
does point B now sit?

00:02:57.690 --> 00:02:59.570
And the realization here is

00:02:59.570 --> 00:03:02.840
that angle measures are preserved.

00:03:02.840 --> 00:03:05.570
And since angle measures are preserved,

00:03:05.570 --> 00:03:10.160
we are either going to have
a situation where this angle,

00:03:10.160 --> 00:03:12.713
let's see, this angle is angle CAB

00:03:14.660 --> 00:03:15.890
gets preserved.

00:03:15.890 --> 00:03:19.770
So then it would be C prime, A prime,

00:03:19.770 --> 00:03:23.206
and then B prime would have
to sit someplace on this ray.

00:03:23.206 --> 00:03:26.783
Or if we're gonna preserve
the measure of angle CAB,

00:03:27.660 --> 00:03:31.910
B prime is going to sit
someplace along that ray.

00:03:31.910 --> 00:03:34.420
Because an angle is defined by two rays

00:03:34.420 --> 00:03:38.030
that intersect at the vertex
or start at the vertex.

00:03:38.030 --> 00:03:40.530
And because this angle is preserved,

00:03:40.530 --> 00:03:43.960
that's the angle that is
formed by these two rays.

00:03:43.960 --> 00:03:46.980
You could say ray CA and ray CB.

00:03:46.980 --> 00:03:49.520
We know that B prime
also has to sit someplace

00:03:49.520 --> 00:03:51.510
on this ray as well.

00:03:51.510 --> 00:03:55.970
So B prime also has to
sit someplace on this ray,

00:03:55.970 --> 00:03:58.290
and I think you see where this is going.

00:03:58.290 --> 00:04:01.890
If B prime, because these
two angles are preserved,

00:04:01.890 --> 00:04:03.990
because this angle and
this angle are preserved,

00:04:03.990 --> 00:04:07.020
have to sit someplace
on both of these rays,

00:04:07.020 --> 00:04:08.650
they intersect at one point,

00:04:08.650 --> 00:04:12.000
this point right over here
that coincides with point E.

00:04:12.000 --> 00:04:14.300
So this is where B prime would be.

00:04:14.300 --> 00:04:16.910
So that's one scenario,
in which case we've shown

00:04:16.910 --> 00:04:19.590
that you can get a series
of rigid transformations

00:04:19.590 --> 00:04:21.580
from this triangle to this triangle.

00:04:21.580 --> 00:04:22.790
But there's another one.

00:04:22.790 --> 00:04:26.550
There is a circumstance where
the angles get preserved.

00:04:26.550 --> 00:04:29.260
But instead of being on,

00:04:29.260 --> 00:04:31.150
instead of the angles being on the,

00:04:31.150 --> 00:04:32.690
I guess you could say
the bottom right side

00:04:32.690 --> 00:04:33.840
of this blue line,

00:04:33.840 --> 00:04:35.900
you could imagine the angles get preserved

00:04:35.900 --> 00:04:38.560
such that they are on the other side.

00:04:38.560 --> 00:04:41.090
So the angles get preserved

00:04:41.090 --> 00:04:45.220
so that they are on the
other side of that blue line.

00:04:45.220 --> 00:04:47.490
And then the question
is, in that situation,

00:04:47.490 --> 00:04:49.560
where would B prime end up?

00:04:49.560 --> 00:04:51.810
Well, actually, let me
draw this a little bit,

00:04:51.810 --> 00:04:53.950
let me do this a little bit more exact.

00:04:53.950 --> 00:04:56.690
Let me replicate these angles.

00:04:56.690 --> 00:05:01.690
So I'm going to draw an arc
like this, an arc like this,

00:05:01.910 --> 00:05:03.823
and then I'll measure this distance.

00:05:05.497 --> 00:05:06.570
It's just like this.

00:05:06.570 --> 00:05:07.940
We've done this in other videos,

00:05:07.940 --> 00:05:10.433
when we're trying to replicate angles.

00:05:11.490 --> 00:05:12.730
So it's like that far,

00:05:12.730 --> 00:05:15.730
and so let me draw that on
this point right over here,

00:05:15.730 --> 00:05:16.890
this far.

00:05:16.890 --> 00:05:19.830
So if the angles are on that side of line,

00:05:19.830 --> 00:05:22.950
I guess we could say
DF or A prime, C prime,

00:05:22.950 --> 00:05:25.100
we know that B prime

00:05:25.100 --> 00:05:28.850
would have to sit someplace on this ray.

00:05:28.850 --> 00:05:32.260
So let me draw that as neatly as I can,

00:05:32.260 --> 00:05:35.023
someplace on this ray.

00:05:36.890 --> 00:05:38.670
And it would have to sit someplace

00:05:38.670 --> 00:05:41.060
on the ray formed by the other angle.

00:05:41.060 --> 00:05:44.720
So let me see if I can draw
that as neatly as possible.

00:05:44.720 --> 00:05:45.730
So let me

00:05:48.760 --> 00:05:51.340
make a arc like this.

00:05:51.340 --> 00:05:53.990
I probably did that a little
bit bigger than I need to,

00:05:53.990 --> 00:05:56.620
but hopefully it serves our purposes.

00:05:56.620 --> 00:05:59.230
I measured this distance right over here.

00:05:59.230 --> 00:06:01.970
If I measure that distance over here,

00:06:01.970 --> 00:06:05.380
it would get us right over there.

00:06:05.380 --> 00:06:10.380
So B prime either sits on
this ray, or it could sit,

00:06:10.500 --> 00:06:15.260
or and it has to sit, I should
really say, on this ray,

00:06:15.260 --> 00:06:17.910
that goes through this
point and this point.

00:06:17.910 --> 00:06:19.730
And it has to sit on this ray.

00:06:19.730 --> 00:06:23.270
And you can see where
these two rays intersect

00:06:23.270 --> 00:06:24.120
is right over there.

00:06:24.120 --> 00:06:26.690
So the other scenario is
if the angles get preserved

00:06:26.690 --> 00:06:29.700
in a way that they're on the
other side of that blue line,

00:06:29.700 --> 00:06:31.750
well, then B prime is there.

00:06:31.750 --> 00:06:34.910
And then we could just add
one more rigid transformation

00:06:34.910 --> 00:06:37.150
to our series of rigid transformations,

00:06:37.150 --> 00:06:41.310
which is essentially or
is a reflection across

00:06:41.310 --> 00:06:44.570
line DF or A prime, C prime.

00:06:44.570 --> 00:06:48.210
Why will that work, to map B prime onto E?

00:06:48.210 --> 00:06:51.460
Well, because reflection is
also a rigid transformation,

00:06:51.460 --> 00:06:53.010
so angles are preserved.

00:06:53.010 --> 00:06:55.360
And so as this angle gets
flipped over, it's preserved.

00:06:55.360 --> 00:06:56.890
As this angle gets flipped over,

00:06:56.890 --> 00:06:59.460
the measure of it, I
should say, is preserved.

00:06:59.460 --> 00:07:02.000
And so that means we'll
go to that first case

00:07:02.000 --> 00:07:04.490
where then these rays would
be flipped onto these rays,

00:07:04.490 --> 00:07:07.170
and B prime would have to
sit on that intersection.

00:07:07.170 --> 00:07:08.430
And there you have it.

00:07:08.430 --> 00:07:09.970
If you have two angles,

00:07:09.970 --> 00:07:12.140
and if you have two angles,
you're gonna know the third,

00:07:12.140 --> 00:07:14.370
if you have two angles and a side

00:07:14.370 --> 00:07:17.130
that have the same measure or length,

00:07:17.130 --> 00:07:18.880
if we're talking about angle or a side,

00:07:18.880 --> 00:07:20.370
well, that means that they are going

00:07:20.370 --> 00:07:22.293
to be congruent triangles.


Proving the SAS triangle congruence criterion using transformations

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:01.970
- [Instructor] What we're
going to do in this video

00:00:01.970 --> 00:00:04.390
is see that if we have
two different triangles,

00:00:04.390 --> 00:00:07.880
and we have two sets
of corresponding sides

00:00:07.880 --> 00:00:10.140
that have the same length,

00:00:10.140 --> 00:00:12.830
for example this blue
side has the same length

00:00:12.830 --> 00:00:14.710
as this blue side here,

00:00:14.710 --> 00:00:16.900
and this orange side
has the same length side

00:00:16.900 --> 00:00:18.827
as this orange side here.

00:00:18.827 --> 00:00:22.150
And the angle that is
formed between those sides,

00:00:22.150 --> 00:00:25.030
so we have two corresponding
angles right over here,

00:00:25.030 --> 00:00:27.420
that they also have the equal measure.

00:00:27.420 --> 00:00:29.900
So we could think about
we have a side, an angle,

00:00:29.900 --> 00:00:33.060
a side, a side, an angle and a side.

00:00:33.060 --> 00:00:36.340
If those have the same
lengths or measures,

00:00:36.340 --> 00:00:38.270
then we can deduce

00:00:38.270 --> 00:00:41.220
that these two triangles must be congruent

00:00:41.220 --> 00:00:44.880
by the rigid motion
definition of congruency.

00:00:44.880 --> 00:00:46.450
Or the short hand is,

00:00:46.450 --> 00:00:48.920
if we have side, angle, side in common,

00:00:48.920 --> 00:00:51.260
and the angle is between the two sides,

00:00:51.260 --> 00:00:54.430
then the two triangles will be congruent.

00:00:54.430 --> 00:00:55.840
So to be able to prove this,

00:00:55.840 --> 00:00:57.210
in order to make this deduction,

00:00:57.210 --> 00:00:58.650
we just have to say that there's always

00:00:58.650 --> 00:01:00.810
a rigid transformation if we have

00:01:00.810 --> 00:01:02.720
a side, angle, side in common

00:01:02.720 --> 00:01:05.270
that will allow us to map
one triangle onto the other.

00:01:05.270 --> 00:01:06.510
Because if there is a series

00:01:06.510 --> 00:01:08.640
of rigid transformations
that allow us to do it,

00:01:08.640 --> 00:01:10.660
then by the rigid
transformation definition

00:01:10.660 --> 00:01:13.130
the two triangles are congruent.

00:01:13.130 --> 00:01:14.750
So the first thing that we could do

00:01:14.750 --> 00:01:17.410
is we could reference back to where we saw

00:01:17.410 --> 00:01:21.130
that if we have two segments
that have the same length,

00:01:21.130 --> 00:01:24.950
like segment AB and segment DE.

00:01:24.950 --> 00:01:27.220
If we have two segments
with the same length

00:01:27.220 --> 00:01:28.600
that they are congruent.

00:01:28.600 --> 00:01:30.850
You can always map one
segment onto the other

00:01:30.850 --> 00:01:33.070
with a series of rigid transformations.

00:01:33.070 --> 00:01:34.900
The way that we could do that in this case

00:01:34.900 --> 00:01:39.850
is we could map point B onto point E.

00:01:39.850 --> 00:01:44.390
So this would be now I'll
put B prime right over here.

00:01:44.390 --> 00:01:47.320
And if we just did a
transformation to do that,

00:01:47.320 --> 00:01:49.320
if we just translated like that,

00:01:49.320 --> 00:01:54.320
then side, woops, then side B A would,

00:01:54.440 --> 00:01:56.730
that orange side would
be something like that.

00:01:56.730 --> 00:01:58.470
But then we could do
another rigid transformation

00:01:58.470 --> 00:02:02.650
that rotates about point E, or B prime,

00:02:02.650 --> 00:02:04.260
that rotates that orange side,

00:02:04.260 --> 00:02:07.970
and the whole triangle with it, onto DE.

00:02:07.970 --> 00:02:10.670
In which case, once we do that
second rigid transformation,

00:02:10.670 --> 00:02:13.280
point A will now coincide with D.

00:02:13.280 --> 00:02:16.400
Or we could say A prime is equal to D.

00:02:16.400 --> 00:02:20.870
But the question is,
where would C now sit?

00:02:20.870 --> 00:02:24.040
Well, we can see the
distance between A and C.

00:02:24.040 --> 00:02:25.940
In fact, we can use our compass for it.

00:02:25.940 --> 00:02:30.930
The distance between A
and C is just like that.

00:02:30.930 --> 00:02:32.170
And so since all

00:02:32.170 --> 00:02:35.570
of these rigid transformations
preserve distance,

00:02:35.570 --> 00:02:39.270
we know that C prime, the
point that C gets mapped to

00:02:39.270 --> 00:02:41.540
after those first two transformations.

00:02:41.540 --> 00:02:43.620
C prime it's distance is going

00:02:43.620 --> 00:02:45.360
to stay the same from A prime.

00:02:45.360 --> 00:02:48.010
So C prime is going to be some place,

00:02:48.010 --> 00:02:53.010
some place along this
curve right over here.

00:02:54.260 --> 00:02:55.530
We also know

00:02:55.530 --> 00:02:59.600
that the rigid transformations
preserve angle measures.

00:02:59.600 --> 00:03:02.770
And so we also know that
as we do the mapping,

00:03:02.770 --> 00:03:04.410
the angle will be preserved.

00:03:04.410 --> 00:03:08.100
So either side AC will be mapped

00:03:08.100 --> 00:03:11.570
to this side right over
here, and if that's the case

00:03:11.570 --> 00:03:13.710
then F would be equal to C prime,

00:03:13.710 --> 00:03:16.200
and we would have found
our rigid transformation

00:03:16.200 --> 00:03:18.730
based on SAS, and so therefore

00:03:18.730 --> 00:03:20.550
the two triangles would be congruent.

00:03:20.550 --> 00:03:21.730
But there's another possibility

00:03:21.730 --> 00:03:23.480
that the angle gets conserved,

00:03:23.480 --> 00:03:28.020
but side AC is mapped down here.

00:03:28.020 --> 00:03:31.020
So there's another
possibility that side AC,

00:03:31.020 --> 00:03:32.540
due to our rigid transformations,

00:03:32.540 --> 00:03:36.370
or after our first set
of rigid transformations,

00:03:36.370 --> 00:03:38.660
looks something like this.

00:03:41.599 --> 00:03:44.160
It looks something like that.

00:03:44.160 --> 00:03:48.320
In which case, C prime would
be mapped right over there.

00:03:48.320 --> 00:03:49.153
And in that case,

00:03:49.153 --> 00:03:51.500
we can just do one more
rigid transformation.

00:03:51.500 --> 00:03:55.830
We can just do a reflection about DE,

00:03:55.830 --> 00:03:59.340
or A prime B prime, to
reflect point C prime

00:03:59.340 --> 00:04:00.990
over that to get right over there.

00:04:00.990 --> 00:04:04.000
How do we know that C prime
would then be mapped to F?

00:04:04.000 --> 00:04:06.410
Well, this angle would be preserved

00:04:06.410 --> 00:04:07.950
due to the rigid transformation.

00:04:07.950 --> 00:04:11.900
So as we flip it over, as we
do the reflection over DE,

00:04:11.900 --> 00:04:13.790
the angle will be preserved.

00:04:13.790 --> 00:04:17.340
And A prime C prime will then map to DF.

00:04:17.340 --> 00:04:18.590
And then we'd be done.

00:04:18.590 --> 00:04:20.980
We have just shown that
there's always a series

00:04:20.980 --> 00:04:22.380
of rigid transformations,

00:04:22.380 --> 00:04:25.270
as long as you meet this SAS criteria,

00:04:25.270 --> 00:04:27.980
that can map one triangle onto the other.

00:04:27.980 --> 00:04:29.933
And therefore, they are congruent.


Proving the SSS triangle congruence criterion using transformations

WEBVTT
Kind: captions
Language: en

00:00:00.220 --> 00:00:02.050
- [Instructor] What we're
going to do in this video

00:00:02.050 --> 00:00:04.220
is see that if we have
two different triangles

00:00:04.220 --> 00:00:06.570
where the corresponding
sides have the same measure,

00:00:06.570 --> 00:00:09.840
so this orange side has the
same length as this orange side,

00:00:09.840 --> 00:00:12.460
this blue side has the same
length as this blue side,

00:00:12.460 --> 00:00:15.960
this gray side has the same
length as this gray side,

00:00:15.960 --> 00:00:18.210
then we can deduce that
these two triangles

00:00:18.210 --> 00:00:19.990
are congruent to each other

00:00:19.990 --> 00:00:24.390
based on the rigid transformation
definition of congruence.

00:00:24.390 --> 00:00:26.100
And to show that, we just have to show

00:00:26.100 --> 00:00:29.200
that there's always a series
of rigid transformations

00:00:29.200 --> 00:00:34.200
that maps triangle ABC onto triangle EDF.

00:00:35.010 --> 00:00:36.620
So how do we do that?

00:00:36.620 --> 00:00:38.200
Well, first of all, in other videos,

00:00:38.200 --> 00:00:41.110
we showed that if we
have two line segments

00:00:41.110 --> 00:00:43.750
that have the same measure,
they are congruent.

00:00:43.750 --> 00:00:47.770
You can map one onto the other
using rigid transformations.

00:00:47.770 --> 00:00:50.570
So let's do a series of
rigid transformations

00:00:50.570 --> 00:00:53.950
that maps AB onto ED.

00:00:53.950 --> 00:00:56.170
And you could imagine how to do that.

00:00:56.170 --> 00:00:57.960
You would translate point A.

00:00:57.960 --> 00:01:00.600
You would translate this
entire left triangle

00:01:00.600 --> 00:01:03.810
so that point A coincides with point E,

00:01:03.810 --> 00:01:07.300
and then side AB would be moving in this,

00:01:07.300 --> 00:01:08.670
would be on this direction over here.

00:01:08.670 --> 00:01:11.640
And then you would rotate around
this point right over here.

00:01:11.640 --> 00:01:12.920
You could call that A prime.

00:01:12.920 --> 00:01:15.200
So this is going to be equal to A prime.

00:01:15.200 --> 00:01:16.770
You rotate around that

00:01:16.770 --> 00:01:21.240
so that side AB coincides with side ED.

00:01:21.240 --> 00:01:23.090
And we've talked about
that in other videos.

00:01:23.090 --> 00:01:26.950
So at that point, D would
be equal to B prime,

00:01:26.950 --> 00:01:29.220
the point to which B is mapped.

00:01:29.220 --> 00:01:31.750
But the question is where is C?

00:01:31.750 --> 00:01:35.900
If we can show that for
sure C is either at point F

00:01:35.900 --> 00:01:37.670
or, with another rigid transformation,

00:01:37.670 --> 00:01:39.480
we can get C to point F,

00:01:39.480 --> 00:01:41.330
then we would have completed our proof.

00:01:41.330 --> 00:01:42.340
We would have been able to show

00:01:42.340 --> 00:01:44.450
that with a series of
rigid transformations,

00:01:44.450 --> 00:01:46.060
you can go from this triangle

00:01:46.060 --> 00:01:49.180
or you can map this
triangle onto that triangle.

00:01:49.180 --> 00:01:50.990
And to think about where point C is,

00:01:50.990 --> 00:01:54.520
this is where this compass
is going to prove useful.

00:01:54.520 --> 00:01:59.520
We know that point C is exactly
this far away from point A.

00:02:00.040 --> 00:02:02.380
I will measure that with my compass.

00:02:02.380 --> 00:02:04.210
So I could do it this way as well.

00:02:04.210 --> 00:02:08.990
Point C is exactly that far from point A.

00:02:08.990 --> 00:02:13.990
And so that means that point
C needs to be someplace,

00:02:14.190 --> 00:02:17.300
someplace on this curve right over here,

00:02:17.300 --> 00:02:18.940
on this arc that I'm doing.

00:02:18.940 --> 00:02:23.520
These are some of the points
that are exactly that far away.

00:02:23.520 --> 00:02:27.730
I can do a complete circle, but
you see where this is going.

00:02:27.730 --> 00:02:30.410
So point C, I guess we
could save C prime or C

00:02:30.410 --> 00:02:31.950
will be mapped to some point

00:02:31.950 --> 00:02:35.090
on that circle if you take
it from A's perspective,

00:02:35.090 --> 00:02:37.240
because that's how far C is from A,

00:02:37.240 --> 00:02:41.150
but then we also know
that C is this far from B.

00:02:41.150 --> 00:02:43.600
So let me adjust my compass again.

00:02:43.600 --> 00:02:47.390
C is that far from B,

00:02:47.390 --> 00:02:49.840
and so if B is mapped to this point,

00:02:49.840 --> 00:02:53.890
this is where B prime is then
C prime where C is mapped

00:02:53.890 --> 00:02:57.370
is going to be someplace along this curve,

00:02:57.370 --> 00:03:00.200
and so you could view those
two curves as constraints

00:03:00.200 --> 00:03:03.940
so we know that C prime has to
sit on both of these curves.

00:03:03.940 --> 00:03:07.160
So it's either going to sit
right over here where F is

00:03:07.160 --> 00:03:09.780
and so if my rigid
transformation got us to a point

00:03:09.780 --> 00:03:11.960
where C sits exactly where F is,

00:03:11.960 --> 00:03:13.740
well then our proof is complete.

00:03:13.740 --> 00:03:16.500
We've come up with a rigid transformation.

00:03:16.500 --> 00:03:17.800
Now another possibility is

00:03:17.800 --> 00:03:19.500
when we do that transformation

00:03:19.500 --> 00:03:22.550
C prime ends up right over here.

00:03:22.550 --> 00:03:24.110
So what could we then do

00:03:24.110 --> 00:03:27.150
to continue to transform rigidly

00:03:27.150 --> 00:03:29.150
so that C prime ends up with F?

00:03:29.150 --> 00:03:30.460
Remember the other two points

00:03:30.460 --> 00:03:35.130
have already coincided on with E and D

00:03:35.130 --> 00:03:38.440
so we just have to get C
prime to coincide with F.

00:03:38.440 --> 00:03:40.570
Well one way to think about it is,

00:03:40.570 --> 00:03:43.560
if we if we think about it,

00:03:43.560 --> 00:03:48.560
E, point E is equidistant
to C Prime and F.

00:03:49.450 --> 00:03:52.123
We see this is going to be equal to,

00:03:53.071 --> 00:03:54.220
we could put three hashtags there,

00:03:54.220 --> 00:03:55.800
because once again that defined the radius

00:03:55.800 --> 00:04:00.130
of this arc and we know that point C Prime

00:04:00.130 --> 00:04:03.960
in this case the point
C prime in this case

00:04:03.960 --> 00:04:08.770
is the same distance
from D as F is, as F is.

00:04:08.770 --> 00:04:10.120
And so one way to think about it,

00:04:10.120 --> 00:04:12.620
imagine a line between F and,

00:04:12.620 --> 00:04:14.170
I could get my straight edge here

00:04:14.170 --> 00:04:16.310
so it looks a little bit neater,

00:04:16.310 --> 00:04:21.310
imagine a line that
connects F and this C Prime,

00:04:21.460 --> 00:04:22.560
and once again we're in the case

00:04:22.560 --> 00:04:24.420
where C prime immediately didn't go to F,

00:04:24.420 --> 00:04:28.990
where C prime ended up being
on this side so to speak,

00:04:28.990 --> 00:04:31.450
and you can see that point E,

00:04:31.450 --> 00:04:35.310
because it is equidistant
to C Prime and F,

00:04:35.310 --> 00:04:38.840
it must sit on the perpendicular bisector

00:04:38.840 --> 00:04:41.560
of the segment FC.

00:04:41.560 --> 00:04:44.910
Same thing about point D or B prime.

00:04:44.910 --> 00:04:47.080
This must be the perpendicular bisector,

00:04:47.080 --> 00:04:48.880
because this point is equidistant to F

00:04:48.880 --> 00:04:51.180
as it is to C prime.

00:04:51.180 --> 00:04:54.010
This point is equidistant
to F as it is to C prime.

00:04:54.010 --> 00:04:55.940
The set of points whose distance

00:04:55.940 --> 00:04:59.970
is equal to F and C prime they will form

00:04:59.970 --> 00:05:02.930
the perpendicular bisector of FC.

00:05:02.930 --> 00:05:04.580
So we know that this orange line

00:05:04.580 --> 00:05:07.200
is a perpendicular bisector of FC.

00:05:07.200 --> 00:05:08.510
Why is that helpful?

00:05:08.510 --> 00:05:09.680
Well that tells us is

00:05:09.680 --> 00:05:11.540
if when we do that first transformation

00:05:11.540 --> 00:05:14.190
to make AB coincide with EF,

00:05:14.190 --> 00:05:16.750
if C Prime doesn't end up
here and it ends up there,

00:05:16.750 --> 00:05:18.500
we just have to do one
more transformation.

00:05:18.500 --> 00:05:21.580
We just have to do a reflection about ED

00:05:21.580 --> 00:05:23.330
or about A prime B prime,

00:05:23.330 --> 00:05:24.200
however you want to view it,

00:05:24.200 --> 00:05:25.580
about this orange line,

00:05:25.580 --> 00:05:28.710
and then C will coincide with F,

00:05:28.710 --> 00:05:31.040
because orange is a
perpendicular bisector.

00:05:31.040 --> 00:05:32.720
So I could do something like this.

00:05:32.720 --> 00:05:35.840
This length is the same as this length,

00:05:35.840 --> 00:05:36.703
and since it's perpendicular bisector,

00:05:36.703 --> 00:05:39.430
when you do the reflection

00:05:39.430 --> 00:05:42.010
C prime will then coincide with F.

00:05:42.010 --> 00:05:44.360
And a reflection is a
rigid transformation,

00:05:44.360 --> 00:05:45.953
so we would be all good.


Geometric constructions: perpendicular line through a point off the line

WEBVTT
Kind: captions
Language: en

00:00:00.270 --> 00:00:02.270
- [Instructor] What I have here is a line,

00:00:02.270 --> 00:00:05.140
and I have a point that
is not on that line,

00:00:05.140 --> 00:00:08.240
and my goal is to draw a new line

00:00:08.240 --> 00:00:10.010
that goes through this point,

00:00:10.010 --> 00:00:12.550
and is perpendicular to my original line.

00:00:12.550 --> 00:00:14.480
How do I do that?

00:00:14.480 --> 00:00:16.740
Well you might imagine that our compass

00:00:16.740 --> 00:00:17.680
will come in handy,

00:00:17.680 --> 00:00:19.140
it's been handy before.

00:00:19.140 --> 00:00:22.390
And so what I will do is,
I'll pick an arbitrary point

00:00:22.390 --> 00:00:23.223
on our original line,

00:00:23.223 --> 00:00:25.400
let's say this point right over here.

00:00:25.400 --> 00:00:27.400
And then I'll adjust my compass,

00:00:27.400 --> 00:00:30.940
so the distance between the pivot point

00:00:30.940 --> 00:00:32.890
and my pencil tip,

00:00:32.890 --> 00:00:35.420
is the same as the distance
between those two points.

00:00:35.420 --> 00:00:38.320
And then I can now use my compass

00:00:38.320 --> 00:00:41.290
to trace out an arc of that radius.

00:00:41.290 --> 00:00:42.963
So there you go.

00:00:43.900 --> 00:00:46.000
Now my next step is to find another point

00:00:46.000 --> 00:00:47.880
on my original line

00:00:47.880 --> 00:00:49.740
that has the same distance

00:00:49.740 --> 00:00:51.960
from that point that is off the line.

00:00:51.960 --> 00:00:55.720
And I can do that by centering my compass

00:00:55.720 --> 00:00:58.170
on that offline point,

00:00:58.170 --> 00:00:59.940
and then drawing another arc.

00:00:59.940 --> 00:01:02.230
And I can see very clearly

00:01:02.230 --> 00:01:05.480
that this point also has the same distance

00:01:05.480 --> 00:01:07.760
from this point up here.

00:01:07.760 --> 00:01:11.200
And then I can center my
compass on that point,

00:01:11.200 --> 00:01:12.880
and notice I haven't changed the radius

00:01:12.880 --> 00:01:14.570
of my compass,

00:01:14.570 --> 00:01:18.910
to draw another arc like this,

00:01:18.910 --> 00:01:21.063
to draw another arc like this.

00:01:22.240 --> 00:01:25.080
And then what I can do is connect

00:01:25.080 --> 00:01:27.623
this point and that point,

00:01:28.770 --> 00:01:30.930
and it at least looks perpendicular,

00:01:30.930 --> 00:01:32.460
but we're going to prove to ourselves

00:01:32.460 --> 00:01:34.320
that it is indeed perpendicular

00:01:34.320 --> 00:01:35.960
to our original line.

00:01:35.960 --> 00:01:40.043
So let me just draw it, so
you have that like that.

00:01:40.950 --> 00:01:42.260
So how do we feel good

00:01:42.260 --> 00:01:44.000
that this new line that I just drew

00:01:44.000 --> 00:01:46.460
is perpendicular to our original one?

00:01:46.460 --> 00:01:48.670
Well let's connect the dots

00:01:48.670 --> 00:01:49.940
that we made.

00:01:49.940 --> 00:01:52.240
So if we connect all the dots

00:01:52.240 --> 00:01:53.720
we're going to get a rhombus.

00:01:53.720 --> 00:01:55.483
We know that this distance,

00:01:56.570 --> 00:02:01.067
this distance is the
same as this distance.

00:02:01.067 --> 00:02:05.150
The same as this one right over here,

00:02:05.150 --> 00:02:08.089
which is the same as this distance,

00:02:08.089 --> 00:02:10.800
so let me make sure I got
my straight edge right.

00:02:10.800 --> 00:02:13.290
Same as that distance,

00:02:13.290 --> 00:02:17.490
which is the same as this distance,

00:02:17.490 --> 00:02:22.490
same as that distance.

00:02:24.100 --> 00:02:28.480
And then, so this is a rhombus,

00:02:28.480 --> 00:02:30.450
and we know that the diagonals

00:02:30.450 --> 00:02:31.790
of a rhombus

00:02:31.790 --> 00:02:34.240
intersect at right angles.

00:02:34.240 --> 00:02:35.073
So there you have it.

00:02:35.073 --> 00:02:36.240
I have drawn a new line

00:02:36.240 --> 00:02:38.750
that goes through that offline point,

00:02:38.750 --> 00:02:41.253
and is perpendicular to our original line.


Geometric constructions: congruent angles

WEBVTT
Kind: captions
Language: en

00:00:01.110 --> 00:00:02.560
- [Instructor] What we're
gonna do in this video

00:00:02.560 --> 00:00:04.970
is learn to construct congruent angles,

00:00:04.970 --> 00:00:06.360
and we're gonna do it, with of course,

00:00:06.360 --> 00:00:08.050
a pen or a pencil here.

00:00:08.050 --> 00:00:11.080
I'm gonna use a ruler as a straight edge.

00:00:11.080 --> 00:00:12.380
And then I'm gonna use a tool

00:00:12.380 --> 00:00:13.600
known as a compass.

00:00:13.600 --> 00:00:15.360
Which looks a little bit fancy,

00:00:15.360 --> 00:00:16.710
but what it allows us to do

00:00:16.710 --> 00:00:19.420
it'll apply using it in a little bit,

00:00:19.420 --> 00:00:21.530
is it allows us to draw perfect circles,

00:00:21.530 --> 00:00:23.030
or arcs, of a given radius.

00:00:23.030 --> 00:00:24.720
You pivot on one point here

00:00:24.720 --> 00:00:26.570
and then you use your pen or your pencil

00:00:26.570 --> 00:00:28.810
to trace out the arc,

00:00:28.810 --> 00:00:30.200
or the circle.

00:00:30.200 --> 00:00:32.570
So let's just start with this angle

00:00:32.570 --> 00:00:33.520
right over here,

00:00:33.520 --> 00:00:35.470
and I'm going to construct an angle

00:00:35.470 --> 00:00:37.750
that is congruent to it.

00:00:37.750 --> 00:00:40.650
So let me make the
vertex of my second angle

00:00:40.650 --> 00:00:42.020
right over there,

00:00:42.020 --> 00:00:44.500
and then let me draw one of the rays

00:00:44.500 --> 00:00:46.690
that originates at that vertex.

00:00:46.690 --> 00:00:47.750
And I'm gonna put this angle

00:00:47.750 --> 00:00:48.850
in a different orientation,

00:00:48.850 --> 00:00:50.010
just to show that they don't even have

00:00:50.010 --> 00:00:53.290
to have the same orientation.

00:00:53.290 --> 00:00:54.260
So it's going to look

00:00:54.260 --> 00:00:57.480
something like that,
that's one of the rays.

00:00:57.480 --> 00:00:58.313
But then we have to figure out

00:00:58.313 --> 00:00:59.410
where do we put,

00:00:59.410 --> 00:01:01.190
where do we put the other ray

00:01:01.190 --> 00:01:03.330
so that the two angles are congruent?

00:01:03.330 --> 00:01:05.890
And this is where our compass

00:01:05.890 --> 00:01:07.910
is going to be really useful.

00:01:07.910 --> 00:01:10.740
So what I'm going to do
is put the pivot point

00:01:10.740 --> 00:01:12.100
of a compass, of the compass,

00:01:12.100 --> 00:01:14.170
right at the vertex of the first angle,

00:01:14.170 --> 00:01:19.170
and I'm going to draw
out an arc like this.

00:01:19.240 --> 00:01:20.355
And what's useful about the compass

00:01:20.355 --> 00:01:24.210
is you can keep the radius constant,

00:01:24.210 --> 00:01:26.810
and you can see it intersects

00:01:26.810 --> 00:01:29.910
our first two rays at points,

00:01:29.910 --> 00:01:33.020
let's just call this B and C.

00:01:33.020 --> 00:01:34.850
And I could call this point A,

00:01:34.850 --> 00:01:35.980
right over here.

00:01:35.980 --> 00:01:37.140
And so let me,

00:01:37.140 --> 00:01:39.390
now that I have my compass with the exact

00:01:39.390 --> 00:01:41.130
right radius right now,

00:01:41.130 --> 00:01:44.290
let me draw that right over here.

00:01:44.290 --> 00:01:47.350
But this alone won't allow us to draw

00:01:47.350 --> 00:01:48.753
the angle just yet,

00:01:49.850 --> 00:01:52.320
but let me draw it like this,

00:01:52.320 --> 00:01:55.073
and that is pretty good.

00:01:56.220 --> 00:01:59.626
And let's call this
point right over here D,

00:01:59.626 --> 00:02:01.550
and I'll call this one E,

00:02:01.550 --> 00:02:02.780
and I wanna figure out where to put

00:02:02.780 --> 00:02:04.630
my third point F,

00:02:04.630 --> 00:02:06.230
so I can define ray E F,

00:02:06.230 --> 00:02:08.390
so that these two angles are congruent.

00:02:08.390 --> 00:02:12.140
And what I can do is take my compass again

00:02:12.140 --> 00:02:15.680
and get a clear sense of the distance

00:02:15.680 --> 00:02:16.990
between C and B,

00:02:16.990 --> 00:02:18.980
by adjusting my compass.

00:02:18.980 --> 00:02:20.420
So one point is on C,

00:02:20.420 --> 00:02:22.610
and my pencil is on B.

00:02:22.610 --> 00:02:24.860
So I have, get this right,

00:02:24.860 --> 00:02:27.430
so I have this distance right over here.

00:02:27.430 --> 00:02:29.280
I know this distance,

00:02:29.280 --> 00:02:31.520
and I've adjusted my compass accordingly,

00:02:31.520 --> 00:02:34.250
so I can get that same distance

00:02:34.250 --> 00:02:35.893
right over there.

00:02:36.860 --> 00:02:38.760
And so you can now image

00:02:38.760 --> 00:02:40.800
where I'm going to draw that second ray.

00:02:40.800 --> 00:02:42.190
That second ray,

00:02:42.190 --> 00:02:46.720
if I put point F right over here,

00:02:46.720 --> 00:02:47.870
my second ray,

00:02:47.870 --> 00:02:51.860
I can just draw between,
starting at point E

00:02:51.860 --> 00:02:52.870
right over here,

00:02:52.870 --> 00:02:55.080
going through point F.

00:02:55.080 --> 00:02:57.620
I could draw a little bit neater,

00:02:57.620 --> 00:03:00.580
so it would look like that, my second ray.

00:03:00.580 --> 00:03:02.210
Ignore that first little line I drew,

00:03:02.210 --> 00:03:03.043
I'm using a pen,

00:03:03.043 --> 00:03:04.390
which I don't recommend for you to do it.

00:03:04.390 --> 00:03:05.450
I'm doing it so that you can see

00:03:05.450 --> 00:03:06.970
it on this video.

00:03:06.970 --> 00:03:09.480
Now how do we know that this angle

00:03:09.480 --> 00:03:11.920
is now congruent to this angle

00:03:11.920 --> 00:03:13.360
right over here?

00:03:13.360 --> 00:03:14.690
Well one way to do it, is to think

00:03:14.690 --> 00:03:17.620
about triangle B A C,

00:03:17.620 --> 00:03:19.440
triangle B A C,

00:03:19.440 --> 00:03:23.230
and triangle, let's just call it D F E.

00:03:23.230 --> 00:03:25.740
So this triangle right over here.

00:03:25.740 --> 00:03:27.570
When we drew that first arc,

00:03:27.570 --> 00:03:30.760
we know that the distance between A C

00:03:30.760 --> 00:03:33.450
is equivalent to the distance between A B,

00:03:33.450 --> 00:03:36.300
and we kept the compass radius the same.

00:03:36.300 --> 00:03:39.110
So we know that's also
the distance between E F,

00:03:39.110 --> 00:03:41.730
and the distance between E D.

00:03:41.730 --> 00:03:43.300
And then the second time,

00:03:43.300 --> 00:03:45.910
when we adjusted our compass radius,

00:03:45.910 --> 00:03:48.580
we now know that the distance between B C

00:03:48.580 --> 00:03:51.940
is the same as the
distance between F and D.

00:03:51.940 --> 00:03:53.180
Or the length of B C

00:03:53.180 --> 00:03:56.380
is the same as the length of F D.

00:03:56.380 --> 00:04:00.430
So it's very clear that we
have congruent triangles.

00:04:00.430 --> 00:04:01.670
All of the three sides

00:04:01.670 --> 00:04:03.380
have the same measure,

00:04:03.380 --> 00:04:06.130
and therefore the corresponding angles

00:04:06.130 --> 00:04:08.153
must be congruent as well.


Geometric constructions: parallel line

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:02.940
- [David] Let's say that we have a line,

00:00:02.940 --> 00:00:05.070
drawing it right over there,

00:00:05.070 --> 00:00:09.470
and our goal is to construct another line

00:00:09.470 --> 00:00:11.340
that is parallel to this line

00:00:11.340 --> 00:00:13.790
that goes through this point.

00:00:13.790 --> 00:00:15.740
How would we do that?

00:00:15.740 --> 00:00:17.840
Well, the way that we can approach it is

00:00:17.840 --> 00:00:20.630
by creating what will
eventually be a transversal

00:00:20.630 --> 00:00:22.950
between the two parallel lines.

00:00:22.950 --> 00:00:24.450
So let me draw that.

00:00:24.450 --> 00:00:26.530
So I'm just drawing a line
that goes through my point

00:00:26.530 --> 00:00:28.878
and intersects my original line.

00:00:28.878 --> 00:00:32.020
Do that, so it's going to look like that.

00:00:32.020 --> 00:00:34.430
And then, I'm really just
going to use the idea

00:00:34.430 --> 00:00:38.470
of corresponding angled
congruents for parallel lines.

00:00:38.470 --> 00:00:41.660
So what I can do is now take my compass

00:00:42.750 --> 00:00:46.363
and think about this
angle right over here.

00:00:47.210 --> 00:00:49.263
So I'll draw it like that.

00:00:50.190 --> 00:00:53.600
And say, all right, if I draw an arc

00:00:53.600 --> 00:00:56.230
of the same radius over here,

00:00:56.230 --> 00:00:59.280
can I reconstruct that angle?

00:00:59.280 --> 00:01:03.440
And so where should the
point be on this left end?

00:01:03.440 --> 00:01:06.480
Well, to do that, I can just measure

00:01:07.640 --> 00:01:09.450
the distance between these two points

00:01:09.450 --> 00:01:12.200
using my compass, so I'm
adjusting it a little bit

00:01:12.200 --> 00:01:14.950
to get the distance
between those two points.

00:01:14.950 --> 00:01:19.550
And then I can use that up
over here to figure out,

00:01:19.550 --> 00:01:21.240
I got a little bit shaky.

00:01:21.240 --> 00:01:23.540
I can figure out that
point right over there.

00:01:23.540 --> 00:01:28.540
And just like that, I now
have two corresponding angles

00:01:29.010 --> 00:01:31.810
to find my transversal and parallel lines,

00:01:31.810 --> 00:01:34.780
so what I can do is take my straightedge

00:01:34.780 --> 00:01:39.290
and make it go through those
points that I just created,

00:01:39.290 --> 00:01:41.350
so let's see, make sure I'm going through,

00:01:41.350 --> 00:01:43.180
and it would look like that,

00:01:43.180 --> 00:01:46.880
and I have just constructed
two parallel lines.

00:01:46.880 --> 00:01:50.090
And once again, how do
I know that this line

00:01:50.090 --> 00:01:51.950
is parallel to this line?

00:01:51.950 --> 00:01:55.160
Because we have a transversal
that intersects both of them

00:01:55.160 --> 00:01:58.240
and these two angles, which
are corresponding angles,

00:01:58.240 --> 00:01:59.073
are congruent.

00:01:59.073 --> 00:02:02.303
So these two lines must be parallel.


Angle congruence equivalent to having same measure

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:01.460
- [Instructor] What we're
going to do in this video

00:00:01.460 --> 00:00:04.500
is demonstrate that angles
are congruent if and only

00:00:04.500 --> 00:00:07.540
if they have the same measure,

00:00:07.540 --> 00:00:10.440
and for our definition of congruence,

00:00:10.440 --> 00:00:13.750
we will use the rigid
transformation definition,

00:00:13.750 --> 00:00:16.420
which tells us two figures
are congruent if and only

00:00:16.420 --> 00:00:20.100
if there exists a series
of rigid transformations

00:00:20.100 --> 00:00:23.620
which will map one figure onto the other.

00:00:23.620 --> 00:00:26.200
And then, what are rigid transformations?

00:00:26.200 --> 00:00:29.010
Those are transformations
that preserve distance

00:00:29.010 --> 00:00:31.860
between points and angle measures.

00:00:31.860 --> 00:00:33.920
So, let's get to it.

00:00:33.920 --> 00:00:36.350
So, let's start with two
angles that are congruent,

00:00:36.350 --> 00:00:39.070
and I'm going to show that
they have the same measure.

00:00:39.070 --> 00:00:42.660
I'm going to demonstrate
that, so they start congruent,

00:00:42.660 --> 00:00:45.380
so these two angles are
congruent to each other.

00:00:45.380 --> 00:00:49.680
Now, this means by the
rigid transformation

00:00:49.680 --> 00:00:54.680
definition of congruence,
there is a series

00:00:56.500 --> 00:00:59.630
of rigid transformations,

00:00:59.630 --> 00:01:04.550
transformations that map

00:01:04.550 --> 00:01:09.023
angle ABC onto angle,

00:01:10.400 --> 00:01:13.073
I'll do it here, onto angle DEF.

00:01:16.408 --> 00:01:20.510
By definition, by definition
of rigid transformations,

00:01:20.510 --> 00:01:25.510
they preserve angle measure,
preserve angle measure.

00:01:26.750 --> 00:01:30.000
So, if you're able to map the left angle

00:01:30.000 --> 00:01:32.730
onto the right angle, and
in doing so, you did it

00:01:32.730 --> 00:01:35.250
with transformations that
preserved angle measure,

00:01:35.250 --> 00:01:38.400
they must now have the same angle measure.

00:01:38.400 --> 00:01:43.160
We now know that the measure
of angle ABC is equal

00:01:43.160 --> 00:01:45.133
to the measure of angle DEF.

00:01:46.430 --> 00:01:50.060
So, we've demonstrated this
green statement the first way,

00:01:50.060 --> 00:01:51.620
that if things are congruent,

00:01:51.620 --> 00:01:53.110
they will have the same measure.

00:01:53.110 --> 00:01:55.550
Now, let's prove it the other way around.

00:01:55.550 --> 00:01:59.453
So now, let's start with the
idea that measure of angle ABC

00:02:01.090 --> 00:02:04.043
is equal to the measure of angle DEF,

00:02:04.930 --> 00:02:07.220
and to demonstrate that these
are going to be congruent,

00:02:07.220 --> 00:02:09.000
we just have to show that
there's always a series

00:02:09.000 --> 00:02:12.880
of rigid transformations
that will map angle ABC

00:02:12.880 --> 00:02:16.130
onto angle DEF, and to help us there,

00:02:16.130 --> 00:02:18.380
let's just visualize these angles,

00:02:18.380 --> 00:02:21.957
so, draw this really fast, angle ABC,

00:02:24.480 --> 00:02:28.690
and angle is defined by two
rays that start at a point.

00:02:28.690 --> 00:02:32.090
That point is the vertex, so that's ABC,

00:02:32.090 --> 00:02:35.453
and then let me draw angle DEF.

00:02:37.158 --> 00:02:39.408
So, that might look
something like this, DEF,

00:02:42.840 --> 00:02:46.000
and what we will now do is let's do

00:02:46.000 --> 00:02:48.350
our first rigid transformation.

00:02:48.350 --> 00:02:53.273
Let's translate, translate angle ABC

00:02:56.780 --> 00:03:01.780
so that B mapped to point E,

00:03:05.330 --> 00:03:08.470
and if we did that, so we're
gonna translate it like that,

00:03:08.470 --> 00:03:12.514
then ABC is going to look something like,

00:03:12.514 --> 00:03:13.347
ABC is gonna look something like this.

00:03:15.753 --> 00:03:17.000
It's going to look something like this.

00:03:17.000 --> 00:03:20.430
B is now mapped onto E.

00:03:20.430 --> 00:03:23.120
This would be where A would get mapped to.

00:03:23.120 --> 00:03:24.610
This would where C would get mapped to.

00:03:24.610 --> 00:03:27.450
Sometimes you might see a
notation A prime, C prime,

00:03:27.450 --> 00:03:30.120
and this is where B would get mapped to,

00:03:30.120 --> 00:03:32.150
and then the next thing I would do

00:03:32.150 --> 00:03:37.150
is I would rotate angle
ABC about its vertex,

00:03:41.040 --> 00:03:45.327
about B, so that ray BC,

00:03:48.600 --> 00:03:53.600
ray BC, coincides, coincides with ray EF.

00:03:57.870 --> 00:04:02.300
Now, you're just gonna rotate
the whole angle that way

00:04:02.300 --> 00:04:07.080
so that now, ray BC coincides with ray EF.

00:04:07.080 --> 00:04:08.657
Well, you might be saying,
"Hey, C doesn't necessarily have

00:04:08.657 --> 00:04:11.017
"to sit on F 'cause they
might be different distances

00:04:11.017 --> 00:04:13.480
"from their vertices,"
but that's all right.

00:04:13.480 --> 00:04:17.410
The ray can be defined by any
point that sits on that ray,

00:04:17.410 --> 00:04:21.780
so now, if you do this
rotation, and ray BC coincides

00:04:21.780 --> 00:04:24.960
with ray EF, now those two
rays would be equivalent

00:04:24.960 --> 00:04:29.550
because measure of angle
ABC is equal to the measure

00:04:29.550 --> 00:04:30.383
of angle DEF.

00:04:32.060 --> 00:04:37.060
That will also tell us that
ray BA, ray BA now coincides,

00:04:41.200 --> 00:04:45.920
coincides with ray ED, and just like that,

00:04:45.920 --> 00:04:47.990
I've given you a series
of rigid transformations

00:04:47.990 --> 00:04:49.500
that will always work.

00:04:49.500 --> 00:04:51.810
If you translate so that
the vertices are mapped

00:04:51.810 --> 00:04:54.160
onto each other and then you rotate it

00:04:54.160 --> 00:04:55.990
so that the bottom ray
of one angle coincides

00:04:55.990 --> 00:04:57.820
with the bottom ray of the other angle,

00:04:57.820 --> 00:05:00.190
then you could say the
top ray of the two angles

00:05:00.190 --> 00:05:02.930
will now coincide because the
angles have the same measure,

00:05:02.930 --> 00:05:06.800
and because of that, the
angles now completely coincide,

00:05:06.800 --> 00:05:11.800
and so we know that angle ABC
is congruent to angle DEF,

00:05:13.200 --> 00:05:14.033
and we're now done.

00:05:14.033 --> 00:05:17.210
We've proven both sides of this statement.

00:05:17.210 --> 00:05:18.920
If they're congruent, they
have the same measure.

00:05:18.920 --> 00:05:21.903
If they have the same measure,
then they are congruent.


Introduction to Khan Academy Kids

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:01.270
- [Voice-over] Khan Academy Kids

00:00:01.270 --> 00:00:04.750
includes thousands of
activities, books, and videos

00:00:04.750 --> 00:00:08.870
helping kids read, play, learn, and grow.

00:00:08.870 --> 00:00:10.730
Narrator Kodi Bear and her friends

00:00:10.730 --> 00:00:12.170
guide kids through the app,

00:00:12.170 --> 00:00:15.950
encouraging creativity,
curiosity, and kindness.

00:00:15.950 --> 00:00:17.380
Because we believe that everyone

00:00:17.380 --> 00:00:19.060
should have access to learning,

00:00:19.060 --> 00:00:21.620
Khan Academy Kids is completely free.

00:00:21.620 --> 00:00:24.170
No ads, no subscriptions.

00:00:24.170 --> 00:00:26.933
Download now, and let the
joyful learning begin!


Khan Academy Kids: Celebrating First Words Through First Grade

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:02.210
- [Narrator] Growing up is exciting.

00:00:02.210 --> 00:00:05.200
There are so many firsts to discover:

00:00:05.200 --> 00:00:08.180
your first word, your first book,

00:00:08.180 --> 00:00:10.230
your first day of school,

00:00:10.230 --> 00:00:12.913
and with each new thing
you learn, you grow.

00:00:13.800 --> 00:00:16.640
At Khan Academy, we embrace
and celebrate learning

00:00:16.640 --> 00:00:18.210
at all ages.

00:00:18.210 --> 00:00:20.000
We created Khan Academy Kids

00:00:20.000 --> 00:00:21.330
to spark the joy of learning

00:00:21.330 --> 00:00:24.010
in kids ages two through seven.

00:00:24.010 --> 00:00:27.440
Through thousands of
activities, books, and videos,

00:00:27.440 --> 00:00:30.933
kids can read, play, learn, and grow.

00:00:31.880 --> 00:00:33.740
Narrator Kodi Bear and her friends

00:00:33.740 --> 00:00:35.470
guide kids through the app,

00:00:35.470 --> 00:00:39.930
encouraging creativity,
curiosity, and kindness.

00:00:39.930 --> 00:00:41.450
Because we believe that everyone

00:00:41.450 --> 00:00:43.300
should have access to learning,

00:00:43.300 --> 00:00:45.940
Khan Academy Kids is completely free,

00:00:45.940 --> 00:00:48.143
no ads, no subscriptions.

00:00:49.070 --> 00:00:50.630
Khan Academy Kids,

00:00:50.630 --> 00:00:53.540
now celebrating first
words through first grade.

00:00:53.540 --> 00:00:56.823
Available in the App Store,
Google Play, and Amazon.


Introduction to meditation to reduce test prep anxiety

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.100
- Hello, Sal here from Khan Academy.

00:00:03.100 --> 00:00:05.590
So when you hear the word meditation,

00:00:05.590 --> 00:00:07.830
for many of y'all, it
might evoke some type of

00:00:07.830 --> 00:00:10.620
new age thing that has nothing to do

00:00:10.620 --> 00:00:12.270
with standardize tests.

00:00:12.270 --> 00:00:14.250
And if you're about to
take a standardize test,

00:00:14.250 --> 00:00:16.890
I'm sure there's many thoughts about

00:00:16.890 --> 00:00:19.300
am I prepared, am I going to do well,

00:00:19.300 --> 00:00:22.180
I hope I don't bomb the exam.

00:00:22.180 --> 00:00:23.624
That stress and that anxiety

00:00:23.624 --> 00:00:26.160
is, if anything, going
to hurt your performance

00:00:26.160 --> 00:00:27.850
on these things that you wanna do well.

00:00:27.850 --> 00:00:31.140
And so, meditation is a
way to help still that.

00:00:31.140 --> 00:00:33.970
But it's really just for you
to observe your thoughts,

00:00:33.970 --> 00:00:35.810
not get stressed out about
it if your thoughts start

00:00:35.810 --> 00:00:37.890
to wander, but then just bring them back

00:00:37.890 --> 00:00:39.440
and try to not think.

00:00:39.440 --> 00:00:42.170
Or think about just
very positive thoughts.

00:00:42.170 --> 00:00:44.420
I think if you do that you're
gonna find you're gonna be

00:00:44.420 --> 00:00:46.620
happier, you're gonna be more positive,

00:00:46.620 --> 00:00:48.870
all sorts of positive energy
is going to be attracted

00:00:48.870 --> 00:00:51.360
to you, and, even though
that's not the most important

00:00:51.360 --> 00:00:54.080
thing in the world, you'll
do just fine on these tests

00:00:54.080 --> 00:00:56.230
that right now you
might be worrying about.


Introduction to meditation for students

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:03.300
- Hi everyone, Sal here from Khan Academy.

00:00:03.300 --> 00:00:05.960
So for most of human history,

00:00:05.960 --> 00:00:09.440
when we were either hunter
gatherers or early farmers,

00:00:09.440 --> 00:00:12.760
we had a lot of time where
we would just look at things

00:00:12.760 --> 00:00:16.890
and not be overstimulated,
there'd just be a bubbling brook.

00:00:16.890 --> 00:00:19.170
But now we're constantly stimulated

00:00:19.170 --> 00:00:22.190
by social media, by things
we have to do at school,

00:00:22.190 --> 00:00:23.870
things that we have to do at work,

00:00:23.870 --> 00:00:26.800
our cell phones ringing,
getting different text messages

00:00:26.800 --> 00:00:28.000
and before we know it,

00:00:28.000 --> 00:00:30.580
in modern society, our brains,

00:00:30.580 --> 00:00:34.370
our thoughts are taking control of us.

00:00:34.370 --> 00:00:36.840
And we actually don't have as much peace

00:00:36.840 --> 00:00:39.450
and as much stillness as we would like.

00:00:39.450 --> 00:00:41.041
And so one thing that I found in my life

00:00:41.041 --> 00:00:43.901
is that everyday you take some time out

00:00:43.901 --> 00:00:46.630
to just let your mind be still,

00:00:46.630 --> 00:00:48.520
you start to observe your thoughts,

00:00:48.520 --> 00:00:50.160
that it can be incredibly powerful,

00:00:50.160 --> 00:00:51.880
it's going to improve your mood,

00:00:51.880 --> 00:00:54.250
and it's going to just
make you a overall happier

00:00:54.250 --> 00:00:55.723
and more functional person.